A two-box-shift morphism between Specht modules

Matthias Künzer

March 13, 2000

Resumé

Soit n ⩾ 1, soit λ une partition de n et soit μ une partition obtenue à partir de λ par décalage à gauche de deux cases situés en bas d’une colonne. Nous donnons une formule pour un morphisme ZS_{n}-linéaire d’ordre m entre les modules de Specht sur Z/(m) correspondants, où m désigne la longueur du décalage des cases (divisée par deux en certains cas, specifiés combinatoirement). En d’autres termes, nous obtenons une extension des modules de Specht sur Z correspondants, d’ordre m en tant qu’élément de Ext^{1}.

Abstract

Let n ⩾ 1, let λ be a partition of n and let μ be a partition arising from λ by a downwards shift of two boxes situated at the bottom of a column. We give a formula for a ZS_{n}-linear morphism of order m between the corresponding Specht modules over Z/(m), where m is the box shift length (divided by two in certain combinatorially specified cases). Reformulated, this yields an extension of the corresponding Specht modules over Z of order m in Ext^{1}.

0 Introduction

0.1 Known results

The irreducible modules of the complex group algebra of the symmetric group on n letters possess the Specht lattices S^{λ} as combinatorially described integral models, indexed by partitions λ of n. Let d ⩾ 1 be an integer. Suppose given a partition λ of n and integers 1 ⩽ g ⩽ k ⩽ λ_{1} − 1 such that the shift of d boxes situated at the bottom of column k + 1 of λ downwards to column g yields a partition again, which we denote by μ. Let λ′ be the partition transposed to λ. The number

m_{0} := λ′_{g} − λ′_{k+1} + (k + 1) − g + d

of steps by which we shift leftwards and downwards is called the box shift length.

For instance, we may shift d = 2 boxes of λ = (5, 4, 4, 2, 1, 1) from column k + 1 = 4 downwards to column g = 2 to obtain the partition μ = (5, 3, 3, 2, 2, 2). We have λ′ = (6, 4, 3, 3, 1), hence the box shift length equals m_{0} = 5.

Theorem. Based on [CL 74], Carter and Payne [CP 80] have shown that

Hom_{KS_{n}}(K ⊗_{Z} S^{λ}, K ⊗_{Z} S^{μ}) ≠ 0,

K being an infinite field of characteristic p such that d < p^{v_{p}(m_{0})} (^{1}), i.e. such that the number of shifted boxes is smaller than the p-part of the box shift length. In case d = 1, this condition on p translates to p | m_{0}. In case d = 2, it translates for p ⩾ 3 to p | m_{0}, and for p = 2 to 4 | m_{0}.

Theorem. Suppose d = 1. In [K 99, 4.3.31, cf. 0.7.1], the existence of an element of order m_{0} in

Hom_{(Z/(m_{0}))S_{n}}(Z/(m_{0}) ⊗_{Z} S^{λ}, Z/(m_{0}) ⊗_{Z} S^{μ})

has been shown via construction of such a morphism.

0.2 Results

We maintain the situation from §0.1.

Suppose d = 2. Let m be either m_{0} or m_{0}/2, the latter value being taken in case m_{0} is even, a similar combinatorial integer is even and the diagram of λ takes a certain shape near the column g.

Theorem (2.37). There exists a morphism of order m in

Hom_{(Z/(m))S_{n}}(Z/(m) ⊗_{Z} S^{λ}, Z/(m) ⊗_{Z} S^{μ}).

As a consequence, we recover the according part of the result of Carter and Payne [CP 80]. An account of the construction of this morphism, including the definition of the modulus m, is given at the end of §1.1. In §0.3, we attempt to give an informal description of that construction. Some examples may be found in §2.4.

1A rank argument shows that [CP 80] also yields this result for K an arbitrary, possibly finite field of characteristic p such that d < p^{v_{p}(m_{0})}, as pointed out by G. Nebe.

The proof by means of our techniques requires consideration of (essentially) 9^{2} cases - the base 9 resulting from the structure of the one-step Garnir relations, the exponent 2 being the number of shifted boxes (cf. A.1). Moreover, the connection between the modulus and the box shift length results from the calculation only.

A morphism modulo the box shift length m_{0} itself need not exist. For instance, we obtain

Hom_{ZS_{5}}(S^{(2,2,1)}, Z/(5!) ⊗_{Z} S^{(1,1,1,1,1)})^{\overline{}^{∼}} Hom_{ZS_{5}}(S^{(2,2,1)}, Z/(2) ⊗_{Z} S^{(1,1,1,1,1)}) ≃ Z/2.

Let d be arbitrary, but suppose g = k. This is to say, we shift d boxes from column g + 1 to column g.

Theorem (4.9). Let ilog_{p}d := max{i ∈ Z_{⩾0} | p^{i} ⩽ d}. There exists a morphism of order

m := m_{0} · ∏_{p prime, p|m_{0}}p^{− min(v_{p}(m_{0}),ilog_{p}(d))} in

Hom_{(Z/(m))S_{n}}(Z/(m) ⊗_{Z} S^{λ}, Z/(m) ⊗_{Z} S^{μ}).

Again, the according part of the result of Carter and Payne [CP 80] is recovered. Roughly speaking, this morphism is given by picking d entries from column g + 1 of a given λ-polytabloid, by placing them at the bottom of column g, and by mapping this λ-polytabloid to an alternating sum of the resulting μ-polytabloids over the various possibilities to pick those entries.

Remark. For an abelian group A and an integer m ⩾ 2, we write A[m] := Kern(A ^{\overline{}^{m}} A). There is an isomorphism

Hom_{(Z/(m))S_{n}}(Z/(m) ⊗_{Z} S^{λ}, Z/(m) ⊗_{Z} S^{μ}) ^{\overline{}^{∼}} Ext_{ZS_{n}}^{1}(S^{λ}, S^{μ})[m]

which translates both results on the existence of a morphism of order m into assertions on the existence of an extension of order m in Ext^{1}. Cf. §0.4.

Remark. Let m ⩾ 2 be an integer. For arbitrary partitions λ, μ of n, we dispose of the transposition isomorphism

Hom_{(Z/(m))S_{n}}(Z/(m)⊗_{Z} S^{λ}, Z/(m)⊗_{Z} S^{μ}) ^{\overline{}^{∼}} Hom_{(Z/(m))S_{n}}(Z/(m)⊗_{Z} S^{μ′}, Z/(m)⊗_{Z} S^{λ′}),

given by dualization, followed by alternation and isomorphic substitution (3.16). Based on the simple assertion (3.2), we give in (3.27) an explicit, but nevertheless not quite satisfactory formula for the transpose of the two-box-shift morphism in (2.37) in terms of tabloids. Whereas the image of a polytabloid under that transpose is known to be an element of the target Specht module, hence known to be expressible as a linear combination of polytabloids, a formula for such an expression is lacking. Some examples are given in §3.4, cf. also (4.15).

0.3 Construction (informal description)

The following description might serve as an intuitive thread through our formalism concerning the two-box-shift morphism in (2.37). We maintain the situation of the beginning of §0.2 and refer to the usual way to depict partitions and tableaux [J 78].

We write the Specht lattice S^{λ} as a quotient of the free ZS_{n}-module F^{λ} on one generator, displayed as having as Z-linear basis the set of λ-tableaux equipped with the action of the S_{n} on the tableau entries. The kernel of the canonical epimorphism F^{λ} \overline{}S^{λ} is generated, over ZS_{n}, by signed column transpositions and one-step Garnir relations. Given a tableau, a signed column transposition is the sum of the tableau and the tableau having two entries of a column interchanged. A one-step Garnir relation depends on subsets of two subsequent columns of a tableau [a] the sizes of which add up to (the length of the left column)+1. After factoring out the signed column transpositions, such a Garnir relation can be written, up to a scalar, as the alternating sum of the tableaux arising from [a] by permutation inside the union of the chosen subsets of the columns, the sign being that of the afforded permutation. Our morphism is constructed as a factorization of a morphism (F^{λ} \overline{}S^{μ} \overline{}S^{μ}/mS^{μ}) over the canonical epimorphism (F^{λ} \overline{}S^{λ} \overline{}S^{λ}/mS^{λ}). To a λ-tableau [a], we may apply certain place operations, indexed by double paths, which produce μ-polytabloids. A double path is a pair of sequences of positions in the union of the diagrams of λ and μ that do not intersect, that run strictly from right to left, that start arbitrarily in column k + 1 and that end at the the position of the shifted boxes in column g. (So at least 4 and at most 4 + 2(k − g) positions are occupied.) Given such a double path, the place operation it gives rise to is obtained by inserting entries of column k + 1 into the double path from the right and by subsequently pushing entries through along each path separately. In particular, the last two entries forced to move are installed at the positions occupied by the shifted boxes in the diagram of μ. Cf. (2.1). Each double path has a weight attached, i.e. a tuple recording the number of positions it occupies in each column, and a sign, depending on the positions it occupies in column k+1. Sending [a] to the accordingly signed sum of the resulting μ-polytabloids indexed by double paths of a fixed weight yields a ZS_{n}-linear map from F^{λ} to S^{μ} that annihilates the signed column transpositions. Forming a linear combination of these maps, indexed over the weights, equipped with certain coefficients which are polynomial in the combinatorial data, and dividing by an essentially polynomial factor of redundancy yields a ZS_{n}-linear map F^{λ} ^{\overline{}^{f′′}} S^{μ} that annihilates all one-step Garnir relations except for those involving column g and column g +1, which vanish only modulo mS^{μ}. Thus (F^{λ} ^{\overline{}^{f′′}} S^{μ} \overline{}S^{μ}/mS^{μ}) factors over a ZS_{n}-linear map S^{λ}/mS^{λ} ^{\overline{}^{f}} S^{μ}/mS^{μ}. Since the image of F^{λ} ^{\overline{}^{f′′}} S^{μ} has not been contained in a strict multiple tS^{μ} ⊂ S^{μ}, t ⩾ 2, the resulting morphism S^{λ}/mS^{λ} ^{\overline{}^{f}} S^{μ}/mS^{μ} is of order m in its Hom-group.

0.4 Motivation

We consider the integral group ring ZS_{n} as a subring of a product of integral matrix rings via Wedderburn’s embedding, sending a group element g ∈ S_{n} to the tuple (ρ^{λ}(g))_{λ} of its operating matrices on the Specht lattices with respect to a choice of integral bases. Let m ⩾ 2 be an integer. A Z-linear map S^{λ} ^{\overline{}^{f}} S^{μ} that becomes ZS_{n}-linear modulo m ⩾ 2, i.e. that satisfies f ∘ ρ^{λ}(g) ≡_{m} ρ^{μ}(g) ∘ f for all g ∈ S_{n}, imposes this congruence as a necessary condition on an element of our product of integral matrix rings to lie in the image of that embedding. More generally, unscrewing the regular lattice into simple lattices via a binary tree of short exact sequences, we obtain a necessary and sufficient system of modular morphisms by means of the correspondence

Hom_{ZS_{n}}(X, Y/mY) ^{\overline{}^{∼}} Ext_{ZS_{n}}^{1}(X, Y)[m],

where X and Y are rationally disjoint ZS_{n}-lattices. This correspondence attaches to a morphism X \overline{}Y/mY the extension obtained as pullback of

0 \overline{}Y ^{\overline{}^{m}} Y \overline{}Y/mY \overline{}0.

Conversely, a retraction up to m to the inclusion of an extension of X by Y yields the corresponding modular morphism as being induced on the cokernels. Since the Ext^{1}-order of a short exact sequence that occurs in that binary tree is not necessarily square-free, modular means modulo prime powers. See [K 99] for an elaboration on this theme.

0.5 Acknowledgements

I thank Prof. Rentschler for kind hospitality during the time of the preparation of this article. Large parts of the work have been supported by the EU TMR-network ‘Algebraic Lie Representations’, grant no. ERB FMRX-CT97-0100. I thank my fellow postdocs in Paris for help of various kinds. I thank Prof. Steffen König for encouragement, advice and support during and after the time of the preparation of my thesis, upon the methods of which the present result relies.

0.6 Leitfaden

[figure: 1 \underline{Garnir}; 1.1 Specht lattices; 2 \underline{A two-box-shift morphism}; 2.1 Double paths; 1.2 A Garnir formula; 3 \underline{Semistandard morphisms}; 3.1 Correspondoids; A. Two lemmata; 2.2 Morphism, unreduced version; 3.2 Remarks on transposition in characteristic 2; 3.3 Paths as correspondoids; 2.3 Morphism, reduced version; 4 \underline{Two columns, several boxes}; 3.4 Horizontal examples; 2.4 Vertical examples]

1 Garnir

1.1 Specht lattices

For a, b ∈ Z, we let [a, b] := {i ∈ Z | a ⩽ i and i ⩽ b}.

Let n ⩾ 1. The symmetric group on the set [1, n] is denoted by S_{n} := Aut_{Sets}[1, n]. The sign of a permutation σ ∈ S_{n} is denoted by ε_{σ}. Maps are written in various manners, on the right, on the left, using indices etc. The symmetric group acts on the right. Occasionally, we shall allow ourselves to treat tuples consisting of pairwise different entries and single elements as sets. Intervals are to be read as subsets of Z. Conjugation in a group is also written as h^{g} := g^{−1}hg. Given a module X over a commutative ring R and an element r ∈ R, we shall usually write X/r := X/rX (cf. e.g. 3.14). If X and Y are modules over an R-algebra A, we identify Hom_{A}(X/r, Y/r) = Hom_{A}(X, Y/r) via composition with the residue class map X \overline{}X/r.

Let

[figure: N; λ; N_{0}; i; λ_{i}]

be a partition of n, i.e. assume ∑_{i} λ_{i} = n and λ_{i} ⩾ λ_{i+1} for i ∈ N. Usually, we write a partition as a tuple, i.e. λ = (λ_{1}, λ_{2}, …). Sometimes, we abbreviate repeated entries by exponents that indicate their multiplicity, such as e.g. (9, 2, 2, 2, 1, 1) = (9, 2^{3}, 1^{2}) ≠ (9, 8, 1). Let [λ] := {i × j ∈ N × N | λ_{i} ⩾ j} be the diagram of λ. A λ-tableau is a bijection

[figure: [λ]; [a]; ∼; [1, n]; i × j; a_{j,i}] .

The transposed partition of λ is denoted by λ′, i.e. i × j ∈ [λ] ⟺ j × i ∈ [λ′]. For a λtableau [a], we denote by [a′] the λ′-tableau obtained by composition with i × j \overline{}j × i, mapping [λ′] onto [λ]. A permutation σ ∈ S_{n} acts on the set of λ-tableaux T^{λ} via composition [a] ^{\overline{}^{σ}} [a]σ. Let F^{λ} be the free Z-module on T^{λ}, endowed with the induced action of the S_{n}. Let

[figure: [λ]; π_{R}^{λ}; N; ,; [λ]; π_{C}^{λ}; N; i × j; i; i × j; j]

denote the projections. To a λ-tableau [a] we attach a λ-tabloid

{a} := {[a]} := [a]^{−1}π_{R}^{λ} ∈ N^{[1,n]}.

In general, X being a set, an element σ ∈ S_{n} acts on f ∈ X^{[1,n]} via (i)(fσ) := (iσ^{−1})f. In particular, the free Z-module on the set of λ-tabloids, denoted by M^{λ}, carries a structure as a ZS_{n}-lattice, viz. {a}σ = {[a]σ}. Let

\begin{aligned}C_{[a]} & := & \{κ ∈ S_{n} | [a]^{−1}π_{C}^{λ} = ([a]κ)^{−1}π_{C}^{λ}\} \\ R_{[a]} & := & \{ρ ∈ S_{n} | [a]^{−1}π_{R}^{λ} = ([a]ρ)^{−1}π_{R}^{λ}\} \end{aligned}

be the column stabilizer and the row stabilizer of [a], respectively. Note that C_{[a]σ} = (C_{[a]})^{σ} and R_{[a]σ} = (R_{[a]})^{σ}. Let the Specht lattice S^{λ} be the ZS_{n}-sublattice of M^{λ} generated Zlinearly by the λ-polytabloids

⟨a⟩ := ⟨[a]⟩ := ∑_{κ∈C_{[a]}}{a} · κε_{κ} ∈ M^{λ}.

Let ζ ⊆ [1, n]. We denote by S_{ζ} := C_{S_{n}}([1, n]\ζ) the subgroup of S_{n} consisting of permutations that act merely on ζ. For an element x of a ZS_{n}-module X we denote

x ∘ ζ := ∑_{σ∈S_{ζ}} x · σε_{σ}.

Given p ⩾ 1 and 1 ⩽ j ⩽ k ⩽ λ′_{p}, we write a_{p,[j,k]} := (a_{p,i})_{i∈[j,k]} and abbreviate a_{p} := a_{p,[1,λ′_{p}]}. Let ξ ⊆ a_{p}, η ⊆ a_{p+1} be such that #ξ +#η = λ′_{p} +1. Letting S_{ξ} ×S_{η}\S_{ξ∪η} denote a chosen set of representatives of right cosets, the expression

G′′_{[a],ξ,η} := ∑_{σ∈S_{ξ}×S_{η}\S_{ξ∪η}}[a]σε_{σ}

is called a one-step Garnir relation. For p ⩾ 1, u, v ∈ a_{p}, u ≠ v, the expression

[a] + [a](u, v)

is called a signed column transposition. From [J 78, 7.2] and from the proof of [J 78, 8.4] we take that the (finite) set

\begin{aligned}& \{[a] + [a](u, v) | [a] ∈ T^{λ}, p ⩾ 1, u, v ∈ a_{p}, u ≠ v\} \\ ∪ & \{G′′_{[a],ξ,η} | [a] ∈ T^{λ}, p ⩾ 1, s, t ⩾ 1, s + t = λ′_{p} + 1, ξ = a_{p,[λ′_{p}−s+1,λ′_{p}]}, η = a_{p+1,[1,t]}\} \end{aligned}

ZS_{n}-linearly generates the kernel of the epimorphism

[figure: F^{λ}; ν^{λ}; S^{λ}; [a]; ⟨a⟩.]

Given a ZS_{n}-module X, its alternated module is written X^{−} := X ⊗_{Z} Z^{−}, Z^{−} being the abelian group Z, equipped with 1 · σ := ε_{σ}, σ ∈ S_{n}. So Z^{−} ≃ S^{(1^{n})}. For each partition λ of n, we fix a λ-tableau [a_{λ}] and let

{[a′_{λ}]σ}^{−} := ({a′_{λ}} ⊗ 1)σ ∈ M^{λ′,−}

for σ ∈ S_{n}. Note that {a′}^{−}σ = {[a′]σ}^{−} and that {a′}^{−}κ = {a′}ε_{κ} for [a] ∈ T^{λ}, σ ∈ S_{n}, κ ∈ C_{[a]}. We obtain a factorization into epimorphisms

( [figure: F^{λ}; ν^{λ}; S^{λ}; [a]; ⟨a⟩] ) = ( [figure: F^{λ}; ν_{M}^{λ}; M^{λ′,−}; ν_{S}^{λ}; S^{λ}; [a]; {a′}^{−}; ⟨a⟩] )

by first factoring out the signed column transpositions. Using this factorization, we may write

G′_{[a],ξ,η} := (G′′_{[a],ξ,η})ν_{M}^{λ} = \frac{1}{#ξ!#η!} · \{a′\}^{−} ∘ (ξ ∪ η).

A λ-tableau [a] is called standard if a_{j,i} ⩽ a_{j′,i′} for i × j, i′ × j′ ∈ [λ] such that i ⩽ i′ and j ⩽ j′. The tuple (⟨a⟩ | [a] is a standard λ-tableau) is a Z-linear basis of S^{λ} [J, 8.4]. Its elements are called standard λ-polytabloids. Let [ǎ_{λ}] be the standard λ-tableau determined by ǎ_{λ,j,i+1} = ǎ_{λ,j,i} + 1 whenever i × j, (i + 1) × j ∈ [λ]. (We do not require a priori that [a_{λ}] = [ǎ_{λ}], a requirement which is convenient for practical purposes, however.)

Account of the result (2.37). The notation deviates from our working notation further down. In particular, the following definitions are valid only in the remainder of this subsection (and will be repeated further down in case they coincide nonetheless). Let 1 ⩽ g < k ⩽ λ_{1} − 1 such that

μ′_{i} := \{ \begin{matrix}λ′_{i} + 2 & for i = g \\ λ′_{i} − 2 & for i = k + 1 \\ λ′_{i} & else \end{matrix}

defines a partition μ. (For the case of g = k, see (2.17)). A double path γ is a pair of integers l(δ) ⩾ 1 together with a pair of maps

[figure: γ_{δ} : [0, l(δ)]; [λ] ∪ [μ]; i; α(δ, i) × β(δ, i),]

where δ ∈ [1, 2], subject to the following conditions.

(i) i < i′ implies β(δ, i) < β(δ, i′) for δ ∈ [1, 2] and i, i′ ∈ [0, l(δ)].

(ii) γ_{1}([0, l(1)]) ∩ γ_{2}([0, l(2)]) = ∅.

(iii) α(δ, 0) × β(δ, 0) = (λ′_{g} + δ) × g for δ ∈ [1, 2].

(iv) β(δ, l(δ)) = k + 1 for δ ∈ [1, 2].

(v) α(1, l(1)) < α(2, l(2)).

The set of double paths is denoted by Γ. Suppose given γ ∈ Γ. Let ε_{γ} := (−1)^{α(1,l(1))+α(2,l(2))}. For j ∈ [g + 1, k], we let

e(γ, j) := ∑_{δ∈[1,2]}#γ_{δ}^{−1}([1, λ′_{j}] × {j}).

Let

[figure: [0, λ′_{k+1}]; [0, μ′_{k+1}]; i; φ; #([1, i]\{α(1, l(1)), α(2, l(2))}); min(φ^{−1}({j})); ψ; j.]

Given a λ-tableau [a], we let the μ-tableau [a^{γ}] be defined by

a_{j,i}^{γ} := a_{j,i}  for i × j ∈ [μ]\(γ_{1}([0, l(1) − 1]) ∪ γ_{2}([0, l(2) − 1]) ∪ N × {k + 1})
a_{β(δ,i),α(δ,i)}^{γ} := a_{β(δ,i+1),α(δ,i+1)} for δ ∈ [1, 2], i ∈ [0, l(δ) − 1]
a_{k+1,i}^{γ} := a_{k+1,ψ(i)}  for i ∈ [1, μ′_{k+1}].

For j ∈ [g, k + 1], we let

X_{j} := (λ′_{j} − j) − (λ′_{k+1} − (k + 1)).

For i ∈ [0, 1], we denote

L(i) := {j ∈ [g + 1, k − 1] | λ′_{j+1} = λ′_{j} − i}.

We write [g + 1, k] = ⋃_{κ∈[1,K]}[p(κ), q(κ)] such that p(1) = g + 1, such that p(κ) ⩽ q(κ), [p(κ), q(κ) − 1] ⊆ L(0) ∪ L(1) and q(κ) ∉ L(0) ∪ L(1) for κ ∈ [1, K], such that q(κ) + 1 = p(κ + 1) for κ ∈ [1, K − 1], and such that q(K) = k. For Z ∈ Z and i ⩾ 0, we write Z^{(i)} := (Z + i − 1)!/(Z − 1)!. We let

R := ( ∏_{i∈[0,1]} ∏_{j∈L(i)}X_{j}^{(2−i)})( ∏_{{κ∈[1,K], \atop [p(κ),q(κ)−1]⊆L(1)}} gcd(2, X_{q(κ)}))

and

m := \{ \begin{matrix}X_{g} + 2 & in case [p(1), q(1) − 1] ⊈ L(1) \\ (X_{g} + 2)/ gcd(2, X_{g}, X_{g+1}) & in case [p(1), q(1) − 1] ⊆ L(1) \end{matrix} .

The ZS_{n}-linear map

[figure: F^{λ}; f′′; S^{μ}; [a]; ∑_{γ∈Γ}( ∏_{j∈[g+1,k]}X_{j}^{(2−e(γ,j))})⟨a^{γ}⟩ε_{γ}]

is divisible by R, and its quotient by R factors ZS_{n}-linearly as

( [figure: F^{λ}; f′′/R; S^{μ}; S^{μ}/m; ⟨b⟩; ⟨b⟩ + mS^{μ}] ) = ( [figure: F^{λ}; ν^{λ}; S^{λ}; f; S^{μ}/m; [a]; ⟨a⟩] ) .

The resulting morphism S^{λ} ^{\overline{}^{f}} S^{μ}/m is of order m as an element of Hom_{ZS_{n}}(S^{λ}, S^{μ}/m).

1.2 A Garnir formula

Let λ be a partition of n, let [a] be a λ-tableau, let 1 ⩽ p < q ⩽ λ_{1}, let d ⩾ 1. Let ξ ⊆ a_{p}, s := #ξ, ξ̄ := a_{p}\ξ and η ⊆ a_{q}, t := #η, η̄ := a_{q}\η, u := #η̄, be such that s + t = λ′_{p} + 1 − d. We choose a disjoint decomposition ξ̄ = ξ̄_{0} ∪ ξ̄_{1} such that #ξ̄_{0} = d − 1 and #ξ̄_{1} = t. Lemma 1.1 (cf. [K 99, 4.3.4]) Assume given x ∈ φ ⊆ a_{p}, y ∈ ψ ⊆ a_{q}. Then

\begin{aligned}⟨a⟩ ∘ (φ ∪ ψ) & = & #φ · ⟨a⟩ ∘ (φ ∪ ψ\x) − #ψ · ⟨a⟩ · (x, y) ∘ (φ ∪ ψ\x) \\ & = & #ψ · ⟨a⟩ ∘ (φ ∪ ψ\y) − #φ · ⟨a⟩ · (x, y) ∘ (φ ∪ ψ\y). \end{aligned}

In fact,

⟨a⟩ ∘ (φ ∪ ψ) = ∑_{z∈φ∪ψ} ∑_{σ∈S_{φ∪ψ}, zσ=x}⟨a⟩ · σε_{σ}
= ( ∑_{z∈φ\x} ∑_{σ∈S_{φ∪ψ}, zσ=x}⟨a⟩ · σε_{σ})
+ (⟨a⟩ ∘ (φ ∪ ψ\x))
+ ( ∑_{z∈ψ\y} ∑_{σ∈S_{φ∪ψ}, zσ=x}⟨a⟩ · σε_{σ})
+ ( ∑_{σ∈S_{φ∪ψ}, yσ=x}⟨a⟩ · σε_{σ})
= ((#φ − 1) · ⟨a⟩ ∘ (φ ∪ ψ\x))
+ (⟨a⟩ ∘ (φ ∪ ψ\x))
− ((#ψ − 1) · ⟨a⟩ · (x, y) ∘ (φ ∪ ψ\x))
− (⟨a⟩ · (x, y) ∘ (φ ∪ ψ\x)).

The second equality follows by symmetry. Suppose given disjoint subsets φ, ψ ⊆ [1, n] and a bijection φ \overline{}^{α}_{∼} ψ. We denote

(φ, ψ)_{α} := ∏_{x∈φ}(x, (x)α) ∈ S_{n}.

In case φ ⊆ a_{p}, ψ ⊆ a_{q}, #φ = #ψ, the element ⟨a⟩ · (φ, ψ) := ⟨a⟩ · (φ, ψ)_{α} is independent of the choice of a bijection φ \overline{}^{α}_{∼} ψ. Lemma 1.2 ([K 99, 4.3.5]) In case d = 1 we have

⟨a⟩ ∘ (ξ ∪ η) = s! t! · ⟨a⟩ · (ξ̄, η).

We calculate

⟨a⟩ ∘ (ξ ∪ η) =^{(1.1)} \frac{t}{s+1} · ⟨a⟩ · (x, y) ∘ (ξ ∪ η)
=^{(1.1)} \frac{t(t−1)}{(s+1)(s+2)} · ⟨a⟩ · (x, y) · (x′, y′) ∘ (ξ ∪ η)
=^{(1.1)} ⋯

\begin{matrix}=^{(1.1)} & \frac{s! t!}{(s+t)!} · ⟨a⟩ · (ξ̄, η) ∘ (ξ ∪ η) \\ = & s! t! · ⟨a⟩ · (ξ̄, η), \end{matrix}

where ξ̄ = {x, x′, …}, η = {y, y′, …}.

Lemma 1.3 Let φ ⊆ a_{p} and ψ ⊆ a_{q} be given such that #φ = #ψ and let σ ∈ S_{a_{p}}. We obtain

⟨a⟩ · (φ, ψ) · σ = ⟨a⟩ · ((φ)σ, ψ)ε_{σ}.

Factorization reduces us to the consideration of a transposition σ = (x, y), where, moreover, we may assume x ∈ φ and y ∈ a_{p}\φ. Choosing z ∈ ψ, we conclude

⟨a⟩ · (φ, ψ) · (x, y) = ⟨a⟩((φ\x), (ψ\z))(x, z)(x, y)
= −⟨a⟩((φ\x), (ψ\z))(y, z)
= −⟨a⟩((φ)(x, y), ψ).

Lemma 1.4 The formula

⟨a⟩ ∘ (ξ ∪ η) = \frac{s!}{(d − 1)!} · ⟨a⟩ · (ξ̄_{1}, η) ∘ ξ̄

holds.

The claimed equality may be reformulated to

⟨a⟩ ∘ (ξ ∪ η) ∘ ξ̄ = \frac{s!(t + d − 1)!}{(s + t)!(d − 1)!} · ⟨a⟩ · (ξ̄_{1}, η) ∘ (ξ ∪ η) ∘ ξ̄.

We perform an induction over d, the case d = 1 being treated in (1.2). We assume d ⩾ 2 and perform an induction over t, the case t = 0 being trivial. We assume t ⩾ 1, choose x_{0} ∈ ξ̄_{0}, x_{1} ∈ ξ̄_{1}, y ∈ η and denote ξ̄_{2} := (ξ̄_{1})(x_{0}, x_{1}). Evaluating in two ways, we obtain

⟨a⟩ ∘ (ξ ∪ x_{1} ∪ η) ∘ (ξ̄\x_{1})
=^{1., case d − 1, t} \frac{(s+1)!(t+d−2)!}{(s+t+1)!(d−2)!} · ⟨a⟩ · (ξ̄_{2}, η) ∘ (ξ ∪ x_{1} ∪ η) ∘ (ξ̄\x_{1})
=  \frac{(s+1)!(t+d−2)!}{(s+t)!(d−2)!} · ⟨a⟩ · (ξ̄_{2}, η) ∘ (ξ ∪ η) ∘ (ξ̄\x_{1})
=^{2., (1.1)} (s + 1) · ⟨a⟩ ∘ (ξ ∪ η) ∘ (ξ̄\x_{1})
−  t[⟨a⟩ · (x_{1}, y) ∘ ((ξ ∪ y) ∪ (η\y)) ∘ (ξ̄\x_{1})]
=^{case d, t − 1}  \frac{s+1}{t+d−1} · ⟨a⟩ ∘ (ξ ∪ η) ∘ ξ̄
−  t[\frac{(s+1)!(t+d−2)!}{(s+t)!(d−1)!} · ⟨a⟩ · (x_{1}, y) · (ξ̄_{1}\x_{1}, η\y) ∘ ((ξ ∪ y) ∪ (η\y)) ∘ (ξ̄\x_{1})].

Therefore,

⟨a⟩ ∘ (ξ ∪ η) ∘ ξ̄ = \frac{s!(t+d−1)!}{(s+t)!(d−2)!} · ⟨a⟩ · (ξ̄_{2}, η) ∘ (ξ ∪ η) ∘ (ξ̄\x_{1})
+ t\frac{s!(t+d−1)!}{(s+t)!(d−1)!} · ⟨a⟩ · (ξ̄_{1}, η) ∘ (ξ ∪ η) ∘ (ξ̄\x_{1})
=^{(1.3)} −(d − 1)\frac{s!(t+d−1)!}{(s+t)!(d−1)!} · ⟨a⟩ · (ξ̄_{1}, η) ∘ (ξ ∪ η) · (x_{0}, x_{1}) ∘ (ξ̄_{0} ∪ ξ̄_{1}\x_{1})
+ t\frac{s!(t+d−1)!}{(s+t)!(d−1)!} · ⟨a⟩ · (ξ̄_{1}, η) ∘ (ξ ∪ η) ∘ (ξ̄_{0} ∪ ξ̄_{1}\x_{1})
=^{(1.1)} \frac{s!(t+d−1)!}{(s+t)!(d−1)!} · ⟨a⟩ · (ξ̄_{1}, η) ∘ (ξ ∪ η) ∘ ξ̄.

Proposition 1.5 (Garnir formula) Assume given a subset φ ⊆ ⋃_{j∈[1,p−1]} a_{j}. We have

⟨a⟩ ∘ (φ ∪ ξ ∪ η) = \frac{s!}{(s + t)!(d − 1)!} · ⟨a⟩ · (ξ̄_{1}, η) ∘ ξ̄∘ (φ ∪ ξ ∪ η).

Let A be the set of injections of φ into φ ∪ ξ ∪ η. We write

\begin{aligned}⟨a⟩ ∘ (φ ∪ ξ ∪ η) & = & ∑_{α∈A} ∑_{σ∈S_{φ∪ξ∪η}, σ|_{φ}=α}⟨a⟩ · σε_{σ} \\ ⟨a⟩ · (ξ̄_{1}, η) ∘ ξ̄∘ (φ ∪ ξ ∪ η) & = & ∑_{α∈A} ∑_{σ∈S_{φ∪ξ∪η}, σ|_{φ}=α}⟨a⟩ · (ξ̄_{1}, η) ∘ ξ̄· σε_{σ} \end{aligned}

and fix an embedding α ∈ A in order to compare the summands. We choose a permutation ρ ∈ S_{φ∪ξ∪η} that restricts to ρ|_{φ} = α. On the one hand, we obtain

∑_{σ∈S_{φ∪ξ∪η}, σ|_{φ}=α}⟨a⟩ · σε_{σ} =^{σ = σ′ρ} ∑_{σ′∈S_{ξ∪η}}⟨a⟩ · σ′ε_{σ′} · ρε_{ρ}
= ⟨a⟩ ∘ (ξ ∪ η) · ρε_{ρ}
=^{(1.4)}  \frac{s!}{(d − 1)!} · ⟨a⟩ · (ξ̄_{1}, η) ∘ ξ̄· ρε_{ρ},

on the other hand, we have

\begin{aligned}∑_{σ∈S_{φ∪ξ∪η}, σ|_{φ}=α}⟨a⟩ · (ξ̄_{1}, η) ∘ ξ̄· σε_{σ} & =^{σ = σ′ρ} & ∑_{σ′∈S_{ξ∪η}}⟨a⟩ · (ξ̄_{1}, η) ∘ ξ̄· σ′ε_{σ′} · ρε_{ρ} \\ & = & (s + t)! · ⟨a⟩ · (ξ̄_{1}, η) ∘ ξ̄· ρε_{ρ}, \end{aligned}

whence the result follows by comparison. Independence of the right hand side of the choice of ξ̄_{1} can also be seen by application of (1.3) to an appropriate permutation τ ∈ Sξ̄, for we have (−) · τ ∘ ξ̄ = (−) ∘ ξ̄ε_{τ}.

Lemma 1.6 For i_{0} ∈ [0, s], we write

B_{[a],ξ,η}(i_{0}) := ∑_{s_{0}∈[i_{0},s]}(−1)^{s_{0}} ∑_{{ξ_{0}⊆ξ, \atop #ξ_{0} = s_{0}}} ∑_{{η_{0}⊆η, \atop #η_{0} = s_{0}−i_{0}}} ∑_{{φ_{0}⊆η̄, \atop #φ_{0} = i_{0}}}⟨a⟩ · (ξ_{0}, η_{0} ∪ φ_{0}).

Let i ∈ [1, u]. We define and obtain

\begin{aligned}C_{[a],ξ,η}(i) & := & ∑_{{φ⊆η̄, \atop #φ = i}}⟨a⟩ ∘ (ξ ∪ η ∪ φ) \\ & = & s!(t + i)! ∑_{i_{0}∈[0,min(i,s)]} (\begin{matrix}u−i_{0} \\ i−i_{0} \end{matrix}) B_{[a],ξ,η}(i_{0}). \end{aligned}

We fix a subset φ ⊆ η̄ of cardinality #φ = i. Indexing by subsets that become interchanged by a certain element of S_{ξ∪η∪φ}, we obtain

\begin{matrix}⟨a⟩ ∘ (ξ ∪ η ∪ φ) \\ = s!(t + i)! ∑_{i_{0}∈[0,min(i,s)]} ∑_{s_{0}∈[i_{0},s]} ∑_{{ξ_{0}⊆ξ, \atop #ξ_{0} = s_{0}}} ∑_{{η_{0}⊆η, \atop #η_{0} = s_{0}−i_{0}}} ∑_{{φ_{0}⊆φ, \atop #φ_{0} = i_{0}}}⟨a⟩ · (ξ_{0}, η_{0} ∪ φ_{0})ε_{(ξ_{0},η_{0}∪φ_{0})}. \end{matrix}

The number of subsets φ of cardinality #φ = i that lie in between φ_{0} ⊆ φ ⊆ η̄, where φ_{0} is a given subset of η̄ of cardinality #φ_{0} = i_{0}, amounts to (\begin{matrix}u−i_{0} \\ i−i_{0} \end{matrix}). Whence the result is obtained by collecting

∑_{{φ⊆η̄, \atop #φ = i}} ∑_{{φ_{0}⊆φ, \atop #φ_{0} = i_{0}}}U(φ_{0}) = (\begin{matrix}u−i_{0} \\ i−i_{0} \end{matrix}) ∑_{{φ_{0}⊆η̄, \atop #φ_{0} = i_{0}}}U(φ_{0}),

U(φ_{0}) being some expression independent of φ.

Lemma 1.7 Suppose given in addition a subset ψ ⊆ ξ, of cardinality #ψ =: v. For i_{0} ∈ [0, s − v], we write

B′_{[a],ξ,ψ,η}(i_{0}) := \frac{(−1)^{i_{0}}}{v!t!} ∑_{{ξ_{0}⊆ξ\ψ, \atop #ξ_{0} = i_{0}}} ∑_{{φ_{0}⊆η, \atop #φ_{0} = i_{0}}} ⟨a⟩ · (ξ_{0}, φ_{0}) ∘ (ψ ∪ η).

Let i ∈ [1, u]. We define and obtain

\begin{aligned}C′_{[a],ξ,ψ,η}(i) & := & ∑_{{φ⊆η, \atop #φ = i}}⟨a⟩ ∘ (ξ ∪ φ) ∘ (ψ ∪ η) \\ & = & t!s!(i + v)! · ∑_{i_{0}∈[0,min(i,s−v)]} (\begin{matrix}t−i_{0} \\ i−i_{0} \end{matrix}) B′_{[a],ξ,ψ,η}(i_{0}). \end{aligned}

We fix a subset φ ⊆ η of cardinality #φ = i and obtain

\frac{1}{s!i!}⟨a⟩ ∘ (ξ ∪ φ) ∘ (ψ ∪ η)
= \frac{1}{(s−v)!i!} (\begin{matrix}i+v \\ v \end{matrix}) ⟨a⟩ ∘ ((ξ\ψ) ∪ φ) ∘ (ψ ∪ η)
= (\begin{matrix}i+v \\ v \end{matrix}) ∑_{i_{0}∈[0,min(i,s−v)]} ∑_{{ξ_{0}⊆ξ\ψ, \atop #ξ_{0} = i_{0}}} ∑_{{φ_{0}⊆φ, \atop #φ_{0} = i_{0}}}(−1)^{i_{0}}⟨a⟩ · (ξ_{0}, φ_{0}) ∘ (ψ ∪ η),

whence we conclude as in (1.6) that

∑_{{φ⊆η, \atop #φ = i}}⟨a⟩ ∘ (ξ ∪ φ) ∘ (ψ ∪ η)

= ∑_{{φ⊆η, \atop #φ = i}} \frac{s!(i+v)!}{v!} ∑_{i_{0}∈[0,min(i,s−v)]} ∑_{{ξ_{0}⊆ξ\ψ, \atop #ξ_{0} = i_{0}}} ∑_{{φ_{0}⊆φ, \atop #φ_{0} = i_{0}}}(−1)^{i_{0}}⟨a⟩ · (ξ_{0}, φ_{0}) ∘ (ψ ∪ η)
= \frac{s!(i+v)!}{v!} ∑_{i_{0}∈[0,min(i,s−v)]} ∑_{{ξ_{0}⊆ξ\ψ, \atop #ξ_{0} = i_{0}}} (\begin{matrix}t−i_{0} \\ i−i_{0} \end{matrix}) ∑_{{φ_{0}⊆η, \atop #φ_{0} = i_{0}}}(−1)^{i_{0}}⟨a⟩ · (ξ_{0}, φ_{0}) ∘ (ψ ∪ η).

2 A two-box-shift morphism

2.1 Double paths

Let λ be a partition of n. We assume given integers g, k such that 1 ⩽ g ⩽ k ⩽ λ_{1} − 1 and such that

μ′_{i} := \{ \begin{matrix}λ′_{i} + 2 & for i = g \\ λ′_{i} − 2 & for i = k + 1 \\ λ′_{i} & else \end{matrix}

defines a partition μ. A weight e is a map

[figure: [1, λ_{1}]; e; [0, 2]; j; e_{j}]

that maps g and k + 1 to e_{g} = e_{k+1} = 2 and that maps each j ∈ [1, λ_{1}]\[g, k + 1] to e_{j} = 0. The set of weights is denoted by E. A pattern Ξ of weight e is a subset Ξ ⊆ [1, 2]×[g, k+1] such that

#(Ξ ∩ ([1, 2] × {j})) = e_{j}

for j ∈ [g, k + 1]. A double path γ of weight e is an injection from a pattern Ξ of weight e to [λ] ∪ [μ] of the form

[figure: Ξ; γ; [λ] ∪ [μ]; i × j; γ̄(j, i) × j]

such that

\begin{matrix}γ̄(g, i) = λ′_{g} + i & for i ∈ [1, 2] \\ γ̄(k + 1, 1) < γ̄(k + 1, 2). \end{matrix}

Sometimes, we denote its pattern by Ξ_{γ} := Ξ. Its sign is given by ε_{γ} := (−1)^{γ̄(k+1,1)+γ̄(k+1,2)}, not to be confused with the sign of a permutation. The set of double paths of weight e is denoted by Γ̇(e).

A double path γ gives rise to a place operation in the following manner. Let

[figure: [0, λ′_{k+1}]; [0, μ′_{k+1}]; i; φ; #([1, i]\{γ̄(k + 1, 1), γ̄(k + 1, 2)}); min(φ^{−1}({j})); ψ; j.]

Given a weight e ∈ E, we define an operation of sets

[figure: Γ̇(e); ×; T^{λ}; T^{μ}; γ; ×; [a]; [a^{γ}]]

by

a_{j,i}^{γ} := a_{j,i} for i × j ∈ [μ]\(Ξ)γ and j ≠ k + 1,
a_{j,γ̄(j,i)}^{γ} := a_{j′,γ̄(j′,i)} for i ∈ [1, 2] and for j ∈ [g, k],
where j′ is minimal in [j + 1, k + 1] with i × j′ ∈ Ξ,
a_{k+1,i}^{γ} := a_{k+1,ψ(i)} for i ∈ [1, μ′_{k+1}].

We define a ZS_{n}-linear map

[figure: F^{λ}; f′′_{e}; S^{μ}; [a]; ∑_{γ∈Γ̇(e)}⟨a^{γ}⟩ε_{γ}.]

Example 2.1 For example, consider the case λ = (7, 7, 7, 4), μ = (7, 6, 6, 4, 1, 1), g = 1, k = 6. Let e = (2, 1, 1, 1, 2, 0, 2), and let γ be the (non-ordered) double path of weight e given pictorially by

x x v x x x u x u x x v x x x x x u u x v x x x x u v,

a ‘u’ indicating the γ-image of some 1 × j, a ‘v’ indicating the γ-image of some 2 × j. The operation of γ yields e.g.

γ × 1 5 9 13 17 20 23 2 6 10 14 18 21 24 3 7 11 15 19 22 25 4 8 12 16  \overline{}1 5 18 13 17 20 24 2 15 10 14 25 21 3 7 11 19 23 22 4 8 12 16 6 9

I.e. the double path γ subsequently pushes 23 \overline{}19 \overline{}15 \overline{}6 \overline{}and 25 \overline{}18 \overline{}9 \overline{}.

Lemma 2.2 (path switch) For q ∈ [g + 1, k + 1], we dispose of an involution

[figure: [1, 2]; ×; [g, k + 1]; ι_{q}; [1, 2] × [g, k + 1]; i; ×; j; \{ \begin{matrix}i · (1, 2) & × & j & for j ∈ [g + 1, q − 1] \\ i & × & j & else \end{matrix} .]

The composition ι_{p}γ of this involution, restricted to ι_{p}^{−1}(Ξ_{γ}), with a double path γ furnishes a double path which we denote by ι_{p}γ, being a slight abuse of notation. So Ξ_{ι_{p}γ} = ι_{p}^{−1}(Ξ_{γ}).

Let [a] ∈ T^{λ}, e ∈ E, let p ∈ [g + 1, k] such that e_{p} = 2, let γ ∈ Γ̇(e), u := a_{p,γ̄(p,1)} and v := a_{p,γ̄(p,2)}. We assert the following equalities.

\begin{aligned}(i) & ⟨a^{γ}⟩ · (u, v) & = & −⟨a^{ι_{p}γ}⟩ \\ (ii) & ⟨a^{ι_{p+1}γ}⟩ & = & ⟨a^{γ}⟩ \\ (iii) & ([a]f′′_{e})(u, v) & = & −[a]f′′_{e} \end{aligned}

The operation of the double path ι_{p}γ on [a] can be performed using the operation of γ, but with entries u and v interchanged, and, moreover, with entries placed by γ in column g interchanged, yielding the sign in (i). Analoguously (ii). Since composition with ι_{p} is an involution on Γ̇(e), multiplication by (u, v) attaches a sign to [a]f′′_{e} by (i), whence (iii).

Proposition 2.3 Let e ∈ E. There is a factorization

(F^{λ} ^{\overline{}^{f′′_{e}}} S^{μ}) = (F^{λ} ^{\overline{}^{ν_{M}^{λ}}} M^{λ′,−} ^{\overline{}^{f′_{e}}} S^{μ}).

We need to show that a signed column transposition [a] + [a](u, v), [a] a λ-tableau, p ⩾ 1, u, v ∈ a_{p}, u ≠ v, vanishes under f′′_{e}, for which we may assume p to lie in [g + 1, k + 1]. We abbreviate the double path image of γ in [a] by I_{γ} := ((Ξ)γ)[a].

Case p ∈ [g + 1, k].

Subcase u, v ∈ I_{γ}. See (2.2, iii).

Subcase u =: a_{p,γ̄(p,i)} ∈ I_{γ}, v =: a_{p,l} ∉ I_{γ}. Abbreviating by τ_{γ} the transposition that interchanges the elements γ̄(p, i) × p and l × p of [λ] ∪ [μ], we conclude from

⟨a^{γ}⟩ · (u, v) = −⟨a^{γτ_{γ}}⟩

that the sum over these summands, multiplied by (u, v), yields minus the sum over the summands of the subcase u ∉ I_{γ}, v ∈ I_{γ}. In fact, γ \overline{}γτ_{γ} is a bijection between the sets of double paths occurring in these subcases which preserves the sign.

Subcase u, v ∉ I_{γ}. ⟨a^{γ}⟩ · (u, v) = −⟨a^{γ}⟩ shows these summands to yield zero.

Case p = k + 1. The arguments of the case p ∈ [g + 1, k] work in the subcases u, v ∈ I_{γ} and u, v ∉ I_{γ} as well.

Subcase u =: a_{k+1,γ̄(k+1,i)} ∈ I_{γ}, v =: a_{k+1,l} ∉ I_{γ}. Writing the transposition τ_{γ} as above, we conclude from

⟨a^{γ}⟩ · (u, v) = (−1)^{γ̄(p,i)−l+1}⟨a^{γτ_{γ}}⟩

that the sum over these summands yields minus the result of the sum over the summands of the subcase u ∉ I_{γ}, v ∈ I_{γ}. In fact, γ \overline{}γτ_{γ} is a bijection between the sets of double paths occurring in these subcases which changes the sign by ε_{γτ_{γ}}/ε_{γ} = (−1)^{l−γ̄(p,i)}.

2.2 Morphism, unreduced version

We shall show the vanishing, modulo our modulus, of the required Garnir relations. Rather than giving short versions of all calculations, we give full versions, but only of the two ‘large’ cases (A.1, A.2), and only in Appendix A.

For i ∈ [g, k + 1], we let

X_{i} := (λ′_{i} − i) − (λ′_{k+1} − (k + 1))

and denote X_{i}^{(j)} := (X_{i} + j − 1)!/(X_{i} − 1)!, j ∈ [0, 2]. Let M^{λ′,−} ^{\overline{}^{f′}} S^{μ} be the ZS_{n}-linear map defined by

f′ := ∑_{e∈E} ( ∏_{i∈[g+1,k]}X_{i}^{(2−e_{i})}) f′_{e}.

Let [a] be a λ-tableau, p ∈ [g, k], ξ ⊆ a_{p}, s := #ξ, ξ̄ := a_{p}\ξ and η ⊆ a_{p+1}, t := #η, η̄:= a_{p+1}\η such that s + t = λ′_{p} + 1. Denote u := #η̄ = λ′_{p+1} − t.

Lemma 2.4 (cf. (A.1)) Suppose g < p < k and s, t ⩾ 2. The map f′ annihilates G′_{[a],ξ,η}.

We fix a map

[figure: [1, λ_{1}]\{p, p + 1}; ẽ; [0, 2]; j; ẽ_{j}]

that maps g and k + 1 to e_{g} = e_{k+1} = 2, and that maps j ∈ [1, λ_{1}]\[g, k + 1] to e_{j} = 0. For α, β ∈ [0, 2], we denote by ẽαβ be the prolongation of ẽ to [1, λ_{1}] defined by ẽαβ|_{[1,λ_{1}]\{p,p+1}} := ẽ, (ẽαβ)_{p} := α and (ẽαβ)_{p+1} := β. We contend that

∑_{α,β∈[0,2]}X_{p}^{(2−α)}X_{p+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽαβ} = 0,

from which the lemma ensues.

There exist elements x, y ∈ ξ, x ≠ y, x′, y′ ∈ η, x′ ≠ y′, z ∈ ξ̄, which we choose and fix. For γ ∈ Γ̇(ẽ00), we let x_{γ} := a_{j,γ̄(j,1)}, where j ∈ [p + 2, k + 1] is minimal with 1 × j ∈ Ξ_{γ}, and y_{γ} := a_{j,γ̄(j,2)}, where j ∈ [p + 2, k + 1] is minimal with 2 × j ∈ Ξ_{γ}. I.e. we pick the entries x_{γ}, y_{γ} that ‘cross the columns’ p and p + 1 under the operation of γ. We write

\begin{aligned}U_{γ} & := & (s + t − 2)!^{−1} · ⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (x_{γ}, x, x′) · (y_{γ}, y, y′) ∘ (ξ ∪ η) \\ V_{1,γ} & := & (s + t − 1)!^{−1} · ⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) ∘ (ξ ∪ η) \\ V_{2,γ} & := & (s + t − 1)!^{−1} · ⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (y_{γ}, y, y′) ∘ (ξ ∪ η), \end{aligned}

and let

A := ∑_{γ∈Γ̇(ẽ00)}U_{γ} · ∑_{w′∈η̄}(w′, z)
B := ∑_{γ∈Γ̇(ẽ00)}U_{γ} · (1 − ∑_{w∈ξ̄\z}(w, z))
C_{1} := ∑_{γ∈Γ̇(ẽ00)}U_{γ} · (z, x_{γ})
C_{2} := ∑_{γ∈Γ̇(ẽ00)}U_{γ} · (z, y_{γ})
D := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ} · ∑_{w∈ξ̄}(w, y_{γ})
= ∑_{γ∈Γ̇(ẽ00)}V_{2,γ} · ∑_{w∈ξ̄}(w, x_{γ})
H := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ} · ∑_{w′∈η̄}(w′, y_{γ})
= ∑_{γ∈Γ̇(ẽ00)}V_{2,γ} · ∑_{w′∈η̄}(w′, x_{γ})
F_{1} := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ}
F_{2} := ∑_{γ∈Γ̇(ẽ00)}V_{2,γ}.

Calculations carried out in (A.1) yield the following table.

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ22} & = & 2(s − u)A + (s − u)(s − u − 1)B + 2(s − u)(s − u + 1)D \\ G′_{[a],ξ,η}f′_{ẽ12} & = & −2A − 2(s − u)B − 2(s − u + 1)D + 2(s − u + 1)H \\ G′_{[a],ξ,η}f′_{ẽ02} & = & B − 2H \\ G′_{[a],ξ,η}f′_{ẽ21} & = & 2A + 2(s − u − 1)B − (s − u − 1)(C_{1} + C_{2}) + 4(s − u)D \\ G′_{[a],ξ,η}f′_{ẽ11} & = & −2B + (C_{1} + C_{2}) − 2D + 2H + (s − u)(F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ01} & = & −(F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ20} & = & B − (C_{1} + C_{2}) + 2D \\ G′_{[a],ξ,η}f′_{ẽ10} & = & (F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ00} & = & 0 \end{aligned}

We note that s − u = X_{p} − X_{p+1} and evaluate the linear combination

∑_{α,β∈[0,2]}X_{p}^{(2−α)}X_{p+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽαβ} ; \begin{aligned}= & 1 & ·1 & ·(2(X_{p} − X_{p+1})A + (X_{p} − X_{p+1})(X_{p} − X_{p+1} − 1)B \\ & & & +2(X_{p} − X_{p+1})(X_{p} − X_{p+1} + 1)D) \\ + & X_{p} & ·1 & ·( − 2A − 2(X_{p} − X_{p+1})B − 2(X_{p} − X_{p+1} + 1)D \\ & & & +2(X_{p} − X_{p+1} + 1)H) \\ + & X_{p}(X_{p} + 1) & ·1 & ·(B − 2H) \\ + & 1 & ·X_{p+1} & ·(2A + 2(X_{p} − X_{p+1} − 1)B \\ & & & −(X_{p} − X_{p+1} − 1)(C_{1} + C_{2}) + 4(X_{p} − X_{p+1})D) \\ + & X_{p} & ·X_{p+1} & ·( − 2B + (C_{1} + C_{2}) − 2D + 2H \\ & & & +(X_{p} − X_{p+1})(F_{1} + F_{2})) \\ + & X_{p}(X_{p} + 1) & ·X_{p+1} & ·( − (F_{1} + F_{2})) \\ + & 1 & ·X_{p+1}(X_{p+1} + 1) & ·(B − (C_{1} + C_{2}) + 2D) \\ + & X_{p} & ·X_{p+1}(X_{p+1} + 1) & ·((F_{1} + F_{2})) \\ + & X_{p}(X_{p} + 1) & ·X_{p+1}(X_{p+1} + 1) & ·(0) \\ = & 0. \end{aligned}

Lemma 2.5 Suppose g < p < k and s = λ′_{p}, t = 1. The map f′ annihilates G′_{[a],ξ,η}. Concerning the map ẽ and its prolongations, as well as concerning the integers x_{γ}, y_{γ}, we continue to use the notation of (2.4). We fix such a map ẽ and need to show that

∑_{α,β∈[0,2]}X_{p}^{(2−α)}X_{p+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽαβ} = 0.

There exist elements x, y ∈ ξ, x ≠ y, which we choose and fix. For γ ∈ Γ̇(ẽ00), we write

\begin{aligned}V_{1,γ} & := & \frac{1}{(s−1)!} · ⟨a^{γ}⟩ε_{γ} · (y_{γ}, y, η) ∘ (ξ ∪ η) \\ V_{2,γ} & := & \frac{1}{(s−1)!} · ⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, η) ∘ (ξ ∪ η) \end{aligned}

and let

A := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ} · ∑_{w′∈η̄}(x_{γ}, x, w′)
= ∑_{γ∈Γ̇(ẽ00)}V_{2,γ} · ∑_{w′∈η̄}(y_{γ}, y, w′)
C_{1} := − ∑_{γ∈Γ̇(ẽ00)}V_{1,γ} · (x_{γ}, x)
C_{2} := − ∑_{γ∈Γ̇(ẽ00)}V_{2,γ} · (y_{γ}, y)

H := s^{−1} · ∑_{γ∈Γ̇(ẽ00)}V_{1,γ} · ∑_{w′∈η̄}(x_{γ}, w′)
= s^{−1} · ∑_{γ∈Γ̇(ẽ00)}V_{2,γ} · ∑_{w′∈η̄}(y_{γ}, w′).
F_{1} := s^{−1} · ∑_{γ∈Γ̇(ẽ00)}V_{1,γ}
F_{2} := s^{−1} · ∑_{γ∈Γ̇(ẽ00)}V_{2,γ}

Calculations similar to those of (A.1) yield the following table, to be compared to the table of (2.4).

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ22} & = & 2(s − u)A \\ G′_{[a],ξ,η}f′_{ẽ12} & = & −2A + 2(s − u + 1)H \\ G′_{[a],ξ,η}f′_{ẽ02} & = & −2H \\ G′_{[a],ξ,η}f′_{ẽ21} & = & 2A − (s − u − 1)(C_{1} + C_{2}) \\ G′_{[a],ξ,η}f′_{ẽ11} & = & (C_{1} + C_{2}) + 2H + (s − u)(F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ01} & = & −(F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ20} & = & −(C_{1} + C_{2}) \\ G′_{[a],ξ,η}f′_{ẽ10} & = & (F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ00} & = & 0 \end{aligned}

Lemma 2.6 Suppose g < p < k and s = 1, t = λ′_{p} = λ′_{p+1}. The map f′ annihilates G′_{[a],ξ,η}.

Concerning ẽ, x_{γ}, y_{γ}, we continue to use the notation of (2.4). We fix such a map ẽ and need to show that

∑_{α,β∈[0,2]}X_{p}^{(2−α)}X_{p+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽαβ} = 0.

There exist elements x′, y′ ∈ η, x′ ≠ y′, z ∈ ξ̄, which we choose and fix. For γ ∈ Γ̇(ẽ00), we write

\begin{aligned}U_{1,γ} & := & −\frac{1}{(t−1)!} · ⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (x_{γ}, ξ, x′) · (y_{γ}, y′) ∘ (ξ ∪ η) \\ U_{2,γ} & := & −\frac{1}{(t−1)!} · ⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (y_{γ}, ξ, y′) · (x_{γ}, x′) ∘ (ξ ∪ η) \\ V_{1,γ} & := & \frac{1}{t!} · ⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, ξ, x′) ∘ (ξ ∪ η) \\ V_{2,γ} & := & \frac{1}{t!} · ⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (y_{γ}, ξ, y′) ∘ (ξ ∪ η) \end{aligned}

and let

B := ∑_{γ∈Γ̇(ẽ00)}U_{1,γ} · (1 − ∑_{w∈ξ̄\z}(w, z))
= ∑_{γ∈Γ̇(ẽ00)}U_{2,γ} · (1 − ∑_{w∈ξ̄\z}(w, z))
C_{1} := ∑_{γ∈Γ̇(ẽ00)}U_{1,γ} · (z, x_{γ})

C_{2} := ∑_{γ∈Γ̇(ẽ00)}U_{2,γ} · (z, y_{γ})
D := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ} · ∑_{w∈ξ̄}(w, y_{γ})
= ∑_{γ∈Γ̇(ẽ00)}V_{2,γ} · ∑_{w∈ξ̄}(w, x_{γ})
F_{1} := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ}
F_{2} := ∑_{γ∈Γ̇(ẽ00)}V_{2,γ}

Calculations similar to those of (A.1), using the Garnir relation B = C_{1} + C_{2}, yield the following table, to be compared to the table of (2.4).

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ22} & = & 4D \\ G′_{[a],ξ,η}f′_{ẽ12} & = & −2B − 4D \\ G′_{[a],ξ,η}f′_{ẽ02} & = & B \\ G′_{[a],ξ,η}f′_{ẽ21} & = & 4D \\ G′_{[a],ξ,η}f′_{ẽ11} & = & −B − 2D + (F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ01} & = & −(F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ20} & = & 2D \\ G′_{[a],ξ,η}f′_{ẽ10} & = & (F_{1} + F_{2}) \\ G′_{[a],ξ,η}f′_{ẽ00} & = & 0 \end{aligned}

Note that s − u = X_{p} − X_{p+1} = 1.

Lemma 2.7 Suppose g < p = k and s, t ⩾ 2, η = a_{k+1,[1,t]}. The map f′ annihilates G′_{[a],ξ,η}.

We fix a map

[figure: [1, λ_{1}]\{k}; ẽ; [0, 2]; j; ẽ_{j}]

that maps g and k + 1 to e_{g} = e_{k+1} = 2, and that maps j ∈ [1, λ_{1}]\[g, k + 1] to e_{j} = 0. For α ∈ [0, 2], ẽα denotes the map which prolongs ẽ to [1, λ_{1}] via (ẽα)_{k} = α. We need to show that

∑_{α∈[0,2]}X_{k}^{(2−α)}G′_{[a],ξ,η}f′_{ẽα} = 0.

There exist elements x, y ∈ ξ, x ≠ y, z ∈ ξ̄, which we choose and fix. Let x′ := a_{g+1,1} and y′ := a_{g+1,2}, so that x′, y′ ∈ η. For v, w ∈ a_{k+1}, v ≠ w, we denote

Γ̇(ẽ, a, v, w) := {γ ∈ Γ̇(ẽ0) | a_{k+1,γ̄(k+1,1)} = v, a_{k+1,γ̄(k+1,2)} = w}

and let

A := \frac{1}{(s+t−2)!} ∑_{γ∈Γ̇(ẽ,a,x′,y′)}⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (x, x′) · (y, y′) ∘ (ξ ∪ η) · ∑_{w′∈η̄}(w′, z)
B := \frac{1}{(s+t−2)!} ∑_{γ∈Γ̇(ẽ,a,x′,y′)}⟨a^{γ}⟩ε_{γ}·

·(ξ̄\z, η\{x′, y′}) · (x, x′) · (y, y′) ∘ (ξ ∪ η) · (1 − ∑_{w∈ξ̄\z}(w, z))

D := \frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ,a,x′,y′)} ∑_{w∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄\w, η\\{x′, y′\}) · (x, x′) · (w, y′) ∘ (ξ ∪ η)
= \frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ,a,x′,y′)} ∑_{w∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄\w, η\\{x′, y′\}) · (y, y′) · (w, x′) ∘ (ξ ∪ η)
H := \frac{1}{(s+t−1)!} ∑_{w′∈η̄} ∑_{γ∈Γ̇(ẽ,a,x′,w′)}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x, x′) ∘ (ξ ∪ η)
= −\frac{1}{(s+t−1)!} ∑_{w′∈η̄} ∑_{γ∈Γ̇(ẽ,a,y′,w′)}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (y, y′) ∘ (ξ ∪ η).

Calculations similar to those of (A.1) yield the following table.

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ2} & = & \frac{1}{2}(2(s − u)A + (s − u)(s − u − 1)B + 2(s − u)(s − u + 1)D) \\ G′_{[a],ξ,η}f′_{ẽ1} & = & \frac{1}{2}( − 2A − 2(s − u)B − 2(s − u + 1)D + 2(s − u + 1)H) \\ G′_{[a],ξ,η}f′_{ẽ0} & = & \frac{1}{2}(B − 2H) \end{aligned}

Comparison with the table in (2.4) is possible since X_{k+1} = 0.

Lemma 2.8 Suppose g < p = k and s = λ′_{k}, t = 1, η = a_{k+1,1}. The map f′ annihilates G′_{[a],ξ,η}.

Concerning ẽ and Γ̇, we continue to use the notation of (2.7). We fix such a map ẽ. There exist elements x, y ∈ ξ, x ≠ y, which we choose and fix. We let

\begin{aligned}A & := & \frac{1}{(s−1)!} ∑_{w′∈η̄} ∑_{γ∈Γ̇(ẽ,a,η,w′)}⟨a^{γ}⟩ε_{γ} · (x, η) · (y, w′) ∘ (ξ ∪ η) \\ H & := & \frac{1}{s!} ∑_{w′∈η̄} ∑_{γ∈Γ̇(ẽ,a,η,w′)}⟨a^{γ}⟩ε_{γ} · (x, η) ∘ (ξ ∪ η) \end{aligned}

Calculations similar to those of (A.1) yield the following table, to be compared to the table of (2.7).

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ2} & = & \frac{1}{2}(2(s − u)A) \\ G′_{[a],ξ,η}f′_{ẽ1} & = & \frac{1}{2}( − 2A + 2(s − u + 1)H) \\ G′_{[a],ξ,η}f′_{ẽ0} & = & \frac{1}{2}( − 2H) \end{aligned}

Lemma 2.9 Suppose g < p = k and s = 1, t = λ′_{k} = λ′_{k+1}. The map f′ annihilates G′_{[a],ξ,η}.

Concerning ẽ and Γ̇, we continue to use the notation of (2.7). We fix such a map ẽ. There exist elements x′ ≠ y′ ∈ η, x′ ≠ y′, z ∈ ξ̄, which we choose and fix. We let

B := \frac{1}{(t−1)!} ∑_{γ∈Γ̇(ẽ,a,x′,y′)}⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) · (1 − ∑_{w∈ξ̄\z}(w, z))
D := \frac{1}{t!} ∑_{γ∈Γ̇(ẽ,a,x′,y′)} ∑_{w∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄\w, η\\{x′, y′\}) · (ξ, x′) · (w, y′) ∘ (ξ ∪ η)
= \frac{1}{t!} ∑_{γ∈Γ̇(ẽ,a,x′,y′)} ∑_{w∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄\w, η\\{x′, y′\}) · (ξ, y′) · (w, x′) ∘ (ξ ∪ η).

Calculations similar to those of (A.1) yield the following table.

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ2} & = & \frac{1}{2}(4D) \\ G′_{[a],ξ,η}f′_{ẽ1} & = & \frac{1}{2}( − 2B − 4D) \\ G′_{[a],ξ,η}f′_{ẽ0} & = & \frac{1}{2}(B) \end{aligned}

In comparison with the table of (2.7), we note that s − u = 1.

Lemma 2.10 (cf. (A.2)) Suppose g = p < k and s, t ⩾ 2. There exist elements x′, y′ ∈ η, x′ ≠ y′, z ∈ ξ̄, which we choose and fix. For γ ∈ Γ̇(ẽ0), we let x_{γ} := a_{j,γ̄(j,1)}, where j ∈ [g + 2, k + 1] is minimal with 1 × j ∈ Ξ_{γ}, and y_{γ} := a_{j,γ̄(j,2)}, where j ∈ [g + 2, k + 1] is minimal with 2 × j ∈ Ξ_{γ}. I.e. we pick the entries x_{γ}, y_{γ} that ‘cross the column’ g + 1 under the operation of γ. The set of maps

[figure: [1, λ_{1}]\{g + 1}; ẽ; [0, 2]; j; ẽ_{j}]

that send g and k + 1 to ẽ_{g} = ẽ_{k+1} = 2, and that map j ∈ [1, λ_{1}]\[g, k + 1] to e_{j} = 0, is denoted by Ẽ. For ẽ ∈ Ẽ and β ∈ [0, 2], we denote by ẽβ the prolongation of ẽ to [g, k + 1] by (ẽβ)_{g+1} = β. For γ ∈ Γ̇(ẽ0), we write

U_{γ} := ⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\{x′, y′}) · (x_{γ}, x′) · (y_{γ}, y′)

and let

\begin{aligned}A_{[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · ∑_{w′∈η̄}(w′, z) \\ B_{[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (1 − ∑_{w∈ξ̄\z}(w, z)) \\ C_{1,[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (z, x_{γ}) \\ C_{2,[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (z, y_{γ}). \end{aligned}

We obtain

\begin{matrix}G′_{[a],ξ,η}f′ = (X_{g} + 2)∑_{ẽ∈Ẽ}( ∏_{j∈[g+2,k]}X_{j}^{(2−ẽ_{j})})(2A_{[a],ξ,η,ẽ} + (X_{g} + 1)B_{[a],ξ,η,ẽ} \\ −X_{g+1}(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ})). \end{matrix}

Calculations carried out in (A.2) yield the following table.

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ2} & = & 2(s − u + 2)A_{[a],ξ,η,ẽ} + (s − u + 2)(s − u + 1)B_{[a],ξ,η,ẽ} \\ G′_{[a],ξ,η}f′_{ẽ1} & = & 2A_{[a],ξ,η,ẽ} + 2(s − u + 1)B_{[a],ξ,η,ẽ} − (s − u + 1)(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}) \\ G′_{[a],ξ,η}f′_{ẽ0} & = & B_{[a],ξ,η,ẽ} − (C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}) \end{aligned}

We note that s − u = X_{g} − X_{g+1} and evaluate the linear combination

∑_{β∈[0,2]}X_{g+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽβ}

= 1  ·(2(X_{g} − X_{g+1} + 2)A_{[a],ξ,η,ẽ}
+(X_{g} − X_{g+1} + 2)(X_{g} − X_{g+1} + 1)B_{[a],ξ,η,ẽ})
+ X_{g+1}  ·(2A_{[a],ξ,η,ẽ} + 2(X_{g} − X_{g+1} + 1)B_{[a],ξ,η,ẽ}
−(X_{g} − X_{g+1} + 1)(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}))
+ X_{g+1}(X_{g+1} + 1) ·(B_{[a],ξ,η,ẽ} − (C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}))

= 2(X_{g} + 2)A_{[a],ξ,η,ẽ} + (X_{g} + 2)(X_{g} + 1)B_{[a],ξ,η,ẽ} − X_{g+1}(X_{g} + 2)(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}). Lemma 2.11 Suppose g = p < k and s = λ′_{g}, t = 1. Concerning ẽ, x_{γ}, y_{γ}, we let the notation be as in (2.10). For γ ∈ Γ̇(ẽ0), we write

\begin{aligned}V_{1,γ} & := & ⟨a^{γ}⟩ε_{γ} · (y_{γ}, η) \\ V_{2,γ} & := & ⟨a^{γ}⟩ε_{γ} · (x_{γ}, η) \end{aligned}

and let

A_{[a],ξ,η,ẽ} := ∑_{γ∈Γ̇(ẽ0)}V_{1,γ} · ∑_{w′∈η̄}(x_{γ}, w′)
= ∑_{γ∈Γ̇(ẽ0)}V_{2,γ} · ∑_{w′∈η̄}(y_{γ}, w′)
B_{[a],ξ,η,ẽ} := 0
C_{1,[a],ξ,η,ẽ} := − ∑_{γ∈Γ̇(ẽ0)}V_{1,γ}
C_{2,[a],ξ,η,ẽ} := − ∑_{γ∈Γ̇(ẽ0)}V_{2,γ}.

We obtain

\begin{matrix}G′_{[a],ξ,η}f′ = (X_{g} + 2)∑_{ẽ∈Ẽ}( ∏_{j∈[g+2,k]}X_{j}^{(2−ẽ_{j})})(2A_{[a],ξ,η,ẽ} + (X_{g} + 1)B_{[a],ξ,η,ẽ} \\ −X_{g+1}(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ})). \end{matrix}

Calculations similar to those of (A.2) yield the following table, to be compared to the table of (2.10).

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ2} & = & 2(s − u + 2)A_{[a],ξ,η,ẽ} \\ G′_{[a],ξ,η}f′_{ẽ1} & = & 2A_{[a],ξ,η,ẽ} − (s − u + 1)(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}) \\ G′_{[a],ξ,η}f′_{ẽ0} & = & −(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}) \end{aligned}

Lemma 2.12 Suppose g = p < k and s = 1, t = λ′_{g} = λ′_{g+1}. Concerning ẽ, x_{γ}, y_{γ}, we let the notation be as in (2.10). There exist elements x′, y′ ∈ η, x′ ≠ y′, z ∈ ξ̄, which we choose and fix. For γ ∈ Γ̇(ẽ0), we write

U_{γ} := ⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\{x′, y′}) · (x_{γ}, x′) · (y_{γ}, y′)

and let

\begin{aligned}A_{[a],ξ,η,ẽ} & := & 0 \\ B_{[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (1 − ∑_{w∈ξ̄\z}(w, z)) \\ C_{1,[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (x_{γ}, z) \\ C_{2,[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (y_{γ}, z). \end{aligned}

We obtain

\begin{matrix}G′_{[a],ξ,η}f′ = (X_{g} + 2)∑_{ẽ∈Ẽ}( ∏_{j∈[g+2,k]}X_{j}^{(2−ẽ_{j})})(2A_{[a],ξ,η,ẽ} + (X_{g} + 1)B_{[a],ξ,η,ẽ} \\ −X_{g+1}(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ})). \end{matrix}

Calculations similar to those of (A.2) yield the following table.

\begin{aligned}G′_{[a],ξ,η}f′_{ẽ2} & = & 6B_{[a],ξ,η,ẽ} \\ G′_{[a],ξ,η}f′_{ẽ1} & = & 4B_{[a],ξ,η,ẽ} − 2(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ})) \\ G′_{[a],ξ,η}f′_{ẽ0} & = & B_{[a],ξ,η,ẽ} − (C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}) \end{aligned}

In comparison to the table of (2.10), we note that X_{g} − X_{g+1} = 1.

Lemma 2.13 Suppose g = p = k and s ⩾ 1, t ⩾ 2, η = a_{g+1,[1,t]}. Let γ^{1} be the double path defined by γ_{g+1,1}^{1} := 1, γ_{g+1,2}^{1} := 2. Let x′ := a_{g+1,1} and y′ := a_{g+1,2}, so that x′, y′ ∈ η. There exists an element z ∈ ξ̄, which we choose and fix. We write

U := ⟨a^{γ^{1}}⟩ε_{γ^{1}} · (ξ̄\z, η\{x, y})

and denote

\begin{aligned}A & := & U · ∑_{w′∈η̄}(w′, z) \\ B & := & U · (1 − ∑_{w∈ξ̄\z}(w, z)) \end{aligned}

We obtain

G′_{[a],ξ,η}f′ = \frac{1}{2}((X_{g} + 1)(X_{g} + 2)B + 2(X_{g} + 2)A),

whence the map f′ annihilates G′_{[a],ξ,η} modulo X_{g} + 2 in case X_{g} + 2 is odd, modulo (X_{g} + 2)/2 in case X_{g} + 2 is even.

Calculations similar to those of (A.2) yield

G′_{[a],ξ,η}f′ = \frac{1}{2}((s − u + 1)(s − u + 2)B + 2(s − u + 2)A).

Note that s − u = X_{g}.

Lemma 2.14 Suppose g = p = k and s = λ′_{g}, t = 1, η = a_{g+1,1}. For i ∈ [2, λ′_{g+1}], we let γ^{i} be the double path defined by γ_{g+1,1}^{i} := 1, γ_{g+1,2}^{i} := i. Let

A := ∑_{i∈[2,λ′_{g+1}]}⟨a^{γ^{i}}⟩ε_{γ^{i}}

We obtain

G′_{[a],ξ,η}f′ = (X_{g} + 2)A,

whence the map f′ annihilates G′_{[a],ξ,η} modulo X_{g} + 2.

Calculations similar to those of (A.2) yield

G′_{[a],ξ,η}f′ = (s − u + 2)A.

Remark 2.15 Suppose g = p = k. The application M^{λ′,−} ^{\overline{}^{f′}} S^{μ} maps \{(ǎ_{λ})′\}^{−} to a linear combination with coefficients ±1 of standard μ-polytabloids.

See Subsection 1.1 for the definition of the λ-tableau [ǎ_{λ}].

We summarize.

Proposition 2.16 (provisional version of (2.37)) Suppose g < k. The ZS_{n}-linear map M^{λ′,−} ^{\overline{}^{f′}} S^{μ} factors over

[figure: M^{λ′,−}; f′; S^{μ}; {a′}^{−}; ⟨a⟩; ν_{S}^{λ}; ⟨b⟩; ⟨b⟩ + m′S^{μ},; S^{λ}; f; S^{μ}/m′]

where m′ is an integer that divides the element

\begin{matrix}G′_{[a],ξ,η}f′ = (X_{g} + 2) · ∑_{ẽ∈Ẽ}( ∏_{j∈[g+2,k]}X_{j}^{(2−ẽ_{j})})(2A_{[a],ξ,η,ẽ} + (X_{g} + 1)B_{[a],ξ,η,ẽ} \\ −X_{g+1}(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ})) \end{matrix}

of S^{μ} for any λ-tableau [a] and any pair of subsets ξ ⊆ a_{g}, η ⊆ a_{g+1} such that #ξ + #η = λ′_{p} +1. The set Ẽ is defined in (2.10). The elements A_{[a],ξ,η,ẽ}, B_{[a],ξ,η,ẽ}, C_{1,[a],ξ,η,ẽ}, C_{2,[a],ξ,η,ẽ} of S^{μ} are defined in (2.10, 2.11, 2.12), the definition depending on whether ξ resp. η contains ⩾ 2 elements.

This is the conclusion we draw from (2.10, 2.11, 2.12) concerning the modulus m′ and from (2.4, 2.5, 2.6, 2.7, 2.8, 2.9) concerning the vanishing of the remaining Garnir relations under f′. The assertion is provisional in that we have neither specified m′ nor commented on the possible divisibility of M^{λ′,−} ^{\overline{}^{f′}} S^{μ} yet.

Proposition 2.17 (to be included in (2.37), cf. (4.9), cf. [K 99, 4.4.3]) Suppose g = k. The ZS_{n}-linear map M^{λ′,−} ^{\overline{}^{f′}} S^{μ} factors over

[figure: M^{λ′,−}; f′; S^{μ}; {a′}^{−}; ⟨a⟩; ν_{S}^{λ}; ⟨b⟩; ⟨b⟩ + mS^{μ},; S^{λ}; f; S^{μ}/m]

where m := X_{g} + 2 in case X_{g} is odd and m := (X_{g} + 2)/2 in case X_{g} is even. The resulting morphism S^{λ} ^{\overline{}^{f}} S^{μ}/m is of order m in Hom_{ZS_{n}}(S^{λ}, S^{μ}/m).

The assertions follow by (2.13, 2.14, 2.15).

We recall that in this case the double path formalism yields

[figure: M^{λ′,−}; f′; S^{μ}; {a′}^{−}; ∑_{1⩽v<w⩽λ′_{g+1}}(−1)^{v+w}⟨a^{v,w}⟩,]

where we let

[figure: [0, λ′_{g+1}]; [0, μ′_{g+1}]; i; φ; #([1, i]\{v, w}); min(φ^{−1}({j})); ψ; j]

in order to define the μ-tableau [a^{v,w}] by

a_{j,i}^{v,w} := a_{j,i}  for (j ∈ [1, λ_{1}]\{g, g + 1} and i ∈ [1, μ′_{j}]) or (j = g and i ∈ [1, λ′_{g}])
a_{g,λ′_{g}+1}^{v,w} := a_{g+1,v}
a_{g,λ′_{g}+2}^{v,w} := a_{g+1,w}
a_{g+1,i}^{v,w} := a_{g+1,ψ(i)} for i ∈ [1, μ′_{g+1}].

2.3 Morphism, reduced version

To achieve that the image of our morphism from M^{λ′,−} to S^{μ} gets is not contained in some tS^{μ} ⊂ S^{μ}, t ⩾ 2, we have to divide f′ by a factor of redundancy.

We assume g < k throughout this section. For g ⩽ p ⩽ q ⩽ k + 1, we let X [p, q] denote the set of partial patterns, i.e. the set of intermediate sets

[1, 2] × ({g, k + 1} ∩ [p, q]) ⊆ Ξ ⊆ [1, 2] × [p, q].

For p > q we let X [p, q] := ∅. For p ⩽ r ⩽ s ⩽ q and Ξ ∈ X [p, q], we let Ξ_{[r,s]} := Ξ ∩ ([1, 2] × [r, s]) ∈ X [r, s]. The partial weight of Ξ is defined to be the map

[figure: [p, q]; [0, 2]; r; e_{Ξ,r} = #Ξ_{[r,r]}.]

Sometimes, we shall write this map as a tuple e_{Ξ} = (e_{Ξ,p}, …, e_{Ξ,q}). An ordered double path of weight e is a double path γ that in addition satisfies the condition

γ̄(p, 1) < γ̄(p, 2) for p ∈ [g + 1, k].

Given a pattern Ξ ∈ X [g, k+1], we let Γ̇(Ξ) be the set of double paths γ of partial pattern Ξ_{γ} = Ξ, and we let Γ⃗(Ξ) be the set of ordered double paths γ of partial pattern Ξ_{γ} = Ξ. We define ZS_{n}-linear maps

[figure: F^{λ}; S^{μ}; [a]; f′′_{Ξ}; ∑_{γ∈Γ̇(Ξ)}⟨a^{γ}⟩ε_{γ}; [a]; f⃗′′_{Ξ}; ∑_{γ∈Γ⃗(Ξ)}⟨a^{γ}⟩ε_{γ}.]

We allow ourselves to write e.g. ^{+}^{−}_{+}_{+} for the partial pattern {1 × u, 2 × u, 2 × (u + 1)} ∈ X [u, u+1] etc. as long as no confusion concerning u can arise. So ‘+’ stands for ‘coordinate contained’ and ‘−’ stands for ‘coordinate not contained’.

Suppose given a λ-tableau [a] and a double path γ. We let x_{γ} := a_{j,γ̄(j,1)}, j ∈ [g + 2, k + 1] being minimal with 1 × j ∈ Ξ_{γ}, and y_{γ} := a_{j,γ̄(j,2)}, j ∈ [g + 2, k + 1] being minimal with 2 × j ∈ Ξ_{γ}. We fix positions u ≠ v ∈ [1, λ′_{g+1}]. For a λ-tableau [a] and a partial pattern Ξ ∈ X [g + 2, k + 1], we write x′ := a_{g+1,u}, y′ := a_{g+1,v} and let

F^{λ}  \overline{}S^{μ}
[a] ^{\overline{}^{A_{Ξ}^{u,v}}}  ∑_{γ∈Γ̇(^{+}^{−}_{+}_{−}∪Ξ)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′)

[a] ^{\overline{}^{A_{Ξ}^{u,−}}}  ∑_{γ∈Γ̇(^{+}^{−}_{+}_{−}∪Ξ)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′)

[a] ^{\overline{}^{A_{Ξ}^{−,v}}}  ∑_{γ∈Γ̇(^{+}^{−}_{+}_{−}∪Ξ)}⟨a^{γ}⟩ε_{γ} · (y_{γ}, y′)

[a] ^{\overline{}^{A⃗_{Ξ}^{u,v}}}  ∑_{γ∈Γ⃗(^{+}^{−}_{+}_{−}∪Ξ)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′)

[figure: [a]; A⃗_{Ξ}^{u,−}; ∑_{γ∈Γ⃗(^{+}^{−}_{+}_{−}∪Ξ)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′); [a]; A⃗_{Ξ}^{−,v}; ∑_{γ∈Γ⃗(^{+}^{−}_{+}_{−}∪Ξ)}⟨a^{γ}⟩ε_{γ} · (y_{γ}, y′).]

For instance, in this definition, it is understood that^{+}^{−}_{+}_{−} = {1×g, 2×g} ⊆ [1, 2]×[g, g+1]. We note that A_{Ξ}^{u,v} = A_{Ξ}^{v,u}.

For i ∈ [0, 1], let L(i) := {j ∈ [g + 1, k − 1] | λ′_{j+1} = λ′_{j} − i} be the set of i-steps. Let

[g, k] = L(0) ∪ L(1) ∪ L̄

be a disjoint union. A partial pattern Ξ ∈ X [p, q], g ⩽ p ⩽ q ⩽ k + 1, is called bulky in case there is some u ∈ [p, q − 1] ∩ L(0) such that

Ξ_{[u,u+1]} ∈ {^{+}^{−}_{+}_{+}, ^{+}^{+}_{+}_{−}, ^{+}^{−}_{−}_{+}, ^{−}^{+}_{+}_{−}, ^{+}^{−}_{−}_{−}, ^{−}^{−}_{+}_{−}, ^{+}^{−}_{+}_{−}},

or in case there is some u ∈ [p, q − 1] ∩ L(1) such that

Ξ_{[u,u+1]} = ^{+}^{−}_{+}_{−}.

Let X_{nb}[p, q] ⊆ X [p, q] be the subset of nonbulky partial patterns in X [p, q].

Lemma 2.18 Let m′ be a number dividing the elements

(X_{g} + 2) · 2· ∑_{Ξ∈X [g+2,k+1]}( ∏_{j∈[g+2,k]}X_{j}^{(2−e_{Ξ,j})})A_{Ξ}^{u,v}
(X_{g} + 2) · (X_{g} + 1)· ∑_{Ξ∈X [g+2,k+1]}( ∏_{j∈[g+2,k]}X_{j}^{(2−e_{Ξ,j})})A_{Ξ}^{u,v}
(X_{g} + 2) · X_{g+1}· ∑_{Ξ∈X [g+2,k+1]}( ∏_{j∈[g+2,k]}X_{j}^{(2−e_{Ξ,j})})(A_{Ξ}^{u,−} + A_{Ξ}^{−,u})

of Hom_{ZS_{n}}(F^{λ}, S^{μ}) for any choice of u ≠ v ∈ [1, λ′_{g+1}] resp. of u ∈ [1, λ′_{g+1}]. Then there is a factorization

(M^{λ′,−} ^{\overline{}^{f′}} S^{μ} \overline{}S^{μ}/m′) = (M^{λ′,−} ^{\overline{}^{ν_{S}^{λ}}} S^{λ} \overline{}S^{μ}/m′).

Let [a] be a λ-tableau, let ξ ⊆ a_{g}, η ⊆ a_{g+1} such that #ξ + #η = λ′_{p} + 1. Let ẽ be a map from [1, λ_{1}]\{g + 1} to [0, 2] that maps g and k + 1 to e_{g} = e_{k+1} = 2, and that maps j ∈ [1, λ_{1}]\[g, k + 1] to e_{j} = 0.

Case #ξ ⩾ 2, #η ⩾ 2. Writing a_{g+1,u} := x′ and a_{g+1,v} := y′, we obtain, in the notation of (2.10), including choices invoved,

A_{[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}([a] · (ξ̄\z, η\{x, y}) · (∑_{w′∈η̄}(w′, z)))A_{Ξ}^{u,v}
B_{[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}([a] · (ξ̄\z, η\{x, y}) · (1 − ∑_{w∈ξ̄\z}(w, z)))A_{Ξ}^{u,v}
C_{1,[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}( − [a] · (ξ̄\z, η\{x, y}) · (x′, z))A_{Ξ}^{−,v}
= ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}( − [a] · (ξ̄\z, η\{x, y}) · (y′, z))A_{Ξ}^{−,u}
C_{2,[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}( − [a] · (ξ̄\z, η\{x, y}) · (y′, z))A_{Ξ}^{u,−},

where the multitransposition (ξ̄\z, η\{x, y}) is to be read as with respect to and independent of a choice of a bijection (cf. §1.2).

Case #ξ = λ′_{g}, #η = 1. Writing {a_{g+1,u}} := η, we obtain, in the notation of (2.11),

A_{[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}} ∑_{v∈[1,λ′_{g+1}], a_{g+2,v}∈η̄}([a])A_{Ξ}^{u,v}
B_{[a],ξ,η,ẽ} = 0
C_{1,[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}( − [a])A_{Ξ}^{−,u}
C_{2,[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}( − [a])A_{Ξ}^{u,−}.

Case #ξ = 1, #η = λ′_{g} = λ′_{g+1}. Writing a_{g+1,u} := x′ and a_{g+1,v} := y′, we obtain, in the notation of (2.12),

A_{[a],ξ,η,ẽ} = 0
B_{[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}([a] · (ξ̄\z, η\{x, y}) · (1 − ∑_{w∈ξ̄\z}(w, z)))A_{Ξ}^{u,v}
C_{1,[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}( − [a] · (ξ̄\z, η\{x, y}) · (x′, z))A_{Ξ}^{−,v}
= ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}( − [a] · (ξ̄\z, η\{x, y}) · (y′, z))A_{Ξ}^{−,u}
C_{2,[a],ξ,η,ẽ} = ∑_{Ξ∈X [g+2,k+1], e_{Ξ} = ẽ|_{[g+2,k+1]}}( − [a] · (ξ̄\z, η\{x, y}) · (y′, z))A_{Ξ}^{u,−},

where (ξ̄\z, η\{x, y}) is to be read as with respect to and independent of a choice of a bijection.

The lemma now follows from (2.16).

Lemma 2.19 Let Ξ ∈ X_{nb}[g, k + 1]. The image

[ǎ_{λ}]f⃗′′_{Ξ}

is a linear combination of standard μ-polytabloids with coefficients ±1. For a pair of different nonbulky patterns, the corresponding pair of sets of occurring standard μ-polytabloids is disjoint.

The summand ±⟨(ǎ_{λ})^{γ}⟩ is a standard polytabloid after having ordered its columns downwards increasingly provided Ξ_{γ} is not bulky. Different ordered double paths yield different fillings of the columns, hence the coefficients remain ±1 and, moreover, the asserted disjointness holds.

For g ⩽ p ⩽ q ⩽ k, a tuple of integers (ϑ_{Ξ})_{Ξ∈X [p,q]} is called a reduced tuple of coefficients if the following conditions hold, expressed using the involution introduced in (2.2).

(I_{p,q}) For Ξ ∈ X [p, q], the invariance ϑ_{(Ξ)ι_{q+1}} = ϑ_{Ξ} holds. Given, in addition, r ∈ [p, q] with e_{Ξ,r} = 2, the invariance ϑ_{(Ξ)ι_{r+1}} = ϑ_{Ξ} holds.

(II_{p,q}) In case Ξ ∈ X [p, q] is bulky, the coefficient ϑ_{Ξ} vanishes, in case not, it does not vanish.

We note that the tuple ( ∏_{j∈[p,q]}X_{j}^{(2−e_{Ξ,j})})_{Ξ∈X [g,p]} satisfies (I_{p,q}) but in general not (II_{p,q}). Let

it be remarked that e.g. (II_{p,p+1}) implies that ϑ_{+}_{−}_{−}_{+} ≠ ϑ_{+}_{+}_{−}_{−} if p ∈ [g +1, k −1] and λ′_{p} = λ′_{p+1}, whereas e_{+}_{−}_{−}_{+} ≠ e_{+}_{+}_{−}_{−}.

Lemma 2.20 Let p ∈ [g+1, k], let (ϑ_{Ξ})_{Ξ∈X [g,p]} (resp. (ϑ_{Ξ})_{Ξ∈X [g+2,p]}) be a tuple of integers satisfying (I_{g,p}) (resp. (I_{g+2,p})), let Φ ∈ X [p + 1, k + 1]. Then

∑_{Ξ∈X [g,p]}ϑ_{Ξ}f′′_{Ξ∪Φ} = ∑_{Ξ∈X [g,p]}( ∏_{j∈[g+1,p]} e_{Ξ,j}!)ϑ_{Ξ}f⃗′′_{Ξ∪Φ}
∑_{Ξ∈X [g+2,p]}ϑ_{Ξ}A_{Ξ∪Φ}^{u,v} = ∑_{Ξ∈X [g+2,p]}( ∏_{j∈[g+2,p]} e_{Ξ,j}!)ϑ_{Ξ}A⃗_{Ξ∪Φ}^{u,v} ,

respectively, and likewise for A^{u,−}, A^{−,v} instead of A^{u,v}.

This follows, separately for each partial weight e_{Ξ}, from (2.2, ii).

Lemma 2.21 Let p ∈ L(0), Ψ ∈ X [g, p − 1], Φ ∈ X [p + 2, k + 1]. We obtain

(i) f′′_{Ψ∪^{+}^{+}_{+}_{+}∪Φ} − 2f′′_{Ψ∪^{+}^{+}_{+}_{−}∪Φ} − 2f′′_{Ψ∪^{+}^{−}_{+}_{+}∪Φ} + 2f′′_{Ψ∪^{+}^{−}_{+}_{−}∪Φ} = 0
(ii)  f′′_{Ψ∪^{+}^{+}_{+}_{+}∪Φ} − f′′_{Ψ∪^{+}^{+}_{+}_{−}∪Φ} − f′′_{Ψ∪^{+}^{−}_{+}_{+}∪Φ} = 0
(iii)  f′′_{Ψ∪^{+}^{+}_{−}_{+}∪Φ} − f′′_{Ψ∪^{+}^{−}_{−}_{+}∪Φ} − f′′_{(Ψ)ι_{p}∪^{−}^{+}_{+}_{−}∪Φ} = 0
(iii′)  f′′_{Ψ∪^{−}^{+}_{+}_{+}∪Φ} − f′′_{(Ψ)ι_{p}∪^{+}^{−}_{−}_{+}∪Φ} − f′′_{Ψ∪^{−}^{+}_{+}_{−}∪Φ} = 0
(iv)  f′′_{Ψ∪^{+}^{+}_{−}_{−}∪Φ} − f′′_{Ψ∪^{+}^{−}_{−}_{−}∪Φ} = 0
(iv′)  f′′_{Ψ∪^{−}^{−}_{+}_{+}∪Φ} − f′′_{Ψ∪^{−}^{−}_{+}_{−}∪Φ} = 0
(v)  2f′′_{Ψ∪^{+}^{−}_{+}_{−}∪Φ} − f′′_{Ψ∪^{+}^{+}_{+}_{+}∪Φ} = 0.

Equation (i) still holds in case p ∈ L(1). Equations (i-v) continue to hold for p ∈ L(0) ∩ [g + 2, k − 1], f′′ replaced by A^{u,v}, A^{u,−} or A^{−,v} and Ψ ∈ X [g, p − 1] replaced by Ψ ∈ X [g + 2, p − 1].

Equation (i) ensues from the Garnir relation that arises, for a given double path γ ∈ Γ̇(Ξ ∪^{+}^{+}_{+}_{+} ∪ Φ), from the join of the elements in column p in the image of γ with column p + 1.

We deduce (ii) from the Garnir relation that arises, for a given double path γ ∈ Γ̇(Ξ ∪ +^{+}_{+}_{+} ∪Φ), from the join of a single element in column p in the image of γ with column p+1, using (2.2 ii) to obtain f′′_{(Ψ)ι_{p}∪^{+}^{+}_{+}_{−}∪Φ} = f′′_{(Ψ∪^{+}^{+}_{+}_{−}∪Φ)ι_{p+1}} = f′′_{Ψ∪^{+}^{+}_{+}_{−}∪Φ}.

Assertion (iii) follows using the Garnir relation that arises, for a given double path γ ∈ Γ̇(Ξ ∪^{+}^{+}_{−}_{+} ∪ Φ), from the join of the element in column p in the image of γ with column p + 1.

Similarly, (iv) can be seen via the Garnir relation that arises, for a given double path γ ∈ Γ̇(Ξ ∪^{+}^{+}_{−}_{−} ∪ Φ), to the join of the element in column p in the image of γ with column p + 1.

As a consequence of (i) and (ii), we obtain (v).

Lemma 2.22 Suppose g + 1 ∈ L(0), Φ ∈ X [g + 3, k + 1]. We obtain

(i) A_{^{+} +∪Φ}^{u,v} − 2A_{^{+} −∪Φ}^{u,v} − 2A_{^{−} +∪Φ}^{u,v} + 2A_{^{−} −∪Φ}^{u,v} = 0
(ii)  A_{^{+} +∪Φ}^{u,v} − A_{^{+} −∪Φ}^{u,v} − A_{^{−} +∪Φ}^{u,v} = 0
(iii)  A_{^{+} +∪Φ}^{u,−} − A_{^{−} +∪Φ}^{u,−} − A_{^{+} −∪Φ}^{−,u} = 0
(iii′)  A_{^{+} +∪Φ}^{−,v} − A_{^{−} +∪Φ}^{v,−} − A_{^{+} −∪Φ}^{−,v} = 0
(iv)  A_{^{+} −∪Φ}^{u,−} − A_{^{−} −∪Φ}^{u,−} = 0
(iv′)  A_{^{−} +∪Φ}^{−,v} − A_{^{−} −∪Φ}^{−,v} = 0
(v)  2A_{^{−} −∪Φ}^{u,v} − A_{^{+} +∪Φ}^{u,v} = 0.

Equation (i) still holds in case g + 1 ∈ L(1).

Lemma 2.23 Let p ∈ L(0), let (ϑ_{Ξ})_{Ξ∈X [g,p]} be a reduced tuple of coefficients, let Φ ∈ X [p + 2, k + 1]. There is a reduced tuple of coefficients (ϑ̂Ξ̂)_{Ξ̂∈X [g,p+1]} such that

∑_{Ξ∈X [g,p]} ∑_{ξ∈X [p+1,p+1]}ϑ_{Ξ} · #ξ! · X_{p+1}^{(2−#ξ)}f⃗′′_{Ξ∪ξ∪Φ} = ∑_{Ξ̂∈X [g,p+1]}ϑ̂Ξ̂f⃗′′_{Ξ̂∪Φ}

Fixing a partial pattern Ξ̌ ∈ X [g, p − 1], ϑ̂ can be obtained from ϑ letting

ϑ̂_{Ξ̌∪^{+}^{+}_{+}_{+}} =  X_{p}(X_{p} + 1)ϑ_{Ξ̌∪^{+} +}
ϑ̂_{Ξ̌∪^{−}^{+}_{+}_{+}} =  (X_{p} + 1)ϑ_{Ξ̌∪^{−} +}
ϑ̂_{Ξ̌∪^{+}^{+}_{−}_{+}} =  (X_{p} + 1)ϑ_{Ξ̌∪^{+} −}
ϑ̂_{Ξ̌∪^{−}^{−}_{+}_{+}} = (X_{p} − 1)(X_{p} + 1)ϑ_{Ξ̌∪^{−} +}
ϑ̂_{Ξ̌∪^{+}^{+}_{−}_{−}} = (X_{p} − 1)(X_{p} + 1)ϑ_{Ξ̌∪^{+} −}
ϑ̂_{Ξ̌∪^{−}^{+}_{−}_{+}} =  2ϑ_{Ξ̌∪^{−} −}
ϑ̂_{Ξ̌∪^{−}^{+}_{−}_{−}} = ϑ̂_{Ξ̌∪^{−}^{−}_{−}_{+}} =  (X_{p} − 1)ϑ_{Ξ̌∪^{−} −}
ϑ̂_{Ξ̌∪^{−}^{−}_{−}_{−}} =  (X_{p} − 1)X_{p}ϑ_{Ξ̌∪^{−} −}.

In case p ∈ L(0) ∩ [g + 2, k + 1], the assertion holds for A⃗^{u,v}, A⃗^{u,−} or A⃗^{−,v} instead of f⃗′′, the reduced tuples of coefficients being indexed by Ξ ∈ X [g + 2, p] on the left hand side and by Ξ̂ ∈ X [g + 2, p + 1] on the right hand side.

Up to a rescaling, we rewrite summands of the left hand side using (2.21), to which the roman numbers refer. To this end, we substitute X_{p+1} = X_{p}−1 and suppose Ξ̌ ∈ X [g, p−1] be given.

ϑ_{Ξ̌∪^{+} +}((X_{p} − 1)X_{p}f′′_{Ξ̌∪^{+}^{−}_{+}_{−}∪Φ}
+(X_{p} − 1)(f′′_{Ξ̌∪^{+}^{+}_{+}_{−}∪Φ} + f′′_{Ξ̌∪^{+}^{−}_{+}_{+}∪Φ}) + f′′_{Ξ̌∪^{+}^{+}_{+}_{+}∪Φ}) =^{(ii, v)} ϑ_{Ξ̌∪^{+} +} · \frac{1}{2}X_{p}(X_{p} + 1)f′′_{Ξ̌∪^{+}^{+}_{+}_{+}∪Φ}
ϑ_{Ξ̌∪^{+} −}((X_{p} − 1)X_{p}f′′_{Ξ̌∪^{+}^{−}_{−}_{−}∪Φ} + (X_{p} − 1)f′′_{Ξ̌∪^{+}^{+}_{−}_{−}∪Φ}) =^{(vi)} ϑ_{Ξ̌∪^{+} −}(X_{p} − 1)(X_{p} + 1)f′′_{Ξ̌∪^{+}^{+}_{−}_{−}∪Φ}
ϑ_{Ξ̌∪^{+} −}(\frac{1}{2}(X_{p} − 1)(f′′_{Ξ̌∪^{+}^{−}_{−}_{+}∪Φ} + f′′_{(Ξ̌)ι_{p}∪^{−}^{+}_{+}_{−}∪Φ})
+f′′_{Ξ̌∪^{+}^{+}_{−}_{+}∪Φ}) =^{(iii)} ϑ_{Ξ̌∪^{+} −} · \frac{1}{2}(X_{p} + 1)f′′_{Ξ̌∪^{+}^{+}_{−}_{+}∪Φ}

For the last equation we made use of (I_{g,p}) in order to have ϑ_{Ξ̌∪^{+} −} = ϑ_{(Ξ̌)ι_{p}∪^{−} +}.

The rewriting of the summands having ^{−} + in column p proceeds by symmetry. Summands of index Ξ = Ξ̌ ∪ ^{−} − remain unchanged. We pass to f⃗′′ by inserting a factor 2 where necessary. So we obtain an array of equations whose left hand sides add up to yield the left hand side of the equation we claimed. After substitution of ϑ by ϑ̂ according to the table above, we sum up the corresponding right hand sides to obtain the right hand side of the equation we claimed.

The arguments are valid verbatim for A⃗^{u,v}, A⃗^{u,−}, A⃗^{−,v} instead of f⃗′′, except that we replace Ξ̌ ∈ X [g, p − 1] by Ξ̌ ∈ X [g + 2, p − 1] and need to suppose that p ⩾ g + 2.

Lemma 2.24 Suppose g + 1 ∈ L(0), let Φ ∈ X [g + 3, k + 1]. We obtain

∑_{ξ∈X [g+2,g+2]}#ξ! · X_{g+2}^{(2−#ξ)} · A⃗_{ξ∪Φ}^{u,v} = X_{g+1}(X_{g+1} + 1)A⃗_{^{+} +∪Φ}^{u,v}
∑_{ξ∈X [g+2,g+2]}#ξ! · X_{g+2}^{(2−#ξ)} · (A⃗_{ξ∪Φ}^{u,−} + A⃗_{ξ∪Φ}^{−,u}) = (X_{g+1} + 1)(A⃗_{^{+} +∪Φ}^{u,−} + A⃗_{^{+} +∪Φ}^{−,u})
+ (X_{g+1} − 1)(X_{g+1} + 1)(A⃗_{^{+} −∪Φ}^{u,−} + A⃗_{^{−} +∪Φ}^{−,u})

This results from (2.22) by the argument in (2.23).

Lemma 2.25 Let p ∈ L(1), let (ϑ_{Ξ})_{Ξ∈X [g,p]} be a reduced tuple of coefficients, let Φ ∈ X [p + 2, k + 1]. There is a reduced tuple of coefficients (ϑ̂Ξ̂)_{Ξ̂∈X [g,p+1]} such that

∑_{Ξ∈X [g,p]} ∑_{ξ∈X [p+1,p+1]}ϑ_{Ξ} · #ξ! · X_{p+1}^{(2−#ξ)}f⃗′′_{Ξ∪ξ∪Φ} = ∑_{Ξ̂∈X [g,p+1]}ϑ̂Ξ̂f⃗′′_{Ξ̂∪Φ}

Fixing a partial pattern Ξ̌ ∈ X [g, p − 1], ϑ and ϑ̂ are related by

ϑ̂_{Ξ̌∪^{+}^{+}_{+}_{+}} =  −X_{p}(X_{p} − 3)ϑ_{Ξ̌∪^{+} +}
ϑ̂_{Ξ̌∪^{+}^{−}_{+}_{+}} = ϑ̂_{Ξ̌∪^{+}^{+}_{+}_{−}} =  X_{p}(X_{p} − 2)ϑ_{Ξ̌∪^{+} +}
ϑ̂_{Ξ̌∪^{+}^{+}_{−}_{+}} =  2ϑ_{Ξ̌∪^{+} −}
ϑ̂_{Ξ̌∪^{+}^{−}_{−}_{+}} = ϑ̂_{Ξ̌∪^{+}^{+}_{−}_{−}} =  (X_{p} − 2)ϑ_{Ξ̌∪^{+} −}
ϑ̂_{Ξ̌∪^{+}^{−}_{−}_{−}} = (X_{p} − 2)(X_{p} − 1)ϑ_{Ξ̌∪^{+} −}
ϑ̂_{Ξ̌∪^{−}^{+}_{+}_{+}} =  2ϑ_{Ξ̌∪^{−} +}

ϑ̂_{Ξ̌∪^{−}^{−}_{+}_{+}} = ϑ̂_{Ξ̌∪^{−}^{+}_{+}_{−}} =  (X_{p} − 2)ϑ_{Ξ̌∪^{−} +}
ϑ̂_{Ξ̌∪^{−}^{−}_{+}_{−}} = (X_{p} − 2)(X_{p} − 1)ϑ_{Ξ̌∪^{−} +}
ϑ̂_{Ξ̌∪^{−}^{+}_{−}_{+}} =  2ϑ_{Ξ̌∪^{−} −}
ϑ̂_{Ξ̌∪^{−}^{−}_{−}_{+}} = ϑ̂_{Ξ̌∪^{−}^{+}_{−}_{−}} =  (X_{p} − 2)ϑ_{Ξ̌∪^{−} −}
ϑ̂_{Ξ̌∪^{−}^{−}_{−}_{−}} = (X_{p} − 2)(X_{p} − 1)ϑ_{Ξ̌∪^{−} −}

In case p ∈ L(1) ∩ [g + 2, k + 1], the assertion holds for A⃗^{u,v}, A⃗^{u,−} or A⃗^{−,v} instead of f′′, the reduced tuples of coefficients being indexed by Ξ ∈ X [g + 2, p] on the left and by Ξ̂ ∈ X [g + 2, p + 1] on the right hand side.

We rewrite summands of the left hand side using (2.21 i). To this end, we substitute X_{p+1} = X_{p} − 2 and suppose Ξ̌ ∈ X [g, p − 1] be given.

ϑ_{Ξ̌∪^{+} +}((X_{p} − 2)(X_{p} − 1)f′′_{Ξ̌∪^{+}^{−}_{+}_{−}∪Φ}
+(X_{p} − 2)(f′′_{Ξ̌∪^{+}^{+}_{+}_{−}∪Φ} + f′′_{Ξ̌∪^{+}^{−}_{+}_{+}∪Φ}) + f′′_{Ξ̌∪^{+}^{+}_{+}_{+}∪Φ}) =^{(i)} ϑ_{Ξ̌∪^{+} +}((X_{p} − 2)X_{p}(f′′_{Ξ̌∪^{+}^{+}_{+}_{−}∪Φ} + f′′_{Ξ̌∪^{+}^{−}_{+}_{+}∪Φ})
−\frac{1}{2}(X_{p} − 3)X_{p}f′′_{Ξ̌∪^{+}^{+}_{+}_{+}∪Φ})

All the other summands remain unchanged.

Lemma 2.26 Suppose g + 1 ∈ L(1), let Φ ∈ X [g + 3, k + 1]. We obtain

∑_{ξ∈X [g+2,g+2]}#ξ! · X_{g+2}^{(2−#ξ)} · A⃗_{ξ∪Φ}^{u,v} = −X_{g+1}(X_{g+1} − 3)A⃗_{^{+} +∪Φ}^{u,v} + X_{g+1}(X_{g+1} − 2)(A⃗_{^{+} −∪Φ}^{u,v} + A⃗_{^{−} +∪Φ}^{u,v}).

This results from (2.22), cf. (2.25).

Henceforth, we denote by ϑ the tuple of tuples ϑ = ((ϑ_{Ξ})_{Ξ∈X [g,r]})_{r∈[g,k]} of integers such that the entry ϑ_{+_{+}} = 1 furnishes the tuple in case r = g, such that each entry (ϑ_{Ξ})_{Ξ∈X [g,r]} is a reduced tuple of coefficients, such that for r ∈ [g, k − 1] the tuples (ϑ_{Ξ})_{Ξ∈X [g,r]} and (ϑ_{Ξ})_{Ξ∈X [g,r+1]} are related by the formulas in (2.23) (resp. 2.25) in case r ∈ L(0) (resp. r ∈ L(1)), and such that in the remaining case r ∈ L̄ the equation

ϑ_{Ξ∪ξ} = #ξ! · X_{r+1}^{(2−#ξ)} · ϑ_{Ξ}

holds ‘unchanged’ for Ξ ∈ X [g, r], ξ ∈ X [r + 1, r + 1]. We reindex the tuple for r = k by ϑ_{Ξ} := ϑ_{Ξ_{[g,k]}} for Ξ ∈ X [g, k + 1]. Furthermore, for r, s ∈ [g + 1, k], we let

R_{[r,s]} := ∏_{i∈[0,1]} ∏_{j∈L(i)∩[r,s]}X_{j}^{(2−i)},

in which a product is to be read to be equal to 1 if no factor is present. For r ∈ Z, we write

ṙ := gcd(r, 2).

Lemma 2.27 Let g + 1 ⩽ r ⩽ q ⩽ k such that r − 1 ∈ L̄, let Ψ ∈ X [g, r − 1], Ξ ∈ X [r, q] such that Ψ ∪ Ξ is nonbulky. Then ϑ_{Ψ}R_{[r,q−1]} divides ϑ_{Ψ∪Ξ}. If, moreover, Ξ_{[r,r]} = ^{+} +, then 2ϑ_{Ξ}R_{[r,q−1]} divides ϑ_{Ψ∪Ξ}.

We perform an induction on q to prove the assertion that ϑ_{Ψ}R_{[r,q−1]} divides ϑ_{Ψ∪Ξ} and that the quotient ϑ_{Ψ∪Ξ}/(ϑ_{Ψ}R_{[r,q−1]}) is divisible by X_{q}^{(2−e_{Ξ,q})}. The factor 2 in the second assertion then follows from the start of the induction.

At q = r, we have R_{[r,r−1]} = 1, and the quotient ϑ_{Ψ∪Ξ}/(ϑ_{Ψ}R_{[r,q−1]}) equals #Ξ! · X_{r}^{(2−#Ξ)} since r − 1 ∈ L̄.

We assume our assertion to hold for q − 1 ⩾ r. Let Ξ̌ := Ξ_{[r,q−1]}.

Case q − 1 ∈ L̄. We have ϑ_{Ψ∪Ξ}/ϑ_{Ψ∪Ξ̌} = #Ξ_{[q,q]}! · X_{q}^{(2−e_{Ξ,q})} and R_{[r,q−1]} = R_{[r,q−2]}.

Case q − 1 ∈ L(0). We have X_{q} = X_{q−1} − 1 and R_{[r,q−1]} = X_{q−1}(X_{q−1} + 1) · R_{[r,q−2]}. By assumption, ϑ_{Ψ∪Ξ̌}/(R_{[r,q−2]}ϑ_{Ψ}) is integral and divisible by X_{q−1}^{(2−e_{Ξ,q−1})}. We need to see that

(ϑ_{Ψ∪Ξ}/ϑ_{Ψ∪Ξ̌}) · (ϑ_{Ψ∪Ξ̌}/(R_{[r,q−2]}ϑ_{Ψ}))

is divisible by X_{q−1}(X_{q−1} + 1) · X_{q}^{(2−e_{Ξ,q})}. The left hand side factor of that expression taken from the table in (2.23), the assertion follows separately for each nonbulky subset Ξ_{[q−1,q]}.

Case q − 1 ∈ L(1). We have X_{q} = X_{q−1} − 2 and R_{[r,q−1]} = X_{q−1} · R_{[r,q−2]}. By assumption, ϑ_{Ψ∪Ξ̌}/(R_{[r,q−2]}ϑ_{Ψ}) is integral and divisible by X_{q−1}^{(2−e_{Ξ,q−1})}. We need to see that

(ϑ_{Ψ∪Ξ}/ϑ_{Ψ∪Ξ̌}) · (ϑ_{Ψ∪Ξ̌}/(R_{[r,q−2]}ϑ_{Ψ}))

is divisible by X_{q−1} · X_{q}^{(2−e_{Ξ,q})}. The left hand side factor of that expression taken from the table in (2.25), the assertion follows separately for each nonbulky subset Ξ_{[q−1,q]}.

Lemma 2.28 Let g + 1 ⩽ r ⩽ q ⩽ k such that r − 1 ∈ L̄ and [r, q − 1] ⊆ L(1), let Ψ ∈ X_{nb}[g, r − 1]. As ideal of coefficients, we obtain

(ϑ_{Ψ∪Ξ} | Ξ ∈ X [r, q]) = (ϑ_{Ψ} · R_{[r,q−1]} · Ẋ_{q}) ⊆ Z.

We shall perform the following calculations.

(a) For Ξ ∈ X_{nb}[r, q] of partial weight e_{Ξ} = (1, 1, …, 1), we have ϑ_{Ψ∪Ξ} = ϑ_{Ψ}·R_{[r,q−1]}·X_{q}.

(b) For Ξ ∈ X_{nb}[r, q] of partial weight e_{Ξ} = (1, 1, …, 1, 2), we have ϑ_{Ψ∪Ξ} = ϑ_{Ψ}·R_{[r,q−1]}·2.

(c) Conversely, ϑ_{Ψ} · R_{[r,q−1]} · 2 or ϑ_{Ψ} · R_{[r,q−1]} · X_{q} divides ϑ_{Ψ∪Ξ} for any Ξ ∈ X_{nb}[r, q].

Ad (a). We perform an induction on q, starting with q = r, R_{[r,r−1]} = 1 and ϑ_{Ψ∪^{+} −} = ϑ_{Ψ∪^{−} +} = ϑ_{Ψ} · X_{q}. Assume the assertion known for Ξ̌ := Ξ_{[r,q−1]}. We calculate

ϑ_{Ψ∪Ξ̌∪^{+} −} =^{(2.25)} ϑ_{Ψ∪Ξ̌} · (X_{q−1} − 2)
= ϑ_{Ψ} · R_{[r,q−2]} · X_{q−1} · (X_{q−1} − 2)
= ϑ_{Ψ} · R_{[r,q−1]} · X_{q},

dito for ξ\Ξ̂ = ^{−} +

Ad (b). The argument for (a) can be applied, except that we obtain a factor 2 instead of a factor (X_{q−1} − 2) = X_{q} in the last step.

Ad (c). We compare to the proof of (2.27). We need to show that there exists a step in the induction in which the quotient ϑ_{Ψ∪Ξ}/(R_{[r,q−1]}ϑ_{Ψ}) is divisible by 2 or by X_{q}. In case e_{Ξ,r} = 2, a factor 2 enters at r already, so that we may assume e_{Ξ,r} ⩽ 1. In case e_{Ξ,q} ⩽ 1, a factor X_{q} enters at q, so that we may assume e_{Ξ,q} = 2. Since Ξ is not bulky, necessarily −^{+}_{+}_{+} or ^{+}^{+}_{−}_{+} occurs as a subpattern of Ξ. But then a factor 2 enters at the induction step that adjoins the right column of that subpattern (2.25).

Lemma 2.29 Let g + 1 ⩽ r ⩽ q ⩽ k such that r − 1 ∈ L̄, [r, q − 1] ⊆ L(0) ∪ L(1) and [r, q − 1] ∩ L(0) ≠ ∅, let Ψ ∈ X_{nb}[g, r − 1]. As ideal of coefficients, we obtain

(ϑ_{Ψ∪Ξ} | Ξ ∈ X [r, q]) = (ϑ_{Ψ} · R_{[r,q−1]}) ⊆ Z.

Let s := max(L(0) ∩ [r, q − 1]) + 1 ∈ [r + 1, q]. We shall show the following assertions.

(a) For t ∈ [s, q] and a partial pattern Ξ(q, t) ∈ X_{nb}[r, q] of partial weight

e_{Ξ(q,t)} = \{ \begin{matrix}2 & for j ∈ [s, t] \\ 1 & for j ∈ [r, q]\[s, t], \end{matrix}

which is a slight abuse of notation since Ξ(q, t) does not only depend on q and t, we have

ϑ_{Ψ∪Ξ(q,t)} = \{ \begin{matrix}±ϑ_{Ψ} · R_{[r,q−1]} · ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q−1} − 2) & for t ∈ [s, q − 1] \\ ±ϑ_{Ψ} · R_{[r,q−1]} · ( ∏_{j∈[s,q−1]}(X_{j} − 3)) & for t = q, \end{matrix}

the sign ± indicating that the equation holds up to sign.

(b) In case s ∈ [r + 1, q − 2], for Ξ ∈ X_{nb}[r, q] of partial weight

e_{Ξ,j} = \{ \begin{matrix}2 & for j ∈ \{s, q\} \\ 1 & for j ∈ [r, q]\\{s, q\}, \end{matrix}

we have

ϑ_{Ψ∪Ξ} = ϑ_{Ψ} · R_{[r,q−1]} · 2.

(c) Assertions (a) and (b) imply the lemma.

Ad (c). The inclusion ⊇ follows from (2.27). In case s = q, the lemma follows from ϑ_{Ψ∪Ξ(q,q)} = ϑ_{Ψ} · R_{[r,q−1]}. In case s = q − 1, the lemma follows from

\begin{aligned}ϑ_{Ψ∪Ξ(q,q−1)} & = & ±ϑ_{Ψ}R_{[r,q−1]}(X_{q−1} − 2) \\ ϑ_{Ψ∪Ξ(q,q)} & = & ±ϑ_{Ψ}R_{[r,q−1]}(X_{q−1} − 3). \end{aligned}

In case s ∈ [r + 1, q − 2], it suffices to show that

𝔞 := (2, ∏_{j∈[s,q−1]}(X_{j} − 3), ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q−1} − 2) | t ∈ [s, q − 1]) = (1).

We perform an induction on t from t = q − 1 downwards to t = s, contending that

∏_{j∈[s,t−1]}(X_{j} − 3) ∈ 𝔞. In fact,

( ∏_{j∈[s,t]}(X_{j} − 3)) − ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q−1} − 2) = ( ∏_{j∈[s,t−1]}(X_{j} − 3))(X_{t} − X_{q−1} − 1)

and X_{t} − X_{q−1} = 2(t − q + 1).

Ad (a). Let q̄ ∈ [r, q]. For t ⩾ q̄ + 1, let Ξ(q̄, t) ∈ X_{nb}[r, q̄] denote a partial pattern of the same partial weight as Ξ(q̄, q̄). We perform an induction on q̄ running from r to q to prove the assertion that

ϑ_{Ψ∪Ξ(q̄,t)} = \{ \begin{matrix}ϑ_{Ψ} · R_{[r,q̄−1]} · Xq̄ & for q̄ ∈ [r, s − 1] \\ ±ϑ_{Ψ} · R_{[r,q̄−1]} · ( ∏_{j∈[s,q̄−1]}(X_{j} − 3)) & for q̄ ∈ [s, t] \\ ±ϑ_{Ψ} · R_{[r,q̄−1]} · ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q̄−1} − 2) & for q̄ ∈ [t + 1, q]. \end{matrix}

To begin with, we recall our situation, viz. r < r + 1 ⩽ s ⩽ t and r ⩽ q̄.

For q̄ = r, we have R_{[r,r−1]} = 1 and e_{Ξ(q̄,t),r} = 1, whence we obtain

ϑ_{Ψ∪Ξ(q̄,t)} = ϑ_{Ψ} · R_{[r,q̄−1]} · Xq̄

since r − 1 ∈ L̄.

Assume the assertion to be true for r ⩽ q̄ − 1 ⩽ s − 2.

Case q̄ − 1 ∈ L(0). We calculate

ϑ_{Ψ∪Ξ(q̄,t)} =^{(2.23)} ϑ_{Ψ∪Ξ(q̄−1,t)} · (X_{q̄−1} − 1)(X_{q̄−1} + 1)
= ϑ_{Ψ} · R_{[r,q̄−2]} · X_{q̄−1} · (X_{q̄−1} − 1)(X_{q̄−1} + 1)
= ϑ_{Ψ} · R_{[r,q̄−1]} · Xq̄.

Case q̄ − 1 ∈ L(1). We calculate

ϑ_{Ψ∪Ξ(q̄,s)} =^{(2.25)} ϑ_{Ψ∪Ξ(q̄−1,t)} · (X_{q̄−1} − 2)
= ϑ_{Ψ} · R_{[r,q̄−2]} · X_{q̄−1} · (X_{q̄−1} − 2)
= ϑ_{Ψ} · R_{[r,q̄−1]} · Xq̄.

Assume the assertion to be true for q̄ − 1 = s − 1. Note that q̄ − 1 ∈ L(0), so

ϑ_{Ψ∪Ξ(q̄,t)} =^{(2.23)} ϑ_{Ψ∪Ξ(q̄−1,t)} · (X_{q̄−1} + 1)
= ϑ_{Ψ} · R_{[r,q̄−2]} · X_{q̄−1} · (X_{q̄−1} + 1)
= ϑ_{Ψ} · R_{[r,q̄−1]}.

Assume the assertion to be true for s ⩽ q̄ − 1 ⩽ t − 1. Note that q̄ − 1 ∈ L(1), so

ϑ_{Ψ∪Ξ(q̄,t)} =^{(2.25)} −ϑ_{Ψ∪Ξ(q̄−1,t)} · X_{q̄−1}(X_{q̄−1} − 3)
= ±ϑ_{Ψ} · R_{[r,q̄−2]} · ( ∏_{j∈[s,q̄−2]}(X_{j} − 3)) · X_{q̄−1}(X_{q̄−1} − 3)
= ±ϑ_{Ψ} · R_{[r,q̄−1]} · ( ∏_{j∈[s,q̄−1]}(X_{j} − 3)).

Assume the assertion to be true for q̄ − 1 = t. Note that q̄ − 1 ∈ L(1), so

ϑ_{Ψ∪Ξ(q̄,t)} =^{(2.25)} ϑ_{Ψ∪Ξ(q̄−1,t)} · X_{q̄−1}(X_{q̄−1} − 2)
= ±ϑ_{Ψ} · R_{[r,q̄−2]} · ( ∏_{j∈[s,q̄−2]}(X_{j} − 3)) · X_{q̄−1}(X_{q̄−1} − 2)
= ±ϑ_{Ψ} · R_{[r,q̄−1]} · ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q̄−1} − 2).

Assume the assertion to be true for t + 1 ⩽ q̄ − 1. Note that q̄ − 1 ∈ L(1), so

ϑ_{Ψ∪Ξ(q̄,t)} =^{(2.25)} ϑ_{Ψ∪Ξ(q̄−1,t)} · (X_{q̄−1} − 2)
= ±ϑ_{Ψ} · R_{[r,q̄−2]} · ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q̄−2} − 2) · (X_{q̄−1} − 2)
= ±ϑ_{Ψ} · R_{[r,q̄−1]} · ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q̄−1} − 2).

Ad (b). We modify the last step in the proof of the formula for ϑ_{Ψ∪Ξ(q,s)} in (a) according to (2.25), i.e. we replace the factor (X_{q−1} − 2) by the factor 2.

A subinterval [r, s] of [g + 1, k] is called connected if [r, s − 1] ⊆ L(0) ∪ L(1). A maximal connected subinterval with respect to the ordering given by inclusion is called a component. We write [g + 1, k] as a disjoint decomposition into components

[g + 1, k] = ⋃_{κ∈[1,K]}[p(κ), q(κ)],

where p(κ) ⩽ q(κ) for κ ∈ [1, K], p(1) = g + 1, q(κ) + 1 = p(κ + 1) for κ ∈ [1, K − 1] and q(K) = k. So {q(κ) | κ ∈ [1, K]} = L̄\{g}.

Lemma 2.30 The ideal of coefficients may be factored as

(ϑ_{Ξ} | Ξ ∈ X [g, k + 1]) = ∏_{κ∈[1,K]} 𝔱_{κ},

where for κ ∈ [1, K] we denote

𝔱_{κ} := (ϑ_{Ψ∪Ξ}/ϑ_{Ψ} | Ξ ∈ X [p(κ), q(κ)]) ⊆ Z

for some choice of a partial pattern Ψ ∈ X_{nb}[g, p(κ) − 1], a choice which does not affect these quotients.

This factorization follows by the independence of choice just mentioned.

Proposition 2.31 Let the factor of redundancy be

R := R_{[g+1,k]} · ( ∏_{{κ∈[1,K], \atop [p(κ),q(κ)−1]⊆L(1)}}Ẋ_{q(κ)}).

The ideal of coefficients is given by

(ϑ_{Ξ} | Ξ ∈ X [g, k + 1]) = (R) ⊆ Z.

In particular, the ZS_{n}-linear map

M^{λ′,−} ^{\overline{}^{f′}} S^{μ}

is divisible exactly by R, i.e. any matrix representing f′ linearly over Z has R as greatest common divisor of its entries.

The assertion on the ideal of coefficients ensues from (2.28, 2.29, 2.30). We conclude by (2.23, 2.25) that the linear combination

f′′ = ∑_{Ξ∈X [g,k+1]}( ∏_{j∈[g+1,k]}(e_{Ξ,j}! · X_{j}^{(2−e_{Ξ,j})})) · f⃗′′_{Ξ} : F^{λ} \overline{}S^{μ},

cf. (2.20), can be rewritten as

f′′ = ∑_{Ξ∈X [g,k+1]}ϑ_{Ξ}f⃗′′_{Ξ},

whose coefficients are divisible by R and vanish for Ξ bulky. Applying f′′ to the tableau [ǎ_{λ}], we see by (2.19) that f′′ is divisible exactly by R. The assertion remains true for f′, being the factorization of f′′ over the epimorphism F^{λ} ^{\overline{}^{ν_{M}^{λ}}} M^{λ′,−}.

The modulus remains to be taken under consideration.

Lemma 2.32 We may rewrite

2 · ∑_{Ξ∈X [g+2,k+1]}( ∏_{j∈[g+2,k]}(e_{Ξ,j}! · X_{j}^{(2−e_{Ξ,j})}))A⃗_{Ξ}^{u,v} = ∑_{Ξ∈X [g+2,k+1]}ϑ_{_{+}_{+}_{+}_{+}∪Ξ}A⃗_{Ξ}^{u,v}

X_{g+1} · ∑_{Ξ∈X [g+2,k+1]}( ∏_{j∈[g+2,k]}(e_{Ξ,j}! · X_{j}^{(2−e_{Ξ,j})}))(A⃗_{Ξ}^{u,−} + A⃗_{Ξ}^{−,u})

= ∑_{Ξ∈X [g+2,k+1]}(ϑ_{_{+}_{+}_{+}_{−}∪Ξ}A⃗_{Ξ}^{u,−} + ϑ_{_{+}_{−}_{+}_{+}∪Ξ}A⃗_{Ξ}^{−,u})

This ensues from (2.24, 2.23, 2.26, 2.25).

Lemma 2.33 Let g + 1 ⩽ q ⩽ k, let ξ = ^{+} − or ξ = ^{−} + in X [g + 1, g + 1]. Suppose [g + 1, q − 1] ⊆ L(1). As ideal of coefficients, we obtain

(ϑ_{Ξ} | Ξ ∈ X [g, q], Ξ_{[g+1,g+1]} = ξ) = (R_{[g+1,q−1]} · Ẋ_{q}).

Consequently, by comparison with (2.31),

(ϑ_{Ξ} | Ξ ∈ X [g, p + 1], Ξ_{[g+1,g+1]} = ξ) = (R).

Consider the partial weights indexing generators of the ideal of coefficients having no restriction imposed on column g + 1. According to assertions (a) and (b) of the proof of (2.28), they can be chosen to have column g + 1 equal to ξ.

Lemma 2.34 Let g + 1 ⩽ q ⩽ k, let ξ = ^{+} − or ξ = ^{−} + in X [g + 1, g + 1]. Suppose [g + 1, q − 1] ⊆ L(0) ∪ L(1) such that [g + 1, q − 1] ∩ L(0) ≠ ∅. As ideal of coefficients we obtain

(ϑ_{Ξ} | Ξ ∈ X [g, q], Ξ_{[g+1,g+1]} = ξ) = (R_{[g+1,q−1]}).

Whence, by comparison with (2.31), we have

(ϑ_{Ξ} | Ξ ∈ X [g, p + 1], Ξ_{[g+1,g+1]} = ξ) = (R).

Consider the partial weights indexing generators of the ideal of coefficients having no restriction imposed on column g + 1. According to assertions (a) and (b) of the proof of (2.29), they can be chosen to have column g + 1 equal to ξ.

Lemma 2.35 Let g + 1 ⩽ q ⩽ k. Suppose [g + 1, q − 1] ⊆ L(1). As ideal of coefficients, we obtain

(ϑ_{Ξ} | Ξ ∈ X [g, q], Ξ_{[g+1,g+1]} = ^{+} +) = (2R_{[g+1,q−1]}).

Therefore, by comparison with (2.31),

(ϑ_{Ξ} | Ξ ∈ X [g, k + 1], Ξ_{[g+1,g+1]} = ^{+} +) = (2R/Ẋ_{q(1)}).

Note that Ẋ_{q(1)} = Ẋ_{g+1}.

We claim the following assertions.

(a) For t ∈ [g + 1, q], we let Ξ(g, t) ∈ X_{nb}[g, q] denote a partial pattern of partial weight

e_{Ξ(q,t),j} = \{ \begin{matrix}2 & for j ∈ [g, t] \\ 1 & for j ∈ [t + 1, q] \end{matrix}

and obtain

ϑ_{Ξ(q,t)} = \{ \begin{matrix}±2R_{[g+1,q−1]} · ( ∏_{j∈[g+1,t−1]}(X_{j} − 3)) · (X_{q−1} − 2) & for t ∈ [g + 1, q − 1] \\ ±2R_{[g+1,q−1]} · ( ∏_{j∈[g+1,q−1]}(X_{j} − 3)) & for t = q. \end{matrix}

(b) In case q ⩾ g + 3, for Ξ ∈ X_{nb}[g, q] of partial weight e_{Ξ} = (2, 2, 1, …, 1, 2), we have ϑ_{Ξ} = 4R_{[g+1,q−1]}.

(c) Assertions (a) and (b) imply the lemma. (a, b, c) follow by the arguments of (2.29). Lemma 2.36 Let g+1 ⩽ q ⩽ k. Suppose [g+1, q−1] ⊆ L(0)∪L(1), [g+1, q−1]∩L(0) ≠ ∅. As ideal of coefficients, we obtain

(ϑ_{Ξ} | Ξ ∈ X [g, q], Ξ_{[g+1,g+1]} = ^{+} +) = (2R_{[g+1,q−1]}).

So, by comparison with (2.31),

(ϑ_{Ξ} | Ξ ∈ X [g, k + 1], Ξ_{[g+1,g+1]} = ^{+} +) = (2R).

Denote

\begin{aligned}p & := & min (([g + 1, q − 1] ∩ L(1)) ∪ \{q\}) & ∈ [g + 1, q] \\ s & := & max ([g + 1, q − 1] ∩ L(0)) + 1 & ∈ [g + 2, q] \end{aligned}

so that we are in the situation g < g + 1 ⩽ p ⩽ s ⩽ q, more precisely, p = s or p ⩽ s − 2. We shall show the following assertions.

(a) For t ∈ [s, q] and a partial pattern Ξ(q, t) ∈ X_{nb}[g, q] of partial weight

e_{Ξ(q,t),j} = \{ \begin{matrix}2 & for j ∈ [g, p] \\ 1 & for j ∈ [p + 1, s − 1] \\ 2 & for j ∈ [s, t] \\ 1 & for j ∈ [t + 1, q], \end{matrix}

we obtain

ϑ_{Ξ(q,t)} = \{ \begin{matrix}±2R_{[g+1,q−1]} · ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q−1} − 2) & for t ∈ [s, q − 1] \\ ±2R_{[g+1,q−1]} · ( ∏_{j∈[s,q−1]}(X_{j} − 3)) & for t = q. \end{matrix}

(b) In case s ∈ [g + 2, q − 2], for Ξ ∈ X_{nb}[g, q] of partial weight

e_{Ξ,j} = \{ \begin{matrix}2 & for j ∈ [g, p] \\ 1 & for j ∈ [p + 1, s − 1] \\ 2 & for j = s \\ 1 & for j ∈ [s + 1, q − 1] \\ 2 & for j = q, \end{matrix}

we obtain

ϑ_{Ξ} = 4R_{[g+1,q−1]}.

(c) Assertions (a) and (b) imply the lemma. Ad (c). As in (2.29). Ad (a). We perform an induction on q̄ ∈ [g + 1, q] to prove

ϑ_{Ξ(q̄,t)} = \{ \begin{matrix}2R_{[g+1,q̄−1]} & for q̄ ∈ [g + 1, p] \\ 2R_{[g+1,q̄−1]} · Xq̄ & for q̄ ∈ [p + 1, s − 1] \\ ±2R_{[g+1,q̄−1]} · ( ∏_{j∈[s,q̄−1]}(X_{j} − 3)) & for q̄ ∈ [s, t] \\ ±2R_{[g+1,q̄−1]} · ( ∏_{j∈[s,t−1]}(X_{j} − 3)) · (X_{q̄−1} − 2) & for q̄ ∈ [t + 1, q], \end{matrix}

where e_{Ξ(q̄,t)} = e_{Ξ(q̄,q̄)} for t ⩾ q̄ + 1. We recall that g < g + 1 ⩽ p ⩽ s ⩽ t. We begin the induction by remarking that for q̄ = g+1 we have R_{[g+1,g]} = 1, e_{Ξ(q̄,t),g+1} = 2, and thus ϑ_{Ξ(q̄,t)} = 2R_{[g+1,q̄−1]}, as required. Case I, p ⩽ s − 2. Assume the assertion known for q̄ − 1 ∈ [g + 1, p − 1]. Note that q̄ − 1 ∈ L(0), so

ϑ_{Ξ(q̄,t)} =^{(2.23)} ϑ_{Ξ(q̄−1,t)} · X_{q̄−1}(X_{q̄−1} + 1)
= 2R_{[g+1,q̄−2]} · X_{q̄−1}(X_{q̄−1} + 1)
= 2R_{[g+1,q̄−1]}.

Assume the assertion known for q̄ − 1 = p. Note that q̄ − 1 ∈ L(1) since p < s, and that therefore [g + 1, q − 1] ∩ L(1) ≠ ∅, so

ϑ_{Ξ(q̄,t)} =^{(2.25)} ϑ_{Ξ(q̄−1,t)} · X_{q̄−1}(X_{q̄−1} − 2)
= 2R_{[g+1,q̄−2]} · X_{q̄−1}(X_{q̄−1} − 2)
= 2R_{[g+1,q̄−1]}Xq̄.

Assume the assertion known for q̄ − 1 ∈ [p + 1, s − 2]. In case q̄ − 1 ∈ L(0), we obtain

ϑ_{Ξ(q̄,t)} =^{(2.23)} ϑ_{Ξ(q̄−1,t)} · (X_{q̄−1} − 1)(X_{q̄−1} + 1)
= 2R_{[g+1,q̄−2]}X_{q̄−1} · (X_{q̄−1} − 1)(X_{q̄−1} + 1)
= 2R_{[g+1,q̄−1]}Xq̄.

In case q̄ − 1 ∈ L(1), we obtain

ϑ_{Ξ(q̄,t)} =^{(2.25)} ϑ_{Ξ(q̄−1,t)} · (X_{q̄−1} − 2)
= 2R_{[g+1,q̄−2]}X_{q̄−1} · (X_{q̄−1} − 2)
= 2R_{[g+1,q̄−1]}Xq̄.

Assume the assertion known for q̄ − 1 = s − 1. Note that q̄ − 1 ∈ L(0), so

ϑ_{Ξ(q̄,t)} =^{(2.23)} ϑ_{Ξ(q̄−1,t)} · (X_{q̄−1} + 1)
= 2R_{[g+1,q̄−2]}X_{q̄−1} · (X_{q̄−1} + 1)
= 2R_{[g+1,q̄−1]}.

Assume the assertion known for q̄ − 1 ∈ [s, t − 1]. Note that q̄ − 1 ∈ L(1), thus

ϑ_{Ξ(q̄,t)} =^{(2.25)} ϑ_{Ξ(q̄−1,t)} · (−X_{q̄−1}(X_{q̄−1} − 3))
= ±2R_{[g+1,q̄−2]}( ∏_{j∈[s,q̄−2]}(X_{j} − 3)) · X_{q̄−1}(X_{q̄−1} − 3)
= ±2R_{[g+1,q̄−1]}( ∏_{j∈[s,q̄−1]}(X_{j} − 3)).

Assume the assertion known for q̄ − 1 = t. Note that q̄ − 1 ∈ L(1), and consequently

ϑ_{Ξ(q̄,t)} =^{(2.25)} ϑ_{Ξ(q̄−1,t)} · X_{q̄−1}(X_{q̄−1} − 2)
= ±2R_{[g+1,q̄−2]}( ∏_{j∈[s,t−1]}(X_{j} − 3)) · X_{q̄−1}(X_{q̄−1} − 2)
= ±2R_{[g+1,q̄−1]}( ∏_{j∈[s,t−1]}(X_{j} − 3))(X_{q̄−1} − 2).

Assume the assertion known for q̄ − 1 ∈ [t + 1, q − 1]. Note that q̄ − 1 ∈ L(1), whence

ϑ_{Ξ(q̄,t)} =^{(2.25)} ϑ_{Ξ(q̄−1,t)} · (X_{q̄−1} − 2)
= ±2R_{[g+1,q̄−2]}( ∏_{j∈[s,t−1]}(X_{j} − 3))(X_{q̄−2} − 2) · (X_{q̄−1} − 2)
= ±2R_{[g+1,q̄−1]}( ∏_{j∈[s,t−1]}(X_{j} − 3))(X_{q̄−1} − 2).

Case II, p = s. I.e. L(0) ∩ [g + 1, q − 1] = [g + 1, p − 1], L(1) ∩ [g + 1, q − 1] = [s, q − 1].

The case q̄ ∈ [g + 1, p] follows from the argument for q̄ − 1 ∈ [g + 1, p − 1] in Case I. The case q̄ ∈ [s + 1, t] follows from the argument for q̄− 1 ∈ [s, t − 1] in Case I. The case q̄ ∈ [t + 1, q] follows from the arguments for q̄− 1 = t and for q̄− 1 ∈ [t + 1, q − 1] in Case I.

Ad (b). We modify the last step in the proof of (a) according to (2.25), i.e. we replace the factor (X_{q−1} − 2) by the factor 2.

Theorem 2.37 Suppose g < k. Let

m := \{ \begin{matrix}X_{g} + 2 & if & [p(1), q(1) − 1] ⊈ L(1) \\ (X_{g} + 2)/ gcd(2, X_{g}, X_{g+1}) & if & [p(1), q(1) − 1] ⊆ L(1). \end{matrix}

Note that if g + 1 ∈ L̄, the case ∅ = [p(1), q(1) − 1] ⊆ L(1) applies. The integer

X_{g} + 2 = (λ′_{g} − g) − (λ′_{k+1} − (k + 1)) + 2

allows an interpretation as box shift length. The quotient f′/R is integral and factors over

[figure: M^{λ′,−}; f′/R; S^{μ}; ν_{S}^{λ}; S^{λ}; f; S^{μ}/m.]

The morphism f has order m as an element of Hom_{ZS_{n}}(S^{λ}, S^{μ}/m). Recall that f′ maps

[figure: M^{λ′,−}; f′; S^{μ}; {a′}^{−}; ∑_{e∈E}( ∏_{i∈[g+1,k]}X_{i}^{(2−e_{i})}) ∑_{γ∈Γ̇(e)}⟨a^{γ}⟩ε_{γ}.]

The map f operates accordingly, division taking place before going modulo m. Alternatively, by (2.31) we may write

[figure: S^{λ}; f; S^{μ}/m; ⟨a⟩; ∑_{Ξ∈X [g,k+1]}(ϑ_{Ξ}/R) ∑_{γ∈Γ̇(Ξ)}⟨a^{γ}⟩ε_{γ},]

the coefficients ϑ_{Ξ}/R being integral.

Suppose g = k. The formula given for f′ in case g < k holds in case g = k as well, E consisting of a single element, R = 1 and m = (X_{g} + 2)/ gcd(2, X_{g}). We have the factorization diagram as above, the resulting morphism S^{λ} ^{\overline{}^{f}} S^{μ}/m being of order m. Cf. (2.17).

Suppose g < k. By (2.31), the ideal generated by Z-linear matrix coefficients of f′/R is (1) ⊆ Z. In case [p(1), q(1) − 1] ⊈ L(1), the integer m · R divides each of the required expressions in (2.18) by (2.32) and by the ideals of coefficients calculated in (2.34, 2.36). In case [p(1), q(1) − 1] ⊆ L(1), the integer m · R divides each of the required expressions in (2.18) by (2.32) and by the ideals of coefficients calculated in (2.33, 2.35).

Suppose g = k. Concerning the assertion, we refer to (2.17) (alternatively, to 4.9). This case behaves as if the first component lay in L(1) since X_{g+1} = 0.

Remark 2.38 In the situation of (2.37), Carter and Payne [CP 80] have shown that

Hom_{KS_{n}}(K ⊗_{Z} S^{λ}, K ⊗_{Z} S^{μ}) ≠ 0,

K being an infinite field of characteristic p such that p divides X_{g} + 2 in case p ⩾ 3, respectively, such that 4 divides X_{g} + 2 in case p = 2. This part of their result is recovered by (2.37).

Question 2.39 I do not know a description of Hom_{ZS_{n}}(S^{λ}, S^{μ}/n!) as an abelian group. See §2.4 for some examples.

2.4 Vertical examples

In the examples that follow, calculated by means of linear algebra (^{2}), we omit to denote the brackets that indicate polytabloids. Example 2.40 Let λ = (3, 3), μ = (2, 2, 1, 1). Then

Hom_{ZS_{6}}(S^{λ}, S^{μ}/6!)^{\overline{}^{∼}} Hom_{ZS_{6}}(S^{λ}, S^{μ}/4) ≃ Z/4,

and a generator of Hom_{ZS_{6}}(S^{λ}/4, S^{μ}/4) is given by

1 3 5 2 4 6  ^{\overline{}f} −2 · 1 5 2 6 3 4 − ( \begin{matrix}1 & 5 \\ 2 & 4 \\ 3 \\ 6 \end{matrix} + \begin{matrix}1 & 3 \\ 2 & 5 \\ 4 \\ 6 \end{matrix} ) − ( \begin{matrix}1 & 6 \\ 2 & 4 \\ 5 \\ 3 \end{matrix} + \begin{matrix}1 & 3 \\ 2 & 6 \\ 5 \\ 4 \end{matrix} ) − 2 · 1 3 2 4 5 6 .

Example 2.41 Let λ = (3, 2, 2), μ = (3, 1, 1, 1, 1). Then

Hom_{ZS_{7}}(S^{λ}, S^{μ}/7!)^{\overline{}^{∼}} Hom_{ZS_{7}}(S^{λ}, S^{μ}/6) ≃ Z/2 × Z/3,

generators of Hom_{ZS_{7}}(S^{λ}/6, S^{μ}/6) being given by

1 4 7 2 5 3 6 ^{\overline{}u^{λ′,t}u^{μ}} 3 · (  ∑_{[b] standard μ-tableau}⟨b⟩)

(cf. 3.19 below) and by

1 4 7 2 5 3 6  ^{\overline{}f} −2 · ( \begin{matrix}1 & 6 & 7 \\ 2 \\ 3 \\ 4 \\ 5 \end{matrix} − 1 5 7 2 3 4 6 + \begin{matrix}1 & 4 & 7 \\ 2 \\ 3 \\ 5 \\ 6 \end{matrix} ) .

Example 2.42 Let λ = (3, 3, 1, 1), μ = (2, 2, 1, 1, 1, 1). Then

Hom_{ZS_{8}}(S^{λ}, S^{μ}/8!)^{\overline{}^{∼}} Hom_{ZS_{8}}(S^{λ}, S^{μ}/6) ≃ Z/6,

and a generator of Hom_{ZS_{8}}(S^{λ}/6, S^{μ}/6) is given by

1 5 7 2 6 8 3 4  ^{\overline{}f} −2 · 1 7 2 8 3 4 5 6 − ( \begin{matrix}1 & 7 \\ 2 & 6 \\ 3 \\ 4 \\ 5 \\ 8 \end{matrix} + \begin{matrix}1 & 5 \\ 2 & 7 \\ 3 \\ 4 \\ 6 \\ 8 \end{matrix} ) − ( \begin{matrix}1 & 8 \\ 2 & 6 \\ 3 \\ 4 \\ 7 \\ 5 \end{matrix} + \begin{matrix}1 & 5 \\ 2 & 8 \\ 3 \\ 4 \\ 7 \\ 6 \end{matrix} ) − 2 · 1 5 2 6 3 4 7 8 .

2I thank the computer staff at the Fakultät für Mathematik in Bielefeld for kind cooperation.

Example 2.43 Let λ = (4, 4), μ = (3, 3, 1, 1). Then

Hom_{ZS_{8}}(S^{λ}, S^{μ}/8!)^{\overline{}^{∼}} Hom_{ZS_{8}}(S^{λ}, S^{μ}/5) ≃ Z/5,

and a generator of Hom_{ZS_{8}}(S^{λ}/5, S^{μ}/5) is given by

1 3 5 7 2 4 6 8 ^{\overline{}f}

−2 · 1 5 7 2 6 8 3 4 − ( \begin{matrix}1 & 6 & 7 \\ 2 & 4 & 8 \\ 5 \\ 3 \end{matrix} + \begin{matrix}1 & 3 & 7 \\ 2 & 6 & 8 \\ 5 \\ 4 \end{matrix} ) − ( \begin{matrix}1 & 5 & 7 \\ 2 & 4 & 8 \\ 3 \\ 6 \end{matrix} + \begin{matrix}1 & 3 & 7 \\ 2 & 5 & 8 \\ 4 \\ 6 \end{matrix} )

− ( \begin{matrix}1 & 5 & 8 \\ 2 & 4 & 6 \\ 7 \\ 3 \end{matrix} + 1 6 5 2 4 8 7 3 + 1 3 8 2 5 6 7 4 + \begin{matrix}1 & 3 & 5 \\ 2 & 6 & 8 \\ 7 \\ 4 \end{matrix} )

− ( \begin{matrix}1 & 5 & 7 \\ 2 & 4 & 6 \\ 3 \\ 8 \end{matrix} + 1 6 5 2 4 7 3 8 + 1 3 7 2 5 6 4 8 + \begin{matrix}1 & 3 & 5 \\ 2 & 6 & 7 \\ 4 \\ 8 \end{matrix} )

−2 · 1 3 7 2 4 8 5 6 − ( \begin{matrix}1 & 3 & 8 \\ 2 & 4 & 6 \\ 7 \\ 5 \end{matrix} + \begin{matrix}1 & 3 & 5 \\ 2 & 4 & 8 \\ 7 \\ 6 \end{matrix} ) − ( \begin{matrix}1 & 3 & 7 \\ 2 & 4 & 6 \\ 5 \\ 8 \end{matrix} + \begin{matrix}1 & 3 & 5 \\ 2 & 4 & 7 \\ 6 \\ 8 \end{matrix} ) − 2 · 1 3 5 2 4 6 7 8 .

Note that R = 6, m̄ = 1 and

\begin{aligned}ϑ_{+}_{+}_{+}_{+}_{+}_{+} & = & 12 \\ ϑ_{+}_{−}_{+}_{+}_{+}_{+} = ϑ_{+}_{+}_{+}_{+}_{−}_{+} & = & 6 \\ ϑ_{+}_{−}_{−}_{+}_{+}_{+} = ϑ_{+}_{+}_{+}_{+}_{−}_{−} & = & 6 \\ ϑ_{+}_{−}_{+}_{+}_{−}_{+} & = & 12 \\ ϑ_{+}_{−}_{−}_{+}_{−}_{+} = ϑ_{+}_{−}_{+}_{+}_{−}_{−} & = & 6 \\ ϑ_{+}_{−}_{−}_{+}_{−}_{−} & = & 12. \end{aligned}

Example 2.44 Let λ = (3, 3, 2), μ = (2, 2, 2, 1, 1). Then Hom_{ZS_{8}}(S^{λ}, S^{μ}/8!)^{\overline{}^{∼}} Hom_{ZS_{8}}(S^{λ}, S^{μ}/5) ≃ Z/5, R = 2, and a generator of Hom_{ZS_{8}}(S^{λ}/5, S^{μ}/5) is given by

\begin{aligned}1 4 7 2 5 8 3 6 & ^{\overline{}f} & − ( \begin{matrix}1 & 7 \\ 2 & 8 \\ 3 & 6 \\ 4 \\ 5 \end{matrix} + 1 7 2 5 3 8 4 6 + \begin{matrix}1 & 4 \\ 2 & 7 \\ 3 & 8 \\ 5 \\ 6 \end{matrix} ) − ( \begin{matrix}1 & 7 \\ 2 & 5 \\ 3 & 6 \\ 4 \\ 8 \end{matrix} + 1 4 2 7 3 6 5 8 + \begin{matrix}1 & 4 \\ 2 & 5 \\ 3 & 7 \\ 6 \\ 8 \end{matrix} ) \\ & & − ( \begin{matrix}1 & 8 \\ 2 & 5 \\ 3 & 6 \\ 7 \\ 4 \end{matrix} + 1 4 2 8 3 6 7 5 + \begin{matrix}1 & 4 \\ 2 & 5 \\ 3 & 8 \\ 7 \\ 6 \end{matrix} ) − 3 · 1 4 2 5 3 6 7 8 . \end{aligned}

Example 2.45 Let λ = (3, 3, 1, 1), μ = (2, 2, 2, 2). Then

Hom_{ZS_{8}}(S^{λ}, S^{μ}/8!)^{\overline{}^{∼}} Hom_{ZS_{8}}(S^{λ}, S^{μ}/3) ≃ Z/3,

and a generator of Hom_{ZS_{8}}(S^{λ}/3, S^{μ}/3) is given by

1 5 7 2 6 8 3 4  ^{\overline{}f} − 1 5 2 6 3 7 4 8 .

Example 2.46 Let λ = (4, 3), μ = (2, 2, 2, 1). Then

Hom_{ZS_{7}}(S^{λ}, S^{μ}/7!)^{\overline{}^{∼}} Hom_{ZS_{7}}(S^{λ}, S^{μ}/4) ≃ Z/4,

and a generator of Hom_{ZS_{7}}(S^{λ}/4, S^{μ}/4) is given by

1 3 5 7 2 4 6  ^{\overline{}g} 2 · 1 5 2 6 3 7 4 + 1 5 2 4 3 7 6 + 1 3 2 5 4 7 6 + 1 6 2 4 5 7 3 + 1 3 2 6 5 7 4 + 2 · 1 3 2 4 5 7 6 .

Example 2.47 Let λ = (4, 4), μ = (2, 2, 2, 2). Then

Hom_{ZS_{8}}(S^{λ}, S^{μ}/8!)^{\overline{}^{∼}} Hom_{ZS_{8}}(S^{λ}, S^{μ}/4) ≃ Z/4,

and a generator of Hom_{ZS_{8}}(S^{λ}/4, S^{μ}/4) is given by

1 3 5 7 2 4 6 8  ^{\overline{}g} 2 · 1 5 2 6 3 7 4 8 + 1 5 2 4 3 7 6 8 + 1 3 2 5 4 7 6 8 + 1 6 2 4 5 7 3 8 + 1 3 2 6 5 7 4 8 + 2 · 1 3 2 4 5 7 6 8 .

Question 2.48 I do not know a generic morphism that specializes to the morphism in (2.46) or to the morphism in (2.47).

3 Semistandard morphisms

The following considerations are based on methods of James [J 78, 13] and follow a hint of Ringel. We transpose our morphism from (2.37), cf. (3.16, 3.27), where transposition is given by dualization, followed by alternation and isomorphic substitution. Moreover, we establish a connection from the morphisms from M^{λ,−} to S^{μ} exhibited in §2.1 to those of the form Θ_{φ}^{−}ν_{S}^{μ} (3.25, 3.26), where the morphisms of type Θ_{φ} occur in [J 78, 13.13].

Informal remark. The original aim to give upper bounds on the Hom-groups is still out of reach (cf. 2.39). It could perhaps be approached by showing that the method that has been sufficient to construct our morphism (2.37) is also necessary in some sense. The connection we establish is to be seen as the first step of this approach, namely that the construction of our building blocks f′′_{e} in §2.1 has been necessary in some sense, except at the prime 2 in certain cases (3.12), and that there is a also reason to exclude bulky patterns by means of a reduced tuple of coefficients (3.28). For a particular case in which this aim could be achieved at least in the regular case, see (4.9).

3.1 Correspondoids

We shall recall the Carter-Lusztig-James construction of the semistandard basis of Hom_{ZS_{n}}(S^{λ}, M^{μ}) [J 78, 13.13], λ and μ partitions of n, in our slightly modified language convenient for the purpose of dualization (cf. 3.2). Moreover, we shall give a recipe for the transposition (3.16).

In this section we write maps on the right. Inverse images of subsets are written on the left. Given a ZS_{n}-lattice X, we denote its dual by X^{∗} = Hom_{Z}(X, Z), and, accordingly, we denote the dual of a morphism X ^{\overline{}^{g}} Y of ZS_{n}-lattices by X^{∗} ^{\overline{}^{g^{∗}}} Y^{∗}. Let λ and μ be arbitrary partitions of n. A correspondence from μ to λ is a bijection

[μ] \overline{}^{φ}_{∼} [λ].

The set of correspondences from μ to λ is denoted by Φ^{μ,λ}. We denote by

\{φ\} := φ^{−1}π_{R}^{μ} : [λ] \overline{}N

what is called a λ-tableau of type μ in [J 78, 13.1] and what in our context might also be called the correspondoid attached to the correspondence φ ∈ Φ^{μ,λ}. The set of correspondoids from μ to λ is denoted by {Φ^{μ,λ}}. James uses the bijection from the set of correspondoids {Φ^{μ,λ}} to the set of μ-tabloids that, for a fixed λ-tableau [a], is given by

\{φ\} = φ^{−1}π_{R}^{μ} ^{\overline{}^{∼}} [a]^{−1}φ^{−1}π_{R}^{μ} = \{φ[a]\}

as identification. Let

\begin{aligned}R_{λ} & := & \{ρ ∈ Φ^{λ,λ} | ρπ_{R}^{λ} = π_{R}^{λ}\} \\ C_{λ} & := & \{κ ∈ Φ^{λ,λ} | κπ_{C}^{λ} = π_{C}^{λ}\} \end{aligned}

be the row stabilizer resp. the column stabilizer of λ. Given a correspondence φ ∈ Φ^{μ,λ}, we write

r_{φ} := #(R_{λ} ∩ φ^{−1}R_{μ}φ)

Note that r_{φ} = r_{φ^{−1}}. Given a correspondence φ ∈ Φ^{μ,λ} and a λ-tableau [a], conjugation with the tableau [a] yields a bijection R_{λ} ∩ φ^{−1}R_{μ}φ ^{\overline{}^{∼}} R_{[a]} ∩ R_{φ[a]}.

Lemma 3.1 Let φ ∈ Φ^{μ,λ} be a correspondence. The map

[figure: M^{λ}; Θ_{φ}; M^{μ}; {a}; r_{φ}^{−1} ∑_{ρ∈R_{[a]}}{φ[a]}ρ = r_{φ}^{−1} ∑_{ρ∈R_{λ}}{φρ[a]}]

is well defined and ZS_{n}-linear. For ρ ∈ R_{μ} and ρ′ ∈ R_{λ}, we have Θ_{ρφρ′} = Θ_{φ}. In particular, Θ_{φ} and Θ_{φ^{−1}} both depend only on the correspondoid {φ}.

First, we note that the element ∑_{ρ∈R_{[a]}}{φ[a]}ρ ∈ M^{μ} is divisible by r_{φ} = #(R_{[a]} ∩ R_{φ[a]})

since {φ[a]}ρ = {φ[a]}ρ_{0}ρ for ρ_{0} ∈ R_{[a]} ∩ R_{φ[a]}. Furthermore, since R_{[a]σ} = (R_{[a]})^{σ} for σ ∈ S_{n}, the map

[figure: F^{λ}; M^{μ}; [a]; r_{φ}^{−1} ∑_{ρ∈R_{[a]}}{φ[a]}ρ]

is ZS_{n}-linear. Finally, since ∑_{ρ∈R_{[a]}}{φ[a]}ρ = ∑_{ρ∈R_{[a]σ}}{φ[a]σ}ρ for σ ∈ R_{[a]}, this map factors

over

[figure: F^{λ}; M^{λ}; [a]; {a}.]

Suppose given ρ ∈ R_{μ} and ρ′ ∈ R_{λ}. We have

{a}Θ_{ρφρ′} = r_{ρφρ′}^{−1} ∑_{ρ′′∈R_{λ}}{ρφρ′ρ′′[a]}
= r_{φ}^{−1} ∑_{ρ′′∈R_{λ}}{φρ′′[a]}
= {a}Θ_{φ}.

A S_{n}-invariant bilinear form on M^{λ} is given by

(\{a\}, \{b\}) = \{ \begin{matrix}1 & for \{a\} = \{b\} \\ 0 & for \{a\} ≠ \{b\}, \end{matrix}

{a}, {b} being λ-tabloids. The resulting dualization isomorphism is denoted by

[figure: M^{λ}; ∂^{λ}; ∼; M^{λ,∗}; {a}; ({a}, −).]

We also write such a λ-cotabloid as {a}^{∗} := ({a}, −).

Lemma 3.2 Given a correspondence φ ∈ Φ^{μ,λ}, the diagram

[figure: M^{λ,∗}; Θ_{φ}^{∗}; M^{μ,∗}; ≀; ∂^{λ}; ≀; ∂^{μ}; M^{λ}; Θ_{φ^{−1}}; M^{μ}]

commutes. The dual of Θ_{φ} sends the μ-cotabloid {b}^{∗} to the map

M^{λ} ^{\overline{}^{\{b\}^{∗}Θ_{φ}^{∗}}} Z
\{a\} \overline{}(\{b\}, \{a\}Θ_{φ})
= r_{φ}^{−1} ∑_{ρ∈R_{[a]}}({b}, {φ[a]}ρ)
= r_{φ}^{−1} ∑_{ρ∈R_{[a]}} ∑_{σ∈R_{[b]}} \{ \begin{matrix}1 & for [b]σ = φ[a]ρ \\ 0 & else \end{matrix} \}
= r_{φ^{−1}}^{−1} ∑_{σ∈R_{[b]}} ∑_{ρ∈R_{[a]}} \{ \begin{matrix}1 & for φ^{−1}[b]σ = [a]ρ \\ 0 & else \end{matrix} \}
= r_{φ^{−1}}^{−1} ∑_{σ∈R_{[b]}}({φ^{−1}[b]}σ, {a})
= ({b}Θ_{φ^{−1}}, {a}).

We denote the inclusion that is given by the definition of the Specht lattice by S^{λ} ^{\overline{}^{ι^{λ}}} M^{λ}. Remark 3.3 Let ν be still another partition of n, let φ ∈ Φ^{μ,λ}, let ψ ∈ Φ^{ν,μ} and let [a] be a λ-tableau. We obtain

\{a\}Θ_{φ}Θ_{ψ} = ∑_{ρ∈R_{μ}} \frac{r_{ψρφ}}{r_{φ}r_{ψ}} \{a\}Θ_{ψρφ}.

In fact,

{a}Θ_{φ}Θ_{ψ} = r_{φ}^{−1} · ( ∑_{ρ′∈R_{[a]}}{φ[a]ρ′})Θ_{ψ}
= r_{φ}^{−1}r_{ψ}^{−1} · ∑_{ρ′∈R_{[a]}} ∑_{ρ∈R_{φ[a]ρ′}}{ψφ[a]ρ′ρ}
= r_{φ}^{−1}r_{ψ}^{−1} · ∑_{ρ∈R_{φ[a]}} ∑_{ρ′∈R_{[a]}}{(ψφ[a]ρ[a]^{−1}φ^{−1}φ)[a]}ρ′
= r_{φ}^{−1}r_{ψ}^{−1} · ∑_{ρ∈R_{φ[a]}}r_{ψφ[a]ρ[a]^{−1}φ^{−1}φ}{a}Θ_{ψφ[a]ρ[a]^{−1}φ^{−1}φ}
= r_{φ}^{−1}r_{ψ}^{−1} · ∑_{ρ∈R_{μ}}r_{ψρφ}{a}Θ_{ψρφ}.

Remark 3.4 X being a ZS_{n}-lattice, we may identify

\begin{matrix}X^{∗,−} & ^{\overline{}^{∼}} & X^{−,∗} \\ f ⊗ a & \overline{} & (x ⊗ b \overline{}xf · ab) \\ (x \overline{}(x ⊗ 1)g) ⊗ 1 & \overline{} & g. \end{matrix}

Proposition 3.5 ([J 78, 6.7]) The map

[figure: S^{λ′}; δ^{λ}; ∼; S^{λ,∗,−}; ⟨a_{λ′}⟩σ; ({(a_{λ′})′}^{∗}|_{S^{λ}} ⊗ 1)σ,]

where σ ∈ S_{n}, is a well defined ZS_{n}-linear isomorphism. The diagram

[figure: M^{λ,∗,−}; ι^{λ,∗,−}; S^{λ,∗,−}; ≀; ∂^{λ,−}; ≀; δ^{λ}; M^{λ,−}; ν_{S}^{λ′}; S^{λ′}]

commutes. We abbreviate [a] = [(a_{λ′})′] in the course of the proof and claim that the kernels of the maps

[figure: M^{λ}; α; S^{λ′,−}; {a}σ; ⟨a′⟩σ ⊗ ε_{σ}; M^{λ}; β; S^{λ,∗}; {a}σ; ({a}σ)δ^{λ}ι^{λ,∗} = {aσ}^{∗}|_{S^{λ}}]

coincide. Suppose given an element ∑_{σ∈S_{n}}x_{σ}{a}σ ∈ M^{λ}, x_{σ} ∈ Z. To be contained in the kernel of α means

0 = ∑_{σ∈S_{n}}x_{σ}⟨a′⟩σ ⊗ ε_{σ}
= ∑_{σ∈S_{n}, ρ∈R_{[a]}}x_{σ}{a′}ρε_{ρ}σ ⊗ ε_{σ}
= ∑_{σ∈S_{n}, ρ∈R_{[a]}}x_{ρ^{−1}σ}{a′}σ ⊗ ε_{σ}.

Comparing coefficients, this is equivalent to saying that for all τ ∈ S_{n} we obtain

∑_{κ∈C_{[a]}, ρ∈R_{[a]}}x_{ρ^{−1}κτ}ε_{κτ} = 0.

To be contained in the kernel of β amounts to

0 = ( ∑_{σ∈S_{n}}x_{σ}{a}σ, ⟨a⟩τ)
= ( ∑_{σ∈S_{n}}x_{σ}{a}στ^{−1} ∑_{κ∈C_{[a]}}κε_{κ}, {a})
= ( ∑_{κ∈C_{[a]}, ρ∈S_{n}}x_{ρκ^{−1}τ}{a}ρε_{κ}, {a})

for all τ ∈ S_{n}, i.e. that for all τ ∈ S_{n}

∑_{κ∈C_{[a]}, ρ∈R_{[a]}}x_{ρκ^{−1}τ}ε_{κ} = 0.

Since α is surjective, we may conclude that the map S^{λ′,−} ^{\overline{}^{δ^{λ},−}} S^{λ,∗} is well defined and injective. To prove surjectivity, it remains to be shown that β = δ^{λ}ι^{λ,∗} is surjective, i.e. that ι^{λ} is a pure monomorphism. But the tuple of standard tableaux gives a basis of the Specht module over F_{p} for any prime p [J 78, 8.4].

Lemma 3.6 Denoting evaluation by X ^{\overline{}^{eva}} X^{∗∗}, X a ZS_{n}-lattice, we may dualize to obtain

[figure: (M^{λ}; eva; M^{λ,∗∗}; ∂^{λ,∗}; M^{λ,∗}); =; (M^{λ}; ∂^{λ}; M^{λ,∗}); (S^{λ}; eva; S^{λ,∗∗}; δ^{λ,−,∗}; S^{λ′,−,∗}); =; (S^{λ}; δ^{λ′}; S^{λ′,−,∗}),]

provided [a_{λ′}] = [(a_{λ})′]. Ignoring this condition, the second equality holds up to sign.

Suppose given λ-tableaux [a] and [b]. Let σ = [a][b]^{−1}. We have

(\{a\}, ⟨b⟩) = \{ \begin{matrix}0 & for σ ∉ R_{λ}C_{λ} \\ ε_{κ} & for σ = ρκ, ρ ∈ R_{λ}, κ ∈ C_{λ}. \end{matrix}

Therefore,

(#) ({a}, ⟨b⟩) = ε_{σ}({b′}, ⟨a′⟩).

We abbreviate [a] = [(a_{λ′})′] = [a_{λ}]. The map S^{λ′} ^{\overline{}^{δ^{λ}}} S^{λ,−,∗} sends ⟨a′⟩τ to (\{[a]τ\}, −) ⊗ ε_{τ}, where τ ∈ S_{n}, so that the composition sends

\begin{matrix}S^{λ} & ^{\overline{}^{eva}} & S^{λ,∗∗} & ^{\overline{}^{δ^{λ,−,∗}}} & S^{λ′,−,∗} \\ ⟨a⟩ & \overline{} & ⟨a⟩eva & \overline{} & (⟨a′⟩τ ⊗ 1 \overline{}ε_{τ}(\{[a]τ\}, ⟨a⟩) =^{(#)} =(\{a′\}, ⟨a′⟩τ)). \end{matrix}

A correspondoid {φ} ∈ {Φ^{μ,λ}} is called semistandard in case the following conditions hold.

(i) For i × j, i × j′ ∈ [λ] with j < j′, we have (i × j){φ} ⩽ (i × j′){φ}.

(ii) For i × j, i′ × j ∈ [λ] with i < i′, we have (i × j){φ} < (i′ × j){φ}.

Let {Φ^{μ,λ}}_{sst} ⊆ {Φ^{μ,λ}} denote the subset of semistandard correspondoids.

We attach to each correspondence φ ∈ Φ^{μ,λ} its column distribution

[figure: N; ×; N; |φ|; N; j; ×; k; |φ|_{k,j} := #({φ}^{−1}(j) ∩ (N × {k})).]

The map from the correspondences to the column distribution factors over the correspondoids, (φ \overline{}|φ|) = (φ \overline{}\{φ\} \overline{}|φ|). For φ ∈ Φ^{μ,λ}, ρ ∈ R_{μ} and κ ∈ C_{λ}, we have |ρφκ| = |φ| because {ρφκ} = {φκ} and because of the bijection, j × k ∈ N × N,

\{φ\}^{−1}(j) ∩ (N × \{k\}) ^{\overline{}^{∼}} \{φκ\}^{−1}(j) ∩ (N × \{k\})

which is given by restriction of κ. Note that the column distribution may be regarded as column equivalence class in the sense of [J 78, 13.8].

A total order on the set of column distributions can be defined as follows. We order N × N lexicographically via j × k < j′ × k′ if j < j′ or (j = j′ and k < k′). Suppose given correspondences φ, φ̃ ∈ Φ^{μ,λ}, |φ| ≠ |φ̃|. Let j × k be minimal with |φ|_{k,j} ≠ |φ̃|_{k,j}. We say that |φ| < |φ̃| in case |φ|_{k,j} < |φ̃|_{k,j}.

The following arguments, needed to obtain (3.12), are taken from [J 78], and we reproduce them here in a slightly adapted manner. I thank M. Härterich for an explanation.

Lemma 3.7 For {φ} ∈ {Φ^{μ,λ}}_{sst} and ρ ∈ R_{λ}, we have |φ| ⩾ |φρ|.

Suppose |φ| ≠ |φρ| and let j × k be minimal with |φ|_{k,j} ≠ |φρ|_{k,j}. By induction on j′ × k′, we see that

(∗) {φρ}^{−1}(j′) ∩ (N × {k′}) = {φ}^{−1}(j′) ∩ (N × {k′})

for 1 × 1 ⩽ j′ × k′ < j × k. In fact, assume that the two inverse images of the value j′ in column k′ differ in spite of having the same cardinality. Then there exists i × k′ ∈ [λ], i × k′′ := (i × k′)ρ^{−1}, such that j′ = (i × k′′){φ} ≠ (i × k′){φ} =: j′′. Assume that j′ > j′′. But since i × k′ ∈ {φ}^{−1}(j′′)\{φρ}^{−1}(j′′), this yields a contradiction to (∗) by induction hypothesis, so that we obtain j′ < j′′.

Let i × k^{′(s)} := (i × k′)ρ^{−s} for s ⩾ 0. We claim that (i × k^{′(s)}){φ} = j′ for all s ⩾ 1, thus deriving a contradiction that establishes (∗). We perform an induction on s. But (i × k^{′(s)}){φ} = j′ implies by semistandardness of {φ} that k^{′(s)} < k′. Thus (∗) applies by induction hypothesis and shows that (i × k^{′(s+1)}){φ} = (i × k^{′(s)}){φρ} = j.

Assume that |φ|_{k,j} < |φρ|_{k,j}. Then there exists i × k ∈ [λ], i × k′ := (i × k)ρ^{−1}, such that j = (i × k′){φ} ≠ (i × k){φ} =: j′. Assume that j > j′. But since i × k ∈ {φ}^{−1}(j′)\{φρ}^{−1}(j′), this yields a contradiction to (∗) and shows j < j′.

Let i × k^{(s)} := (i × k)ρ^{−s} for s ⩾ 0. We claim that (i × k^{(s)}){φ} = j for all s ⩾ 1, thus deriving a contradiction that shows |φ| ⩾ |φρ|. We perform an induction on s. But (i × k^{(s)}){φ} = j implies by semistandardness of {φ} that k^{(s)} < k. Thus (∗) shows that (i × k^{(s+1)}){φ} = (i × k^{(s)}){φρ} = j.

Lemma 3.8 Restricted to \{Φ^{μ,λ}\}_{sst}, the map \{φ\} \overline{}|φ| becomes injective.

To determine a semistandard correspondoid {φ}, it suffices to know the set of its values on column k for each k ⩾ 1, which can be written as {j ∈ N | |φ|_{k,j} ⩾ 1}.

Lemma 3.9 Assume λ to be a 2-regular partition of n, i.e. assume that i < i′ implies λ_{i} > λ_{i′} for i, i′ ∈ [1, λ′_{1}]. Suppose given κ, κ′ ∈ C_{λ}. Let

[figure: [λ]; ι; ∼; [λ]; i × k; i × (λ_{i} − k).]

If κικ′ ∈ R_{λ} then κ = 1, hence also κ′ = 1.

Assume that κ ≠ 1, i.e. assume given some position i × k ∈ [λ] such that i × k ≠ (i × k)κ. First, we choose i minimal such that there exists a k such that i × k ≠ (i × k)κ. Having chosen our i, we choose k maximal such that i × k ≠ (i × k)κ. We write i′ × k := (i × k)κ and remark that i < i′. We obtain

[figure: i; ×; k; κ; i′; ×; k; ι; i′; ×; (λ_{i′} − k); κ′; i′′′; ×; (λ_{i′} − k); i; ×; (k + λ_{i} − λ_{i′}); κ; i; ×; (k + λ_{i} − λ_{i′}); ι; i; ×; (λ_{i′} − k); κ′; i′′; ×; (λ_{i′} − k),] where i′′ ≠ i′′′, contradicting κικ′ ∈ R_{λ}.

Lemma 3.10 Suppose given σ ∉ R_{λ} · C_{λ}. There exists a transposition κ_{σ} ∈ C_{λ} such that σκ_{σ}σ^{−1} ∈ R_{λ}.

Assume that the conclusion does not hold, i.e. that σ^{−1} sends different entries of each given column to different rows. We claim that this implies that σ ∈ R_{λ} · C_{λ}. Given a permutation τ of [λ], we let k_{τ} be the maximal column position k ⩾ 0 such that τ becomes the identity when restricted to [λ] ∩ (N × [1, k]). We perform a downwards induction on k_{σ}, starting with k_{σ} = λ_{1}.

Different entries of column k_{σ} + 1 are mapped to different rows. Hence there is a column permutation κ ∈ C_{λ} acting trivially except perhaps in column k_{σ} + 1 such that (i × (k_{σ} + 1))κσ^{−1}π_{R}^{λ} = i for i ∈ [1, λ′_{k_{σ}+1}]. Thus there is a row permutation ρ ∈ R_{λ} such that k_{κσ^{−1}ρ} > k_{σ^{−1}}, i.e. k_{ρ^{−1}σκ^{−1}} > k_{σ}. By induction, ρ^{−1}σκ^{−1} ∈ R_{λ} · C_{λ} ensues.

Lemma 3.11 Suppose given a semistandard correspondoid {φ} ∈ {Φ^{μ,λ}}_{sst} and a λtableau [a]. Writing

⟨a⟩Θ_{φ} = ∑_{{ψ}∈{Φ^{μ,λ}}}z_{ψ}{ψ[a]} ∈ M^{μ},

we obtain z_{{φ}} = 1 and z_{{ψ}} = 0 for |ψ| > |φ|.

Note that

\begin{aligned}⟨a⟩Θ_{φ} & = & ∑_{κ∈C_{[a]}} ∑_{ρ∈(R_{λ}∩φ^{−1}R_{μ}φ)\R_{λ}}\{φρ[a]\}κε_{κ} \\ & = & ∑_{κ∈C_{λ}} ∑_{ρ∈(R_{λ}∩φ^{−1}R_{μ}φ)\R_{λ}}\{φρκ[a]\}ε_{κ}. \end{aligned}

Since {φ} is semistandard, we have |φ| ⩾^{(3.7)} |φρ| = |φρκ| for ρ ∈ R_{λ}, κ ∈ C_{λ}.

The equality {φ} = {φρκ} is equivalent to the existence of a ρ′ ∈ R_{μ} such that κ = ρ(φ^{−1}ρ′φ). This condition supposed to hold and given i × k ∈ [λ], we obtain

i × k ^{\overline{}^{ρ}} i × k′ ^{^{φ^{−1}ρ′φ} \overline{}}i′ × k,

where (i×k′)\{φ\} = (i′ ×k)\{φ\}. Note that the resulting map i \overline{}i′ is a bijection for each k. We perform an induction on k to show that i ⩽ i′. Suppose i ⩽ i′ and hence i = i′ for k_{0} < k (supposition possibly empty). Therefore, (i × k_{0}){φ} = (i′ × k_{0}){φ} = (i × k′_{0}){φ}. Since {φ} is semistandard, this inhibits (i × k){φ} > (i × k′){φ}, for this would mean that ρ could not act on the set

{i × k_{1} | k_{1} ∈ [1, λ_{i}], (i × k_{1}){φ} = (i × k′){φ}} ⊆ {i} × [1, k − 1],

the inclusion given by semistandardness of {φ}. Thus (i × k){φ} ⩽ (i × k′){φ} = (i′ × k){φ}, hence, by semistandardness of {φ}, i ⩽ i′, forcing i = i′ for all i × k ∈ [λ]. We conclude that κ = 1 and that thus φ^{−1}ρ′φ inverts ρ so that ρ represents the trivial coset.

Let m ⩾ 0. λ is called 2-singular if there exists an i ⩾ 1 such that λ_{i} = λ_{i+1} > 0. The case of m ⩾ 1, m ∈ (2) and λ being 2-singular shall be called the singular case. The complementary case shall be called the regular case, comprising in particular the case m = 0.

Theorem 3.12 (Carter, Lusztig [CL 74, 3.5], James [J 78, 13.13]) In the regular case, the tuple

(Θ_{φ}|_{S^{λ}} ⊗_{Z} Z/m | {φ} ∈ {Φ^{μ,λ}}_{sst})

furnishes a Z/m-linear basis of Hom_{ZS_{n}}(S^{λ}/m, M^{μ}/m).

In the singular case, the tuple

(2 · Θ_{φ}|_{S^{λ}} ⊗_{Z} Z/m | {φ} ∈ {Φ^{μ,λ}}_{sst})

furnishes a Z/(m/2)-linear basis of 2 · Hom_{ZS_{n}}(S^{λ}/m, M^{μ}/m).

We claim that the respective tuple is linearly independent. Suppose given a λ-tableau ⟨a⟩. We write down a matrix whose positions are indexed by {Φ^{μ,λ}}_{sst} × {Φ^{μ,λ}}, and whose entry at position {φ} × {ψ} is given by the multiplicity of the μ-tabloid {ψ[a]} in ⟨a⟩Θ_{φ} ∈ M^{μ}. It suffices to show that the {Φ^{μ,λ}}_{sst} × {Φ^{μ,λ}}_{sst}-part of our matrix is unipotent. But this follows from (3.11) in view of (3.8).

We claim that the respective tuple generates. Suppose given a morphism S^{λ}/m ^{\overline{}^{Θ}} M^{μ}/m such that Θ ≠ 0 in the regular case resp. such that 2 · Θ ≠ 0 in the singular case. We fix a λ-tableau [a] and write

⟨a⟩Θ = ∑_{{φ}∈{Φ^{μ,λ}}}x_{{φ}}{φ[a]},

where all elements of S^{λ} and of M^{μ} which occur in this expression and in the remainder of the proof are to be read as representing their residue classes modulo m. Suppose given {ψ} ∈ {Φ^{μ,λ}} that allows for {ψτ} = {ψ} for some i × k, i′ × k ∈ [λ], i ≠ i′, τ := (i × k, i × k′). We wish to see that x_{{ψ}} = 0 in the regular case and that 2 · x_{{ψ}} = 0 in the singular case. In both cases, we may compare the coefficients of

∑_{{φ}∈{Φ^{μ,λ}}}x_{{φτ}}{φ[a]} = ∑_{{φ}∈{Φ^{μ,λ}}}x_{{φ}}{φ[a]}(a_{k,i}, a_{k,i′})
= ⟨a⟩(a_{k,i}, a_{k,i′})Θ
= − ∑_{{φ}∈{Φ^{μ,λ}}}x_{{φ}}{φ[a]}

to obtain

x_{{ψ}} = x_{{ψτ}} = −x_{{ψ}}.

Now suppose the partition λ to be 2-regular. We need to see that x_{{ψ}} = 0. Let ι be the permutation of [λ] defined by i × k \overline{}^{ι}_{∼} i × (λ_{i} − k), i × k ∈ [λ]. We abbreviate ι^{[a]} := [a]^{−1}ι[a] ∈ S_{n} and calculate

\begin{aligned}∑_{κ′∈C_{[a]}}⟨a⟩ι^{[a]}κ′ε_{κ′} & = & ∑_{κ,κ′∈C_{[a]}}\{a\}κι^{[a]}κ′ε_{κκ′} \\ & = & ∑_{κ,κ′∈C_{λ}}\{κικ′[a]\}ε_{κκ′}, \end{aligned}

where κ ∈ C_{λ} has a sign ε_{κ} as a permutation of [λ], i.e. as an element of S_{[λ]}. Writing the resulting expression in the form ∑_{σ∈R_{λ}\S_{[λ]}}y_{σ}{σ[a]}, y_{σ} ∈ Z, where we choose our coset representatives σ to lie in C_{λ} whenever (uniquely) possible, we obtain

y_{σ} = ∑_{κ,κ′∈C_{λ}, κικ′∈R_{λ}σ}ε_{κκ′}.

In case σ ∈ C_{λ}, the condition on the indexing elements yields κ = 1, κ′ = σ and thus y_{σ} = ε_{σ} (3.9). In case σ ∉ R_{λ} · C_{λ}, we obtain, using (3.10), and choosing a fixed system of representatives of C_{λ}/⟨κ_{σ}⟩,

y_{σ} = ∑_{κ,κ′∈C_{λ}, κικ′∈R_{λ}σ}ε_{κκ′}
= ∑_{κ∈C_{λ}, κ′∈C_{λ}/⟨κ_{σ}⟩, κικ′∈R_{λ}σ}ε_{κκ′} + ∑_{κ∈C_{λ}, κ′∈C_{λ}/⟨κ_{σ}⟩, κικ′κ_{σ}∈R_{λ}σ}ε_{κκ′κ_{σ}}
= 0.

Hence

∑_{κ′∈C_{[a]}}⟨a⟩ι^{[a]}κ′ε_{κ′} = ⟨a⟩,

and therefore

⟨a⟩Θ =^{1.} ∑_{{φ}∈{Φ^{μ,λ}}}x_{{φ}}{φ[a]}
=^{2.} ∑_{κ∈C_{[a]}}⟨a⟩Θι^{[a]}κε_{κ}
= ∑_{{φ}∈{Φ^{μ,λ}}}x_{{φ}}( ∑_{κ∈C_{λ}}{φκ[a]}ε_{κ}).

For {φ} ∈ {Φ^{μ,λ}} and κ ∈ C_{λ}, the equality {ψ} = {φκ} is equivalent to φκψ^{−1} ∈ R_{μ}. Hence, as multiplicity of {ψ[a]} in ∑_{κ∈C_{λ}}{φκ[a]} we get, choosing a fixed system of

representatives for C_{λ}/⟨τ⟩,

\begin{aligned}∑_{κ∈C_{λ}, φκψ^{−1}∈R_{μ}}ε_{κ} & = & ∑_{κ∈C_{λ}/⟨τ⟩, φκψ^{−1}∈R_{μ}}ε_{κ} + ∑_{κ∈C_{λ}/⟨τ⟩, φκτψ^{−1}∈R_{μ}}ε_{κτ} \\ & = & 0. \end{aligned}

Therefore, a comparison of 1. and 2. yields x_{{ψ}} = 0.

Given κ ∈ C_{λ}, we obtain

∑_{{φ}∈{Φ^{μ,λ}}}x_{{φκ^{−1}}}{φ[a]} = ⟨a⟩Θ[a]^{−1}κ[a]
= ⟨a⟩Θε_{κ}
= ∑_{{φ}∈{Φ^{μ,λ}}}ε_{κ}x_{{φ}}{φ[a]},

whence, up to sign, x_{{φ}} depends only on |φ|. Let |φ| be the maximal column distribution such that x_{{φ}} ≠ 0 in the regular case, resp. such that 2 · x_{{φ}} ≠ 0 in the singular case. By our assertion on x_{{ψ}}, we may assume that (i × k){φ} < (i′ × k){φ} for i × k, i′ × k ∈ λ, i < i′. In order to show that {φ} is semistandard, we assume the contrary, i.e. the existence of positions i × k, i × (k + 1) ∈ [λ] such that (i × k){φ} > (i × (k + 1)){φ}. Let ξ := a_{k,[i,λ′_{i}]}, let η := a_{k+1,[1,i]}. For σ in S_{ξ∪η} but not in S_{ξ} × S_{η}, we have |φ| < |φ[a]σ[a]^{−1}| since the minimal value of {φ} changing columns under σ cannot move to the right. Summing up over cosets, a Garnir relation gives

\begin{aligned}0 & = & ∑_{σ∈S_{ξ}×S_{η}\S_{ξ∪η}}⟨a⟩Θσ \\ & = & ∑_{\{χ\}∈\{Φ^{μ,λ}\}}( ∑_{σ∈S_{ξ}×S_{η}\S_{ξ∪η}}x_{\{χ([a]σ^{−1}[a]^{−1})\}})\{χ[a]\}, \end{aligned}

whence ∑_{σ∈S_{ξ}×S_{η}\S_{ξ∪η}}x_{{φ([a]σ^{−1}[a]^{−1})}} = 0. For each σ in S_{ξ∪η} but not in S_{ξ} × S_{η}, maximality

of |φ| forces x_{{φ([a]σ[a]^{−1})}} = 0 in the regular case, resp. 2 · x_{{φ([a]σ[a]^{−1})}} = 0 in the singular case, contradicting x_{{φ}} ≠ 0 in the regular case, resp. 2 · x_{{φ}} ≠ 0 in the singular case. Hence {φ} is semistandard, and its column distribution |φ| shall be called the leading term of Θ. Given a correspondoid {χ} ∈ {Φ^{μ,λ}}, we write Θ̄_{χ} := Θ_{χ}|_{S^{λ}} ⊗_{Z} Z/m.

We consider the regular case. By (3.11), the difference Θ − x_{{φ}} · Θ̄_{φ} has either a strictly smaller leading term than Θ or it vanishes. Both alternatives allow to assume, by induction on the leading term of Θ, or, respectively, directly, that Θ − x_{{φ}} · Θ̄_{φ} is in the linear span of the tuple given above in the regular case.

We consider the singular case. By (3.11), the difference Θ − x_{{φ}} · Θ̄_{φ} has either a strictly smaller leading term than Θ or it vanishes under multiplication with 2. Both alternatives allow to assume, by induction on the leading term of Θ or, respectively, directly, that 2Θ − x_{{φ}} · 2Θ̄_{φ} is in the linear span of the tuple given above in the singular case.

Corollary 3.13 In the regular case, the tuple

(Θ_{φ^{−1}}^{−}ν_{S}^{λ′} ⊗_{Z} Z/m | {φ} ∈ {Φ^{μ,λ}}_{sst})

furnishes a Z/m-linear basis of Hom_{ZS_{n}}(M^{μ,−}/m, S^{λ′}/m).

In the singular case, the tuple

(2 · Θ_{φ^{−1}}^{−}ν_{S}^{λ′} ⊗_{Z} Z/m | {φ} ∈ {Φ^{μ,λ}}_{sst})

furnishes a Z/(m/2)-linear basis of 2 · Hom_{ZS_{n}}(M^{μ,−}/m, S^{λ′}/m).

We write (−)^{∗} := Hom_{Z}(−, Z/m), neglect to denote (−) ⊗_{Z} Z/m for maps and obtain Z/m-linear isomorphisms

Hom_{ZS_{n}}(S^{λ}/m, M^{μ}/m)  \overline{}^{∂^{μ}·(−)^{∗}}_{∼}  Hom_{ZS_{n}}(M^{μ}/m, S^{λ,∗}/m)
ι^{λ}Θ_{φ}  \overline{}∂^{μ}Θ_{φ}^{∗}ι^{λ,∗} =^{(3.2)} = Θ_{φ^{−1}}∂^{λ}ι^{λ,∗}
⋯  \overline{}^{(−)^{−}·(δ^{λ})^{−1}}_{∼}  Hom_{ZS_{n}}(M^{μ,−}/m, S^{λ′}/m)
⋯  \overline{}Θ_{φ^{−1}}^{−}∂^{λ,−}ι^{λ,∗,−}(δ^{λ})^{−1} =^{(3.5)} Θ_{φ^{−1}}^{−}ν_{S}^{λ′}

which send the bases of (3.12) to the tuples just described.

Corollary 3.14 In the regular case, the induced maps

Hom_{ZS_{n}}(M^{λ}/m, M^{μ}/m) ^{\overline{}^{ι^{λ}(−)}} Hom_{ZS_{n}}(S^{λ}/m, M^{μ}/m)
Hom_{ZS_{n}}(M^{μ,−}/m, M^{λ,−}/m) ^{\overline{}^{(−)ν_{S}^{λ′}}} Hom_{ZS_{n}}(M^{μ,−}/m, S^{λ′}/m)

are surjective. In the singular case, their cokernels are annihilated by multiplication by 2. Or, slightly more precise, when passing to 2 · Hom, the maps become surjective.

Question 3.15 I do not know a basis of Hom_{F_{2}S_{n}}(S^{λ}/2, M^{μ}/2) in case λ is 2-singular.

Proposition 3.16 Suppose given a ZS_{n}-linear map of the form

[figure: S^{λ}/m; f; S^{μ}/m; ⟨a⟩; ∑_{{φ}∈{Φ^{μ′,λ′}}}x_{{φ}}{a′}^{−}Θ_{φ}^{−}ν_{S}^{μ},]

where x_{{φ}} ∈ Z/m. Neglecting to denote (−)⊗_{Z} Z/m for maps and using the isomorphism δ from (3.5), we obtain the transposition isomorphism

Hom_{ZS_{n}}(S^{λ}/m, S^{μ}/m) \overline{}^{tra}_{∼} Hom_{ZS_{n}}(S^{μ′}/m, S^{λ′}/m)
f \overline{}f^{t} := δ^{μ}f^{∗,−}(δ^{λ})^{−1}.

This definition can be rephrased via the formula

⟨b′⟩f^{t} = ∑_{{φ}∈{Φ^{μ′,λ′}}}x_{{φ}}⟨b′⟩Θ_{φ^{−1}},

note that Θ_{φ^{−1}} is applicable to ⟨b′⟩ ∈ S^{μ′}/m ⊆ M^{μ′}/m. We have (f^{t})^{t} = f.

We dualize via (−)^{∗} := Hom_{Z}(−, Z/m) to obtain the following diagram.

[figure: M^{λ′,∗}/m; ∑ x_{{φ}}Θ_{φ}^{∗}; M^{μ′,∗}/m; ν_{S}^{λ,∗,−}; ν_{S}^{μ,∗,−}; S^{λ,∗,−}/m; f^{∗,−}; S^{μ,∗,−}/m; ∂^{λ′}; ∂^{μ′}; δ^{λ}; δ^{μ}; M^{λ′}/m; ∑ x_{{φ}}Θ_{φ^{−1}}; M^{μ′}/m; ι^{λ′}; ι^{μ′}; S^{λ′}/m; ft; S^{μ′}/m]

The commutativity on top follows from (3.2). On the left and on the right, commutativity follows by dualizing the commutativity in (3.5) in the sense of (3.6). The commutativity in the back is the dualized assumption on f and the commutativity at the bottom is the definition of f^{t}. The claimed formula is equivalent to the commutativity in the front face. Moreover, (3.6) allows to conclude that

(f^{t})^{t} = δ^{λ′} · f^{t,∗,−} · (δ^{μ′})^{−1}
= δ^{λ′} · (δ^{λ,∗,−})^{−1} · f^{∗∗} · δ^{μ,∗,−} · (δ^{μ′})^{−1}
= eva · f^{∗∗} · eva^{−1}
= f.

Question 3.17 (main technical obstacle) Given S^{λ}/m ^{\overline{}^{f}} S^{μ}/m as in (3.16), I do not know a formula for S^{μ′}/m ^{\overline{}^{ft}} S^{λ′}/m in terms of a factorization over M^{μ,−}/m ^{\overline{}^{ν_{S}^{μ′}}} S^{μ′}/m of a linear combination of maps of the form Θ_{ψ}^{−}ν_{S}^{λ′}, {ψ} ∈ {Φ^{λ,μ}}. I.e., I do not know a formula ‘in terms of polytabloids instead of tabloids’.

3.2 Remarks on transposition in characteristic 2

Given an integer a ⩾ 1, we denote by a_{2} := 2^{v_{2}(a)} its 2-part. From the more general assertion in (4.2) below, but nevertheless readable independently, we take that for b ⩾ 1 we have v_{2}((\begin{matrix}a \\ i \end{matrix})) > 0 for all i ∈ [1, b] if and only if b < a_{2}. We say that a partition λ of n is 2-convergent if λ′_{p+1} < (λ′_{p} + 1)_{2} for all p ∈ [1, λ_{1} − 1]. For instance, if n = 8, the list of 2-convergent partitions is (1^{8})′, (3, 1^{5})′, (3^{2}, 1^{2})′, (5, 1^{3})′, (3^{2}, 2)′, (7, 1)′, (8)′. Let [x] = [1 2 ⋯ n] denote the standard (n)-tableau.

Lemma 3.18 (cf. [J 78, 24.4], [K 99, 4.3.35]) Let λ be a partition of n. We have

dim Hom_{F_{2}S_{n}}(S^{λ}/2, S^{(n)}/2) = \{ \begin{matrix}1 & if λ is 2-convergent \\ 0 & else. \end{matrix}

In case λ is 2-convergent, the nonzero morphism is of the form

[figure: S^{λ}/2; S^{(n)}/2; ⟨a⟩; ⟨x⟩]

By transposition (3.16), we conclude that

dim Hom_{F_{2}S_{n}}(S^{(n)}/2, S^{λ}/2) = \{ \begin{matrix}1 & if λ′ is 2-convergent \\ 0 & else. \end{matrix}

By S_{n}-linearity, a morphism S^{λ}/2 \overline{}S^{(n)}/2, if existent, is necessarily of the form just given. It remains to be seen that this map is well defined if and only if λ is 2-convergent.

Welldefinedness may be rephrased as the existence of a factorization of the map

[figure: F^{λ}; S^{(n)}/2; [a]; ⟨x⟩]

over F^{λ} ^{\overline{}^{ν^{λ}}} S^{λ}. A one-step Garnir relation

G′′_{[a],ξ,η} = ∑_{σ∈S_{ξ}×S_{η}\S_{ξ∪η}}[a]σε_{σ},

where ξ ⊆ a_{p} and η ⊆ a_{p+1} are subsets such that #ξ +#η = λ′_{p} +1 (cf. §1.1), is mapped to

the element (\begin{matrix}λ′_{p}+1 \\ #η \end{matrix})·⟨x⟩ ∈ S^{(n)}/2. Thus the map is well defined if and only if (\begin{matrix}λ′_{p}+1 \\ i \end{matrix}) ≡_{2} 0 for all p ∈ [1, λ_{1} − 1] and for all i ∈ [1, λ′_{p+1}].

Lemma 3.19 Let λ be a partition of n. Let χ^{λ} be the characteristic function of the subset C_{[a_{λ}]}R_{[a_{λ}]} ⊆ S_{n}, i.e. χ_{σ}^{λ} = 1 for σ ∈ C_{[a_{λ}]}R_{[a_{λ}]}, and χ_{σ}^{λ} = 0 for σ ∈ S_{n}\C_{[a_{λ}]}R_{[a_{λ}]}. Let P^{λ} ⊆ S_{n} be the subset of permutations σ for which [a_{λ}]σ is a standard tableau. We suppose given a nonzero F_{2}S_{n}-linear map (unique if existent, cf. 3.18)

[figure: S^{(n)}/2; u^{λ}; S^{λ}/2; ⟨x⟩; ∑_{τ∈P^{λ}}u_{τ}^{λ}⟨a_{λ}⟩τ.]

For each σ ∈ P^{λ}, we obtain

1 ≡_{2} ∑_{τ∈P^{λ}} u_{τ}^{λ}χ_{τσ^{−1}}^{λ}.

Moreover, (χ_{τσ^{−1}}^{λ})_{τ×σ∈P^{λ}×P^{λ}} ∈ GL_{rk S^{λ}}(F_{2}).

Note that the transpose of u^{λ} maps each polytabloid to the nonzero element (3.18).

Disregarding alternation, we obtain on the one hand, σ ∈ S_{n},

⟨x⟩ ^{\overline{}^{⟨a′_{λ}⟩σδ^{λ}u^{λ,∗}}} ∑_{τ∈P^{λ}}u_{τ}^{λ}(\{a_{λ}\}σ, ⟨a_{λ}⟩τ)

and on the other hand

⟨x⟩^{\overline{}^{⟨a′_{λ}⟩σu^{λ,t}δ^{(n)}}} (\{x\}, ⟨x⟩) = 1.

It remains to be shown that ({a_{λ}}σ, ⟨a_{λ}⟩τ) ≡_{2} χ_{τσ^{−1}}^{λ} for τ, σ ∈ P^{λ}. But ({a_{λ}}σ, ⟨a_{λ}⟩τ) ≡_{2} 1 is equivalent to the existence of κ ∈ C_{[a_{λ}]} and ρ ∈ R_{[a_{λ}]} such that [a_{λ}]ρ = [a_{λ}]κτσ^{−1}. The invertibility of the matrix (χ_{τσ^{−1}}^{λ})_{τ×σ∈P^{λ}×P^{λ}} follows from (3.5). Cf. [K 99, 6.2.8].

For instance, in case λ = (3, 2), [a_{λ}] = [ǎ_{λ}], and thus P^{(3,2)} = {1, (45), (23), (23)(45), (2453)}, this invertible matrix over F_{2} turns out to be

(χ_{τσ^{−1}}^{(3,2)})_{τ×σ∈P^{(3,2)}×P^{(3,2)}} = [\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{matrix}] .

In case λ is a hook partition, i.e. in case λ_{i} ⩽ 1 for i ⩾ 2, the matrix (χ_{τσ^{−1}}^{λ})_{τ×σ∈P^{λ}×P^{λ}} is the identity matrix. In particular, we recover [K 99, 4.2.11] by composition of a morphism as in (3.18) and the transpose, as given by (3.19), of a morphism as in (3.18). Cf. (3.31).

3.3 Paths as correspondoids

In order to transpose the vertical two-box-shift morphism in (2.37) via (3.16) to obtain a horizontal twobox-shift morphism, we translate the language of double paths to the language of correspondoids (3.25, 3.26) in order to be able to use our transposition formula (3.16) to obtain the provisional result (3.27, cf. 3.29). We shall use a slightly modified and generalized variant of the setup given in §2.1.

In this section maps are written on the right, with some exceptions made. Suppose given a partition λ of n, an integer d ⩾ 1 and integers 1 ⩽ g ⩽ k ⩽ λ_{1} − 1 such that

μ′_{i} := \{ \begin{matrix}λ′_{i} + d & for i = g \\ λ′_{i} − d & for i = k + 1 \\ λ′_{i} & else \end{matrix}

defines a partition μ. A weight e is a map

[figure: [1, λ_{1}]; e; [0, d]; j; e_{j}]

that maps g and k + 1 to e_{g} = e_{k+1} = d, and that maps j to e_{j} = 0 in case j ∈ [1, λ_{1}]\[g, k + 1]. A pattern Ξ of weight e is a subset Ξ ⊆ [1, d] × [g, k + 1] that has

#(Ξ ∩ ([1, d] × {j})) = e_{j}

for j ∈ [g, k + 1]. A d-fold path γ of weight e is an injection from a pattern Ξ of weight e to [λ] ∪ [μ] of the form

[figure: Ξ; γ; [λ] ∪ [μ]; i × j; γ̄(j, i) × j]

such that

γ̄(g, i) = λ′_{g} + i for i ∈ [1, d].

Sometimes, we denote its pattern by Ξ_{γ} := Ξ. The set of d-fold paths of weight e is denoted by Γ(e).

An ordered d-fold path of weight e is a d-fold path γ for which the application

\{i ∈ [1, d] | i × j ∈ Ξ_{γ}\} ^{^{γ̄(j,−)} \overline{}}[1, λ′_{j}]

is increasing for each j ∈ [g + 1, k + 1]. The set of ordered d-fold paths of weight e is denoted by Γ⃗(e).

Suppose given a weight e. For a d-fold path γ of weight e, we let the successor permutation be defined by

Ξ_{γ}  \overline{}^{suc^{γ}}_{∼} Ξ_{γ}
i × j \overline{}\{ \begin{matrix}i × min\{j′ ∈ [j + 1, k + 1] | i × j′ ∈ Ξ_{γ}\} & for j ∈ [g, k] \\ i × g & for j = k + 1. \end{matrix}

Since γ is an injection, we may define the permutation

[λ′] ∪ [μ′] \overline{}^{γ̂}_{∼} [λ′] ∪ [μ′]

as the completion to a commutative diagram

[figure: Ξ_{γ}; γ; [λ] ∪ [μ]; τ; ∼; i×j; j×i; [λ′] ∪ [μ′]; [λ′] ∪ [μ′]\(Ξ_{γ})γτ; ^{suc^{γ}}; ≀; γ̂; ≀; Ξ_{γ}; γ; [λ] ∪ [μ]; ∼; τ; [λ′] ∪ [μ′]; [λ′] ∪ [μ′]\(Ξ_{γ})γτ]

in which the unlabeled arrows denote inclusions. Thus, roughly speaking, γ̂ is the identical prolongation of suc_{γ} operating via γ. Given a d-fold path γ of weight e, the permutation σ(γ) of [λ′] ∪ [μ′] is defined as being determined by the rule

[λ′] ∪ [μ′] \overline{}^{σ(γ)}_{∼} [λ′] ∪ [μ′]
j × i \overline{}\begin{cases}j & × & i & for j ≠ k + 1 \\ (k + 1) & × & γ̄(k + 1, i − μ′_{k+1}) & for j = k + 1 and i ∈ [μ′_{k+1} + 1, λ′_{k+1}] \end{cases}

and by the requirement that its restriction to {k + 1} × [1, μ′_{k+1}] be of constant value k + 1 in the first component and strictly increasing in the second component. Hence, the [μ′_{k+1} + 1, λ′_{k+1}]-part of the row k + 1 of [λ′] ∪ [μ′] is mapped under σ(γ) to the image of γτ, whereas its [1, μ′_{k+1}]-part is distributed, in a strictly increasing manner, over the complement of this image in that row. We define an element γ̌ ∈ Φ^{μ′,λ′} as the completion to a commutative diagram

[figure: [μ′]; [λ′] ∪ [μ′]; σ(γ); ∼; [λ′] ∪ [μ′]; γ̌; ≀; ≀; γ̂; [λ′]; [λ′] ∪ [μ′],]

the unlabeled arrows denoting inclusions.

Given κ ∈ C_{λ}, we denote its transposition by κ′ ∈ R_{λ′}, mapping j × i ∈ [λ′] to (j × i)κ′ := (i × j)κτ. The permutation of [λ] ∪ [μ] that restricts to κ on [λ] and to the identity on [μ]\[λ] is denoted by κ̂. Similarly, the identical prolongation of κ′ to a permutation of [λ′] ∪ [μ′] is denoted by κ̂′. Likewise for μ instead of λ.

Lemma 3.20 Let γ be a d-fold path of weight e, let κ ∈ C_{λ}. We note that the composition γκ̂ is again a d-fold path of weight e, of the same pattern as γ, and obtain

\begin{aligned}(γκ̂) & ˆ = & (κ̂′)^{−1}γ̂κ̂′ \\ (γκ̂) & ˇ = & (κ′_{0})^{−1}γ̌κ′ \end{aligned}

for some κ_{0} ∈ C_{μ}.

The first equality holds since right conjugation of a cycle product in the symmetric group is performed by an application of the conjugating element to the cycle entries. To see the second equality, we consider the diagram

[figure: [μ′]; [λ′] ∪ [μ′]; σ(γκ̂); ∼; [λ′] ∪ [μ′]; κ_{0}^{′−1}; ≀; ≀; κ̂^{′−1}; [μ′]; [λ′] ∪ [μ′]; σ(γ); ∼; [λ′] ∪ [μ′]; γ̌; ≀; ≀; γ̂; [λ′]; [λ′] ∪ [μ′]; κ′; ≀; ≀; κ̂′; [λ′]; [λ′] ∪ [μ′],]

in which κ_{0}^{′−1} is the completion by restriction and in which the unlabeled arrows denote inclusions. Note that σ(γκ̂)κ̂^{′−1}σ(γ)^{−1} restricts identically to the row k+1 of [λ′] ∪ [μ′].

For each weight e, we fix a d-fold path γ_{e} of weight e and write

ε(e) := ε_{σ(γ_{e})}ε_{[a′_{μ}]^{−1}γ̌_{e}[a′_{λ}]}.

Lemma 3.21 The sign ε(e) is independent of the choice of γ_{e}. More precisely, given a d-fold path γ of weight e, we obtain

ε_{}_{γ̌_{e}^{−1}}γ̌ = ε_{σ(γ_{e})}ε_{σ(γ)}.

Note that εγ̌_{}_{γ̌_{e}^{−1}} = ε_{σ(γ)γ̂γ̂_{e}^{−1}σ(γ_{e})^{−1}} since σ(γ)γ̂γ̂_{e}^{−1}σ(γ_{e})^{−1} restricts to the identity on [λ′]\[μ′]. Moreover, we claim that εγ̂ depends only on the weight of γ. First of all, it depends only on the pattern of γ, since this pattern determines the cycle type of γ̂. The sign of this cycle type in turn depends only on the cardinality ∑_{j∈[g,k+1]} e_{j} of that pattern, since the number of cycles of length ⩾ 2 that occur in γ̂ equals d. Altogether, after a reordering we obtain ε_{}_{γ̌_{e}^{−1}}γ̌ = ε_{σ(γ)}ε_{σ(γ_{e})}.

Lemma 3.22 Given a d-fold path γ of weight e and a λ-tableau [a], we obtain

({γ̌[a′]} ⊗ 1)ν_{S}^{μ} = ⟨(γ̌[a′])′⟩ε_{σ(γ)}ε(e)ε_{[a]^{−1}[a_{λ}]} ∈ S^{μ}.

We may conclude

({γ̌[a′]} ⊗ 1)ν_{S}^{μ} = ⟨(γ̌[a′])′⟩ε_{[a′_{μ}]^{−1}γ̌[a′]}
= ⟨(γ̌[a′])′⟩ε_{[a′_{μ}]^{−1}γ̌_{e}[a′_{λ}]}ε_{[a′_{λ}]^{−1}[a′]}ε_{[a′]^{−1}γ̌_{e}^{−1}γ̌[a′]}
=^{(3.21)} ⟨(γ̌[a′])′⟩ε(e)ε_{[a]^{−1}[a_{λ}]}ε_{σ(γ)}.

Lemma 3.23 Let γ be a d-fold path of weight e. For ξ ∈ S_{d} and j ∈ [g + 1, k + 1] we let ξ operate on the columns ⩽ j via

[figure: Ξ_{γ}; ξ_{j}; [1, d] × [g, k + 1]; i × j′; \{ \begin{matrix}i & × & j′ & for j′ ∈ \{g\} ∪ [j + 1, k + 1] \\ (i)ξ & × & j′ & for j′ ∈ [g + 1, j]. \end{matrix}]

We suppose in addition that ξ_{j} restricts identically to [1, d]×{j}\Ξ_{γ}. Then the composition ((Ξ_{γ})ξ_{j} \overline{}^{ξ_{j}^{−1}}_{∼} Ξ_{γ} ^{\overline{}^{γ}} [λ] ∪ [μ]) is a d-fold path of weight e. Given a λ-tableau [a], we obtain

(\{(ξ_{j}^{−1}γ)ˇ [a′]\} ⊗ 1)ν_{S}^{μ} = \{ \begin{matrix}(\{γ̌[a′]\} ⊗ 1)ν_{S}^{μ} & for j ∈ [g + 1, k] \\ (\{γ̌[a′]\} ⊗ 1)ν_{S}^{μ}ε_{ξ} & for j = k + 1. \end{matrix}

We may suppose ξ to be a transposition, ξ = (s, t), s × j, t × j ∈ Ξ_{j}, s ≠ t. In case j ∈ [g + 1, k] we obtain, composing with transpositions permuting [μ′],

((s, t)_{j} · γ)ˇ = γ̌ · ((s × g)γ, (t × g)γ) · ((s × j)γ, (t × j)γ),

yielding the required equality by two column permutations applied to the resulting polytabloid.

In case j = k + 1, we obtain

((s, t)_{k+1} · γ)ˇ = γ̌ · ((s × g)γ, (t × g)γ),

yielding the required equality by a single column permutation applied to the resulting polytabloid.

Given a weight e, we write

\begin{aligned}e! & := & ∏_{j∈[1,λ_{1}]\\{g\}}e_{j}! \\ λ′! & := & ∏_{j∈[1,λ_{1}]\\{g\}}λ′_{j}! \\ (λ′ − e)! & := & ∏_{j∈[1,λ_{1}]\\{g\}}(λ′_{j} − e_{j})! \end{aligned}

Corollary 3.24 Let e be a weight, let [a] be a λ-tableau. We obtain

∑_{γ∈Γ(e)}({γ̌[a′]} ⊗ 1)ν_{S}^{μ}ε_{σ(γ)} = e! · ∑_{γ∈Γ⃗(e)}({γ̌[a′]} ⊗ 1)ν_{S}^{μ}ε_{σ(γ)}.

We rewrite this sum columnwise from right to left in that we use (3.23) to impose an ordering condition on γ in column j, starting in column j = k + 1 and ending in column j = g + 1. So we obtain a factor e_{j}! in column j.

Given a weight e, we let

[figure: F^{λ}; 𝔣′′_{e}; S^{μ}; [a]; [a]𝔣′′_{e} := ∑_{γ∈Γ⃗(e)}⟨(γ̌[a′])′⟩ε_{σ(γ)}.]

Remark 3.25 (connection to §2.1) In case of d = 2, we obtain that the set of ordered double paths, defined in §2.1, and the set of ordered 2-fold paths, defined here, coincide. Moreover, [a] being a λ-tableau, e being a weight, we obtain

[a]f′′_{e} = −\frac{e!}{2}[a]𝔣′′_{e},

the left hand side written in the notation of §2.1, the right hand side written in the notation introduced here.

The set Γ(e) is the disjoint union of Γ̇(e) and (1, 2)_{k+1} · Γ̇(e). We may rewrite the right hand side as

−( ∏_{j∈[g+1,k]}e_{j}!) ∑_{γ∈Γ⃗(e)}⟨(γ̌[a′])′⟩ε_{σ(γ)} =^{(3.24)} −(e_{k+1}!)^{−1} ∑_{γ∈Γ(e)}⟨(γ̌[a′])′⟩ε_{σ(γ)}
=^{(3.23)} − ∑_{γ∈Γ̇(e)}⟨(γ̌[a′])′⟩ε_{σ(γ)}
= ∑_{γ∈Γ̇(e)}⟨(γ̌[a′])′⟩ε_{γ}
= [a]f′′_{e},

where ε_{γ} = (−1)^{γ̄(k+1,1)+γ̄(k+1,2)} as introduced in §2.1, and where for the last equality we translate [a^{γ}] = (γ̌[a′])′.

Proposition 3.26 Let e be a weight. We may reformulate

[a]𝔣′′_{e} = \frac{1}{e!(λ′ − e)!λ′_{g}} ∑_{γ∈Γ(e)/C_{λ}}rγ̌\{a′\}^{−}Θ− ; γ̌ν_{S}^{μ} · ε(e),

where γ ∈ Γ(e)/C_{λ} signifies that γ runs over a set of orbit representatives of Γ(e) under the operation of C_{λ} in the sense of (3.20). In particular, there is a factorization

(F^{λ} ^{\overline{}^{𝔣′′_{e}}} S^{μ}) = (F^{λ} ^{\overline{}^{ν_{M}^{λ}}} M^{λ,−} ^{\overline{}^{𝔣′_{e}}} S^{μ}).

Given a d-fold path γ of weight e, the stabilizer {κ ∈ C_{λ} | γκ̂ = γ} of γ in C_{λ} has cardinality (λ′ − e)! · λ′_{g}. So we calculate

[a]𝔣′′_{e} = ∑_{γ∈Γ⃗(e)}⟨(γ̌[a′])′⟩ε_{σ(γ)}
=^{(3.24)} \frac{1}{e!} ∑_{γ∈Γ(e)}⟨(γ̌[a′])′⟩ε_{σ(γ)}
= \frac{1}{e!(λ′−e)!λ′_{g}} ∑_{γ∈Γ(e)/C_{λ}}rγ̌r−1 ; γ̌ ∑_{κ∈C_{λ}}⟨(γκ̂)ˇ [a′])′⟩ε_{σ(γ)}
=^{(3.22)} \frac{1}{e!(λ′−e)!λ′_{g}} ∑_{γ∈Γ(e)/C_{λ}}rγ̌r−1 ; γ̌ ∑_{κ∈C_{λ}}(\{(γκ̂)ˇ [a′]\} ⊗ 1)ν_{S}^{μ} · ε(e)ε_{[a]^{−1}[a_{λ}]}

\begin{matrix}=^{(3.20)} & \frac{1}{e!(λ′−e)!λ′_{g}} ∑_{γ∈Γ(e)/C_{λ}}rγ̌((r−1 ; γ̌ ∑_{κ′∈R_{λ′}}\{γ̌κ′[a′]\}) ⊗ 1)ν_{S}^{μ} · ε(e)ε_{[a]^{−1}[a_{λ}]} \\ =^{(3.1)} & \frac{1}{e!(λ′−e)!λ′_{g}} ∑_{γ∈Γ(e)/C_{λ}}rγ̌(\{a′\}Θγ̌ ⊗ 1)ν_{S}^{μ} · ε(e)ε_{[a]^{−1}[a_{λ}]} \\ = & \frac{1}{e!(λ′−e)!λ′_{g}} ∑_{γ∈Γ(e)/C_{λ}}rγ̌\{a′\}^{−}Θ− ; γ̌ν_{S}^{μ} · ε(e). \end{matrix}

Corollary 3.27 (to 2.37 via 3.26) Let d = 2. The transpose in the sense of (3.16) of the morphism given in (2.37) is obtained as a factorization

[figure: F^{μ′}; f^{t,0}; S^{λ′}; [b′]; ⟨b′⟩ + mS^{μ′}; ⟨a′⟩; ⟨a′⟩ + mS^{λ′}; S^{μ′}/m; ft; S^{λ′}/m,]

where

[b′]f^{t,0} := −∑_{e∈E} \frac{1}{(λ′−e)!·2Rλ′_{g}} ( ∏_{i∈[g+1,k]}X_{i}^{(2−e_{i})}) ∑_{γ∈Γ(e)/C_{λ}}rγ̌⟨b′⟩Θ_{}_{γ̌^{−1}} · ε(e),

and where f, E, R, m and X_{i} are defined as for (2.37).

Given t, T ⩾ 2, Z-lattices X and Y and a Z-linear map X/t ^{\overline{}^{u}} Y/t, we denote

(X/(tT)^{\overline{}^{u • T}} Y/(tT)) := (X/(tT) \overline{}X/t ^{\overline{}^{u}} Y/t ^{\overline{}^{T}} Y/(tT)).

Let M := R · λ′! · 2λ′_{g} play the role of a large enough integer and consider the morphism

S^{λ}/(mM) ^{\overline{}^{f • M}} S^{μ}/(mM)
⟨a⟩  \overline{}λ′! · 2λ′_{g} · ∑_{e∈E}( ∏_{i∈[g+1,k]}X_{i}^{(2−e_{i})})\{a′\}^{−}f′_{e}
=^{(3.25, 3.26)} −λ′!λ′_{g} · ∑_{e∈E}( ∏_{i∈[g+1,k]}X_{i}^{(2−e_{i})}){a′}^{−}𝔣′_{e}
=^{(3.26)} −∑_{e∈E} \frac{λ′!}{(λ′−e)!}( ∏_{i∈[g+1,k]}X_{i}^{(2−e_{i})}) ∑_{γ∈Γ(e)/C_{λ}}rγ̌\{a′\}^{−}Θ− ; γ̌ν_{S}^{μ} · ε(e).

For φ ∈ Φ^{μ′,λ′}, the coefficient x_{{φ}} in the sense of (3.16) can be written as

x_{\{φ\}} = −∑_{e∈E} \frac{λ′!}{(λ′−e)!}( ∏_{i∈[g+1,k]}X_{i}^{(2−e_{i})}) ∑_{γ∈Γ(e)/C_{λ}}rγ̌ · \{ \begin{matrix}1 & for \{γ̌\} = \{φ\} \\ 0 & else \end{matrix} \} · ε(e).

Thus, by (3.16), the transpose of S^{λ}/(mM)^{\overline{}^{f • M}} S^{μ}/(mM) maps

[figure: S^{μ′}/(mM); (f • M)t; S^{λ′}/(mM); ⟨b′⟩; −∑_{e∈E} \frac{λ′!}{(λ′−e)!}( ∏_{i∈[g+1,k]}X_{i}^{(2−e_{i})}) ∑_{γ∈Γ(e)/C_{λ}}rγ̌⟨b′⟩Θ_{}_{γ̌^{−1}} · ε(e)]

Finally, there is a commutative diagram

[figure: Hom_{ZS_{n}}(S^{λ}/m, S^{μ}/m); tra; ∼; Hom_{ZS_{n}}(S^{μ′}/m, S^{λ′}/m); Hom_{ZS_{n}}(S^{λ}/(mM), S^{μ}/(mM)); tra; ∼; Hom_{ZS_{n}}(S^{μ′}/(mM), S^{λ′}/(mM))]

with vertical injections given by multiplication by M, so that f^{t} • M = (f • M)^{t}.

Remark 3.28 Consider the set Γ⃗_{0}(e) of ordered d-fold paths γ of weight e that have γ̄(j, i) ∈ [λ′_{j} − e_{j}, λ′_{j}] for i × j ∈ Ξ_{[g+1,k+1]}, i.e. that have their positions sitting at the bottoms of the columns. We say that a correspondoid {φ} ∈ {Φ^{μ′,λ′}} is semistandard up to row permutation if there exists a row permutation ρ ∈ R_{μ′} such that {ρφ} ∈ {Φ^{μ′,λ′}}_{sst} becomes semistandard. Note that Θ_{φ} = Θ_{ρφ} by (3.1). For γ ∈ Γ⃗_{0}(e), the correspondoid {γ̌^{−1}} is semistandard up to row permutation as long as Ξ_{γ} is not bulky in the sense introduced at the beginning of §2.3. Cf. the bases in (3.13), cf. the reduced tuple of coefficients employed in (2.37).

Note that Γ⃗_{0}(e) is a set of representatives of Γ(e)/C_{λ}.

Question 3.29 I do not know a formula for the transpose f^{t} given in (3.27) in terms of λ′-polytabloids. Cf. (3.17).

3.4 Horizontal examples

We shall transpose some examples of §2.4 with respect to the convention that [a_{λ}] = [ǎ_{λ}] for all partitions λ, cf. §1.1. Again, we omit to denote the brackets that indicate polytabloids, moreover, we omit commas in cycles. If possible (cf. 3.13), we sort the resulting linear combination of polytabloids into images under Θ_{φ}^{−}ν_{S}^{λ′} for various (implicitely present) correspondoids {φ}. The formula in (3.27) does not suffice to give the coefficients of this expression (cf. 3.17, 4.15).

Example 3.30 Let λ = (3, 3), μ = (2, 2, 1, 1). The transpose of the morphism given in (2.40) is

[figure: S^{μ′}/4; f^{t}; S^{λ′}/4; 1 3 5 6 2 4; − 1 4 2 5 3 6 · (1 − (34)) + 2 · 1 3 2 4 5 6 .]

Example 3.31 Let λ = (3, 2, 2), μ = (3, 1, 1, 1, 1). The transposes of the morphisms given in (2.41) are

[figure: S^{μ′}/3; f^{t}; S^{λ′}/3; 1 4 5 6 7 2 3; − 1 4 6 2 5 7 3 − 1 4 5 2 6 7 3 .]

and

S^{μ′}/2 ^{\overline{}u^{μ,t}u^{λ′}} S^{λ′}/2
1 4 5 6 7 2 3  \overline{}1 2 6 3 4 7 5 + 1 4 5 2 6 7 3 + 1 3 5 2 6 7 4 + 1 2 5 3 6 7 4 + 1 3 4 2 6 7 5

+ 1 2 4 3 6 7 5 + 1 2 3 4 6 7 5 + 1 2 5 3 4 7 6 + 1 3 4 2 5 7 6 + 1 2 4 3 5 7 6

+ 1 2 3 4 5 7 6 + 1 2 5 3 4 6 7 + 1 3 4 2 5 6 7 + 1 2 4 3 5 6 7 + 1 2 3 4 5 6 7 .

Let us consider the morphism u^{μ,t}u^{λ′}. Using [a_{λ′}] = [ǎ_{λ′}], we obtain, in the notation of (3.19),

\begin{matrix}P^{λ′} = \{1, (56), (34), (34)(56), (354), (3564), (3654), (364), (37654), (3764), (234), (234)(56), \\ (2354), (23564), (23654), (2364), (237654), (23764), (24)(356), (24)(36), (24)(376)\} ⊆ S_{7}. \end{matrix}

Hence, with respect to the ordering of P^{λ′} just indicated, the matrix (χ_{τσ^{−1}}^{λ′})_{τ×σ∈P^{λ′}×P^{λ′}} takes the form

[\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] .

By labelling our morphism as u^{μ,t}u^{λ′}, we implicitely claimed the tuple (u_{τ}^{λ′})_{τ∈P^{λ′}} to be represented by the vector

[0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1].

But this row vector yields a row vector of constant entry 1 by right multiplication with that matrix, which confirms this claim.

Furthermore, as remarked after (3.19), the matrix (χ_{τσ^{−1}}^{μ})_{τ×σ∈P^{μ}×P^{μ}} equals the identity matrix.

Example 3.32 Let λ = (3, 3, 1, 1), μ = (2, 2, 1, 1, 1, 1). The transpose of the morphism given in (2.42) is

S^{μ′}/6 ^{\overline{}f^{t}} S^{λ′}/6
1 3 5 6 7 8 2 4  \overline{}− 1 4 7 8 2 5 3 6 · (1 − (34)) − 2 · 1 3 7 8 2 4 5 6

− 1 4 6 8 2 5 3 7 · (1 − (34)) − 2 · 1 3 6 8 2 4 5 7

− 1 4 5 8 2 6 3 7 · (1 − (34)) − 2 · 1 3 5 8 2 4 6 7

− 1 4 6 7 2 5 3 8 · (1 − (34)) − 2 · 1 3 6 7 2 4 5 8

− 1 4 5 7 2 6 3 8 · (1 − (34)) − 2 · 1 3 5 7 2 4 6 8

− 1 4 5 6 2 7 3 8 · (1 − (34)) − 2 · 1 3 5 6 2 4 7 8 .

Example 3.33 Let λ = (4, 4), μ = (3, 3, 1, 1). The transpose of the morphism given in (2.41) is

[figure: S^{μ′}/5; f^{t}; S^{λ′}/5; 1 4 7 8 2 5 3 6; − 1 5 2 6 3 7 4 8 · (1 − (45) − (46)) − 3 · 1 4 2 5 3 6 7 8 .]

Example 3.34 Let λ = (3, 3, 2), μ = (2, 2, 2, 1, 1). The transpose of the morphism given in (2.41) is

S^{μ′}/5 ^{\overline{}f^{t}} S^{λ′}/5
1 3 5 7 8 2 4 6  \overline{}− 1 4 7 2 5 8 3 6 · (1 − (34)) − 2 · 1 4 6 2 5 8 3 7 · (1 − (34))(1 − (56))

− 1 3 6 2 4 8 5 7 · (1 − (56)) + 1 4 6 2 5 7 3 8 · (1 − (34))(1 − (56))

− 1 3 6 2 4 7 5 8 · (1 − (56)) + 2 · 1 3 5 2 4 6 7 8 .

Example 3.35 Let λ = (3, 3, 1, 1), μ = (2, 2, 2, 2). The transpose of the morphism given in (2.45) is

S^{μ′}/3 ^{\overline{}f^{t}} S^{λ′}/3
1 3 5 7 2 4 6 8  \overline{}1 4 6 8 2 5 3 7 · (1 − (34))(1 − (56))(1 − (78))

− 1 3 6 8 2 4 5 7 · (1 − (56))(1 − (78)).

Remark 3.36 The morphisms in (2.46, 2.47) are transposed to their respective negative, g^{t} = −g. For instance, the morphism S^{(4,3)}/4 ^{\overline{}^{g}} S^{(2,2,2,1)}/4 given in (2.46) can be obtained as

ν_{S}^{λ} · g ≡_{4} (2 · Θ_{φ_{1}}^{−} − Θ_{φ_{2}}^{−} + 2 · Θ_{φ_{3}}^{−}) · ν_{S}^{μ},

\{φ_{1}\} := \{ \begin{matrix}1 & 3 \\ 1 & 3 \\ 2 & 4 \\ 2 \end{matrix} \} , \{φ_{2}\} := \{ \begin{matrix}1 & 2 \\ 1 & 3 \\ 2 & 4 \\ 3 \end{matrix} \} , \{φ_{3}\} := \{ \begin{matrix}1 & 2 \\ 1 & 2 \\ 3 & 4 \\ 3 \end{matrix} \} .

which allows to apply (3.16) directly and to compare linear combinations of tabloids.

4 Two columns, several boxes

We shall generalize (2.17), treating the case of essentially two columns and two shifted boxes, to the case of essentially two columns and an arbitrary number of shifted boxes.

Lemma 4.1 Let x, y ⩾ 0. As polynomials in the free variable t, we obtain

∑_{j∈[0,x]}(−1)^{x−j} (\begin{matrix}y+j \\ j \end{matrix}) (\begin{matrix}t \\ x−j \end{matrix}) = (\begin{matrix}y−t+x \\ x \end{matrix}) ∈ Q[t],

where

(\begin{matrix}f(t) \\ z \end{matrix}) := z!^{−1} · ∏_{i∈[0,z−1]}(f(t) − i)

for f(t) ∈ Q[t], z ⩾ 0. So, in particular, (\begin{matrix}f(t) \\ 0 \end{matrix}) = 1.

We name the claimed equality Eq(x, y) and shall prove the following assertions.

(i) Eq(0, y) holds for y ⩾ 0.

(ii) Eq(x, 0) holds for x ⩾ 0.

(iii) Eq(x, y − 1) and Eq(x − 1, y) together imply Eq(x, y) for x, y ⩾ 1.

Ad (i). Eq(0, y) writes 1 = 1.

Ad (ii). Eq(x, 0) writes

∑_{j∈[0,x]}(−1)^{j} (\begin{matrix}t \\ j \end{matrix}) = (\begin{matrix}x−t \\ x \end{matrix}) .

Proceeding by induction on x and using (i) we are reduced to consider differences, i.e. to see that

(−1)^{x} (\begin{matrix}t \\ x \end{matrix}) = (\begin{matrix}−(t−x+1) \\ x \end{matrix}) .

Ad (iii). The right hand side of Eq(x, y) equals

(\begin{matrix}y−t+x \\ x \end{matrix})  =  (\begin{matrix}y−t+(x−1) \\ (x−1) \end{matrix}) + (\begin{matrix}(y−1)−t+x \\ x \end{matrix})

by assumption =  ∑_{j∈[0,x−1]}(−1)^{x−1−j} (\begin{matrix}y+j \\ j \end{matrix}) (\begin{matrix}t \\ x−1−j \end{matrix}) + ∑_{j∈[0,x]}(−1)^{x−j} (\begin{matrix}y−1+j \\ j \end{matrix}) (\begin{matrix}t \\ x−j \end{matrix})

=  ∑_{j∈[1,x]}(−1)^{x−j} (\begin{matrix}y+j−1 \\ j−1 \end{matrix}) (\begin{matrix}t \\ x−j \end{matrix}) + ∑_{j∈[0,x]}(−1)^{x−j} (\begin{matrix}y+j−1 \\ j \end{matrix}) (\begin{matrix}t \\ x−j \end{matrix})
=  ∑_{j∈[0,x]}(−1)^{x−j} (\begin{matrix}y+j \\ j \end{matrix}) (\begin{matrix}t \\ x−j \end{matrix}) ,

being the left hand side.

Lemma 4.2 Let 1 ⩽ k ⩽ x. We have

gcd((\begin{matrix}x \\ 1 \end{matrix}) , (\begin{matrix}x \\ 2 \end{matrix}) , …, (\begin{matrix}x \\ k \end{matrix})) = x · ∏_{p prime, p|x}p^{− min(v_{p}(x), ilog_{p}(k))},

where

ilog_{p}(k) := max{i ⩾ 0 | p^{i} ⩽ k} = max v_{p}([1, k]).

Let p be a prime, let j ∈ [0, x]. For an integer w ⩾ 0, written p-adically as w = ∑_{i⩾0}w_{i}p^{i},

w_{i} ∈ [0, p − 1], we denote its Quersumme by q_{p}(w) := ∑_{i⩾0}w_{i}. We obtain

v_{p}(j!) = ∑_{i⩾0}w_{i} · \frac{p^{i} − 1}{p − 1} = \frac{j − q_{p}(j)}{p − 1},

and thus

v_{p}((\begin{matrix}x \\ j \end{matrix})) = (p − 1)^{−1}(q_{p}(x − j) − (q_{p}(x) − q_{p}(j))).

In particular, in case j ⩽ p^{v_{p}(x)} we get q_{p}(x − j) = (q_{p}(x) − 1) + (p − 1)(v_{p}(x) − v_{p}(j) − 1) + p · 1 − q_{p}(j), whence

v_{p}((\begin{matrix}x \\ j \end{matrix})) = v_{p}(x) − v_{p}(j).

Thus the minimal such valuation for j ∈ [1, k] takes the value claimed above.

Let n ⩾ 1, d ⩾ 1. Assume given a partition λ of n and an integer g ∈ [1, λ_{1} − 1] such that

μ′_{i} := \{ \begin{matrix}λ′_{i} + d & for i = g \\ λ′_{i} − d & for i = g + 1 \\ λ′_{i} & else \end{matrix}

defines a partition μ. Let Z′ be the set of injective maps

[1, d] ^{\overline{}^{ζ′}} [1, λ′_{g+1}].

Let Z ⊆ Z′ be the subset of strictly monotone maps. Let the sign of ζ ∈ Z be given by ε_{ζ} := (−1)^{∑_{i∈[1,d]} ζ(i)}. For ζ′ ∈ Z′ there is a unique factorization ζ′(i) = ζ⃗′(iσ), i ∈ [1, d], with σ ∈ S_{d} and ζ⃗′ ∈ Z. Let the sign of ζ′ ∈ Z′ be given by ε_{ζ′} := ε_{σ}εζ⃗′. Suppose given a λ-tableau [a] and an injection ζ′ ∈ Z′. We let

[figure: [0, λ′_{g+1}]; [0, μ′_{g+1}]; i; φ; #([1, i]\ζ′([1, d])); min(φ^{−1}({j})); ψ; j]

and define the μ-tableau [a^{ζ′}] by

a_{j,i}^{ζ′} := a_{j,i}  for (j ∈ [1, λ_{1}]\{g, g + 1} and i ∈ [1, μ′_{j}]) or (j = g and i ∈ [1, λ′_{g}])
a_{g,λ′_{g}+i}^{ζ′} := a_{g+1,ζ′(i)} for i ∈ [1, d]
a_{g+1,i}^{ζ′} := a_{g+1,ψ(i)} for i ∈ [1, μ′_{g+1}].

Using this place operation, we define the ZS_{n}-linear map

[figure: F^{λ}; f′′; S^{μ}; [a]; ∑_{ζ∈Z}⟨a^{ζ}⟩ε_{ζ} = \frac{1}{d!} ∑_{ζ′∈Z′}⟨a^{ζ′}⟩ε_{ζ′}.]

For a λ-tableau [a] and for an injection ζ′ ∈ Z′, we denote the ‘[a]-realization’ of ζ′ by

([1, d] ^{\overline{}^{ζ′_{[a]}}} [1, n]) := ([1, d] ^{\overline{}^{ζ′}} [1, λ′_{g+1}] ^{\overline{}^{∼}} ([1, λ′_{g+1}] × \{g + 1\}) [figure] [λ] ^{\overline{}^{[a]}} [1, n]).

Lemma 4.3 We have a factorization

(F^{λ} ^{\overline{}^{f′′}} S^{μ}) = (F^{λ} ^{\overline{}^{ν_{M}^{λ}}} M^{λ′,−} ^{\overline{}^{f′}} S^{μ}).

We need to show that given v, w ∈ a_{g+1}, v ≠ w, and a λ-tableau [a], we have ([a] · (v, w))f′′ = −[a]f′′. Suppose given ζ ∈ Z. In case v, w ∈ ζ_{[a]}([1, d]) or in case v, w ∉ ζ_{[a]}([1, d]), we obtain ⟨a^{ζ}⟩ε_{ζ} ·(v, w) = −⟨a^{ζ}⟩ε_{ζ}. The maps ζ ∈ Z that satisfy v ∈ ζ_{[a]}([1, d]), w ∉ ζ_{[a]}([1, d]) furnish a subset Z_{10} ⊆ Z, the maps ζ ∈ Z that satisfy v ∉ ζ_{[a]}([1, d]), w ∈ ζ_{[a]}([1, d]) furnish a subset Z_{01} ⊆ Z. We have a bijection ι that sends ζ ∈ Z_{10} to the map ι(ζ) ∈ Z_{01} determined by ι(ζ)_{[a]}([1, d]) = (ζ_{[a]}([1, d])) · (v, w). We claim that

⟨a^{ζ}⟩ε_{ζ} · (v, w) = −⟨a^{ι(ζ)}⟩ε_{ι(ζ)},

thus proving the lemma. Let v =: a_{g+1,i}, let w =: a_{g+1,j} and assume i < j. Comparing both sides, the entries in column g of [μ] yield a sign (−1)^{#([i+1,j−1]∩ζ([1,d]))}, the entries in column g+1 of [μ] yield a sign (−1)^{#([i+1,j−1]\ζ([1,d]))}, so, altogether, the entries yield a sign (−1)^{j−i−1}. On the other hand, the quotient of the signs of the maps is ε_{ι(ζ)}/ε_{ζ} = (−1)^{j−i}.

Remark 4.4 Let the correspondence φ ∈ Φ^{λ′,μ′} be defined by

[λ′] \overline{}^{φ}_{∼} [μ′]
j × i \overline{}j × i  for j × i ∈ [λ′] ∩ [μ′]
(g + 1) × (μ′_{g+1} + i) \overline{}g × (λ′_{g+1} + i) for i ∈ [1, d]

The semistandard correspondoid {φ} is in fact the only element in {Φ^{λ′,μ′}}_{sst} (cf. after 3.6). With respect to [a_{λ′}] = [ǎ_{λ′}] and [a_{μ′}] = [ǎ_{μ′}], we obtain f′ = −Θ_{φ^{−1}}^{−} · ν_{S}^{μ}, which reproves (4.3) by means of (3.1).

From (3.14, 3.13) we take the following assertions. Suppose given m̃ ⩾ 1.

In case μ′ is 2-regular, the map

Hom_{ZS_{n}}(M^{λ′,−}/m̃, M^{μ′,−}/m̃) ^{\overline{}^{(−)ν_{S}^{μ}}} Hom_{ZS_{n}}(M^{λ′,−}/m̃, S^{μ}/m̃)

is surjective. In particular, the group Hom_{ZS_{n}}(S^{λ}/m̃, S^{μ}/m̃) is cyclic since we may embed

Hom_{ZS_{n}}(S^{λ}/m̃, S^{μ}/m̃) ^{\overline{}^{ν_{S}^{λ}(−)}} Hom_{ZS_{n}}(M^{λ′,−}/m̃, S^{μ}/m̃).

A generator is a factorization of some multiple of f′ ⊗_{Z} Z/m̃ over ν_{S}^{λ}.

In case μ′ is 2-singular, the map

2 · Hom_{ZS_{n}}(M^{λ′,−}/m̃, M^{μ′,−}/m̃) ^{\overline{}^{(−)ν_{S}^{μ}}} 2 · Hom_{ZS_{n}}(M^{λ′,−}/m̃, S^{μ}/m̃)

is surjective. In particular, the group 2 · Hom_{ZS_{n}}(S^{λ}/m̃, S^{μ}/m̃) is cyclic since we may embed

2 · Hom_{ZS_{n}}(S^{λ}/m̃, S^{μ}/m̃) ^{\overline{}^{ν_{S}^{λ}(−)}} 2 · Hom_{ZS_{n}}(M^{λ′,−}/m̃, S^{μ}/m̃).

A generator is a factorization of some multiple of 2 · f′ ⊗_{Z} Z/m̃ over ν_{S}^{λ}.

Let ξ ⊆ a_{g}, s := #ξ, ξ̄:= a_{g}\ξ, t ∈ [1, λ′_{g+1}], η := a_{g+1,[1,t]}, η̄:= a_{g+1,[t+1,λ′_{g+1}]} be such that s + t = λ′_{g} + 1. Let u := #η̄ = λ′_{g+1} − t.

For x ∈ [0, min(d, t)], we denote

Z′[x] := {ζ′ ∈ Z′ | ζ′(i) = i for i ∈ [1, x], ζ′(i) ⩾ t + 1 for i ∈ [x + 1, d]}
Z[x] := Z ∩ Z′[x]
Λ([a], x) := \frac{1}{s!t!} ∑_{ζ∈Z[x]}⟨a^{ζ}⟩ε_{ζ} ∘ (ξ ∪ η)
= \frac{1}{s!t!(d−x)!} ∑_{ζ′∈Z′[x]}⟨a^{ζ′}⟩ε_{ζ′} ∘ (ξ ∪ η)

so that we can recover

G′_{[a],ξ,η}f′ = ∑_{x∈[0,min(d,t)]} (\begin{matrix}t \\ x \end{matrix}) Λ([a], x).

For 0 ⩽ x ⩽ y ⩽ min(d, t), there is a rectification map

[figure: Z′[x]; [y]; Z′[y]; ζ′; ζ′[y]]

defined by ζ′[y](i) := i for i ∈ [1, y] and ζ′[y](i) := ζ′(i) for i ∈ [y + 1, d]. In case t ⩾ d, we denote by ζ[d] the unique element in Z[d] = Z′[d], mapping each i ∈ [1, d] to i ∈ [1, λ′_{g+1}].

Lemma 4.5 Suppose t ⩾ d. Let ξ̄ = ξ̄_{−1} ∪ ξ̄_{2} be a disjoint decomposition such that #ξ̄_{−1} = d − 1 and such that #ξ̄_{2} = t − d. Let η_{2} := a_{g+1,[d+1,t]}. We obtain, in the notation of (1.6),

G′_{[a],ξ,η}f′ = ε_{ζ[d]} · ∑_{i∈[0,d−1]}(−1)^{i} (\begin{matrix}s−u+d \\ d−i \end{matrix}) · B_{[a^{ζ[d]}]·(ξ̄_{2},η_{2})_{α},ξ̄_{−1},ξ̄_{2}}(i),

independent of the chosen bijection ξ̄_{2} \overline{}^{α}_{∼} η_{2}.

Given d_{1} ∈ [0, d] and a map ζ ∈ Z[d − d_{1}], we denote

\begin{aligned}η_{1} & := & a_{g+1,[d−d_{1}+1,d]} \\ η̄_{1}(ζ) & := & ζ_{[a]}([d − d_{1} + 1, d]). \end{aligned}

We obtain

Λ([a], d − d_{1}) = \frac{1}{s!t!} ∑_{ζ∈Z[d−d_{1}]}⟨a^{ζ}⟩ε_{ζ} ∘ (ξ ∪ η)
=^{(1.4)}  \frac{(s+d−d_{1})!}{s!t!(d−1)!} ∑_{ζ∈Z[d−d_{1}]}⟨a^{ζ}⟩ε_{ζ} · (ξ̄_{2}, η_{2}) · (η̄_{1}(ζ), η_{1}) ∘ (ξ̄∪ η̄_{1}(ζ))
= (−1)^{d_{1}} · \frac{(s+d−d_{1})!}{s!t!(d−1)!} ∑_{{η̄_{1}⊆η̄, \atop #η̄_{1} = d_{1}}}⟨a^{ζ[d]}⟩ε_{ζ[d]} · (ξ̄_{2}, η_{2}) ∘ (ξ̄∪ η̄_{1})

=^{(cf. 1.6)} (−1)^{d_{1}} · \frac{(s+d−d_{1})!}{s!t!(d−1)!} · ε_{ζ[d]} · C_{[a^{ζ[d]}]·(ξ̄_{2},η_{2})_{α},ξ̄_{−1},ξ̄_{2}}(d_{1})
=^{(1.6)} (−1)^{d_{1}} · \frac{(s+d−d_{1})!(d−1)!(t−d+d_{1})!}{s!t!(d−1)!} · ε_{ζ[d]}·

· ∑_{i∈[0,min(d_{1},d−1)]} (\begin{matrix}u−i \\ d_{1}−i \end{matrix}) B_{[a^{ζ[d]}]·(ξ̄_{2},η_{2})_{α},ξ̄_{−1},ξ̄_{2}}(i),

whence our one-step Garnir relation is mapped to

G′_{[a],ξ,η}f′
= ε_{ζ[d]} · ∑_{d_{1}∈[0,d]} ∑_{i∈[0,min(d_{1},d−1)]} (\begin{matrix}t \\ d−d_{1} \end{matrix}) (−1)^{d_{1}} · \frac{(s+d−d_{1})!(t−d+d_{1})!}{s!t!} · (\begin{matrix}u−i \\ d_{1}−i \end{matrix}) ·
·B_{[a^{ζ[d]}]·(ξ̄_{2},η_{2})_{α},ξ̄_{−1},ξ̄_{2}}(i)
= ε_{ζ[d]} · ∑_{i∈[0,d−1]} ( ∑_{j∈[0,d−i]}(−1)^{i+d−i−j} (\begin{matrix}s+j \\ j \end{matrix}) (\begin{matrix}u−i \\ d−i−j \end{matrix})) B_{[a^{ζ[d]}]·(ξ̄_{2},η_{2})_{α},ξ̄_{−1},ξ̄_{2}}(i)
=^{(4.1)} ε_{ζ[d]} · ∑_{i∈[0,d−1]}(−1)^{i} (\begin{matrix}s−u+d \\ d−i \end{matrix}) B_{[a^{ζ[d]}]·(ξ̄_{2},η_{2})_{α},ξ̄_{−1},ξ̄_{2}}(i).

Lemma 4.6 Suppose 2d − 1 ⩽ λ′_{g+1}. Let m_{0} := λ′_{g} − λ′_{g+1} + d + 1 be the box shift length, let

m := m_{0} · ∏_{p prime, p|m_{0}}p^{− min(v_{p}(m_{0}), ilog_{p}(d))},

In case μ′ is 2-regular, the injection

Hom_{ZS_{n}}(S^{λ}, S^{μ}/m) ^{\overline{}^{(−)·(n!/m)}} Hom_{ZS_{n}}(S^{λ}, S^{μ}/n!)

is surjective. In case μ′ is 2-singular, the cokernel of the injection

Hom_{ZS_{n}}(S^{λ}, S^{μ}/m) ^{\overline{}^{(−)·(n!/m)}} Hom_{ZS_{n}}(S^{λ}, S^{μ}/n!)

is annihilated by multiplication by 2.

Consider (4.5) in the case [a] = [ǎ_{λ}], t = d, ξ = a_{g,[1,s]}, ξ̄_{−1} = ξ̄ and ξ̄_{2} = ∅. We obtain

\begin{aligned}G′_{[ǎ_{λ}],ξ,η}f′ & = & ε_{ζ[d]} · ∑_{i∈[0,d−1]}(−1)^{i} (\begin{matrix}s−u+d \\ d−i \end{matrix}) · B_{[(ǎ_{λ})^{ζ[d]}],ξ̄,∅}(i) \\ & =^{(1.6)} & ε_{ζ[d]} · ∑_{i∈[0,d−1]} (\begin{matrix}m_{0} \\ d−i \end{matrix}) · ∑_{{ξ̄_{0}⊆ξ̄, \atop #ξ̄_{0} = i}} ∑_{{φ_{0}⊆η̄, \atop #φ_{0} = i}}⟨(ǎ_{λ})^{ζ[d]}⟩(ξ̄_{0}, φ_{0}). \end{aligned}

Since s = λ′_{g} + 1 − d ⩾ μ′_{g+1} + 1, the elements ⟨(ǎ_{λ})^{ζ[d]}⟩(ξ̄_{0}, φ_{0}) occurring in this expression are standard up to column permutation, i.e. up to sign. Moreover, they are pairwise different because of different column fillings. We note that each i ∈ [0, d − 1] indexes a nonzero summand of this expresseion, namely a nonzero linear combination of different standard polytabloids equipped with coefficients ± (\begin{matrix}m_{0} \\ d−i \end{matrix}), since we assumed 2d−1 ⩽ λ′_{g+1}, i.e. since d − 1 ⩽ #η̄ leaves space for a subset φ_{0} ⊆ η̄ of cardinality i. Therefore, by (4.2), the element G′_{[ǎ_{λ}],ξ,η}f′ ∈ S^{μ} is exactly divisible by m.

Suppose given a ZS_{n}-linear map S^{λ} ^{\overline{}^{f̃}} S^{μ}/n!. Let S^{μ} ^{\overline{}^{φ}} S^{μ}/n! denote the residue class morphism.

Suppose μ′ to be 2-regular. By (4.4), there is an integer z such that ν_{S}^{λ}f̃ = z · f′φ. Thus z · G′_{[ǎ_{λ}],ξ,η}f′ is divisible by n!, and so z · m is divisible by n!. Hence, each element in the image of f̃ is contained in (n!/m)S^{λ}/n!S^{λ}, i.e. there exists a factorization

(S^{λ} ^{\overline{}^{f̃}} S^{μ}/n!) = (S^{λ} \overline{}S^{μ}/m^{\overline{}^{n!/m}} S^{μ}/n!).

Suppose μ′ to be 2-singular. By (4.4), there is an integer z such that 2 · ν_{S}^{λ}f̃ = 2z · f′φ. Thus 2z · G′_{[ǎ_{λ}],ξ,η}f′ is divisible by n!, and so 2z · m is divisible by n!. Hence, each element in the image of 2f̃ is contained in (n!/m)S^{λ}/n!S^{λ}, i.e. there exists a factorization

(S^{λ} ^{\overline{}^{2f̃}} S^{μ}/n!) = (S^{λ} \overline{}S^{μ}/m^{\overline{}^{n!/m}} S^{μ}/n!).

Lemma 4.7 Suppose t ⩽ d − 1. We obtain, in the notation of (1.7),

G′_{[a],ξ,η}f′ = ε_{ζ[d]} · ∑_{i∈[0,t−1]}(−1)^{i} (\begin{matrix}s−u+d \\ t−i \end{matrix}) B′_{[a^{ζ[d]}],ξ̄∪η̄_{2},η̄_{2},η̄\η̄_{2}}(i).

Given d_{0} ∈ [0, t] and an injection ζ′ ∈ Z′[d_{0}], we denote

\begin{aligned}η_{1} & := & ζ′_{[a]}([1, d_{0}]) = a_{g+1,[1,d_{0}]} \\ η̄_{1}(ζ′) & := & ζ′_{[a]}([d_{0} + 1, t]) \\ η̄_{2}(ζ′) & := & ζ′_{[a]}([t + 1, d]) \\ η̄_{2} & := & a_{g+1,[t+1,d]}. \end{aligned}

Given ζ′ ∈ Z′[t], we get

(∗) ⟨a^{ζ′}⟩ε_{ζ′} = ⟨a^{ζ′[d]}⟩σε_{σ}ε_{ζ′[d]}

for any permutation σ ∈ Sη̄ that maps (a_{g+1,i})σ = ζ′_{[a]}(i) for i ∈ [t + 1, d]. Note that #{σ ∈ Sη̄ | (a_{g+1,i})σ = ζ′_{[a]}(i) for i ∈ [t + 1, d]} = (u − d + t)!. For d_{0} ∈ [0, t], we obtain

Λ([a], d_{0})

= \frac{1}{s!t!(d−d_{0})!} ∑_{ζ′∈Z′[d_{0}]}⟨a^{ζ′}⟩ε_{ζ′} ∘ (ξ ∪ η)
=^{(1.4)} \frac{(s+d_{0})!}{s!t!(d−d_{0})!(d−1)!} ∑_{ζ′∈Z′[d_{0}]}⟨a^{ζ′}⟩ε_{ζ′} · (η\η_{1}, η̄_{1}(ζ′)) ∘ (ξ̄∪ η̄_{1}(ζ′) ∪ η̄_{2}(ζ′))
= (−1)^{t−d_{0}} \frac{(s+d_{0})!(t−d_{0})!}{s!t!(d−d_{0})!(d−1)!} ∑_{ζ′∈Z′[t]} ∑_{{η̄_{1}⊆η̄\η̄_{2}(ζ′), \atop #η̄_{1} = t−d_{0}}}⟨a^{ζ′}⟩ε_{ζ′} ∘ (ξ̄∪ η̄_{1} ∪ η̄_{2}(ζ′))
=^{(∗)} (−1)^{t−d_{0}} \frac{(s+d_{0})!(t−d_{0})!}{s!t!(d−d_{0})!(d−1)!(u−d+t)!} ∑_{{η̄_{1}⊆η̄\η̄_{2}, \atop #η̄_{1} = t−d_{0}}}⟨a^{ζ[d]}⟩ε_{ζ[d]} ∘ (ξ̄∪ η̄_{1} ∪ η̄_{2}) ∘ η̄
=^{(cf. 1.7)} (−1)^{t−d_{0}} · ε_{ζ[d]} · \frac{(s+d_{0})!(t−d_{0})!}{s!t!(d−d_{0})!(d−1)!(u−d+t)!} · C′_{[a^{ζ[d]}],ξ̄∪η̄_{2},η̄_{2},η̄\η̄_{2}}(t − d_{0})
=^{(1.7)} (−1)^{t−d_{0}} · ε_{ζ[d]} · \frac{(s+d_{0})!(t−d_{0})!}{s!t!(d−d_{0})!(d−1)!(u−d+t)!}·

·(u−d+t)!(d−1)!(d−d_{0})! · ∑_{i∈[0,min(t−d_{0},t−1)]} (\begin{matrix}u−d+t−i \\ t−d_{0}−i \end{matrix}) B′_{[a^{ζ[d]}],ξ̄∪η̄_{2},η̄_{2},η̄\η̄_{2}}(i),

whence

G′_{[a],ξ,η}f′

= ε_{ζ[d]} · ∑_{d_{0}∈[0,t]} ∑_{i∈[0,min(t−d_{0},t−1)]} (\begin{matrix}t \\ d_{0} \end{matrix}) (−1)^{t−d_{0}} · \frac{(s+d_{0})!(t−d_{0})!}{s!t!}·

· (\begin{matrix}u−d+t−i \\ t−d_{0}−i \end{matrix}) B′_{[a^{ζ[d]}],ξ̄∪η̄_{2},η̄_{2},η̄\η̄_{2}}(i)

= ε_{ζ[d]} · ∑_{i∈[0,t−1]} ( ∑_{d_{0}∈[0,t−i]}(−1)^{i+t−i−d_{0}} · (\begin{matrix}s+d_{0} \\ d_{0} \end{matrix}) · (\begin{matrix}u−d+t−i \\ t−i−d_{0} \end{matrix})) B′_{[a^{ζ[d]}],ξ̄∪η̄_{2},η̄_{2},η̄\η̄_{2}}(i)
=^{(4.1)} ε_{ζ[d]} · ∑_{i∈[0,t−1]}(−1)^{i} (\begin{matrix}s−u+d \\ t−i \end{matrix}) B′_{[a^{ζ[d]}],ξ̄∪η̄_{2},η̄_{2},η̄\η̄_{2}}(i).

Remark 4.8 The morphism F^{λ} ^{\overline{}^{f′′}} S^{μ} maps [ǎ_{λ}] to a linear combination of standard μ-polytabloids with coefficients ±1.

We summarize.

Theorem 4.9 Let m_{0} := λ′_{g} − λ′_{g+1} + d + 1 be the box shift length, let

m := m_{0} · ∏_{p prime, p|m_{0}}p^{− min(v_{p}(m_{0}), ilog_{p}(d))},

where ilog_{p}(k) = max\{i ⩾ 0 | p^{i} ⩽ k\}. The ZS_{n}-linear map M^{λ′,−} ^{\overline{}^{f′}} S^{μ} factors over

[figure: M^{λ′,−}; f′; S^{μ}; {a′}^{−}; ⟨a⟩; ν_{S}^{λ}; ⟨b⟩; ⟨b⟩ + mS^{μ},; S^{λ}; f; S^{μ}/m]

The resulting morphism S^{λ} ^{\overline{}^{f}} S^{μ}/m is of order m in Hom_{ZS_{n}}(S^{λ}, S^{μ}/m). In case μ′ is 2-regular, the group Hom_{ZS_{n}}(S^{λ}, S^{μ}/m) is generated by f. In case μ′ is 2-singular, the group 2 · Hom_{ZS_{n}}(S^{λ}, S^{μ}/m) is generated by 2f. In case 2d − 1 ⩽ λ′_{g+1} and μ′ is 2-regular, there is an isomorphism

Hom_{ZS_{n}}(S^{λ}, S^{μ}/m)  \overline{}^{(−)·(n!/m)}_{∼}  Hom_{ZS_{n}}(S^{λ}, S^{μ}/n!).

In case 2d − 1 ⩽ λ′_{g+1} and μ′ is 2-singular, the cokernel of the inclusion

Hom_{ZS_{n}}(S^{λ}, S^{μ}/m)  \overline{}^{(−)·(n!/m)}_{∼}  Hom_{ZS_{n}}(S^{λ}, S^{μ}/n!),

is annihilated by multiplication with 2. This follows by (4.5, 4.7, 4.2, 4.8, 4.4, 4.6).

Example 4.10 We have Hom_{ZS_{8}}(S^{(2^{3},1^{2})}, S^{(1^{8})}/8!) = 0.

Question 4.11 In case 2d−1 > λ′_{g+1}, I do not know of a counterexample to the assertions in (4.9) that presuppose 2d − 1 ⩽ λ′_{g+1}.

Remark 4.12 In the situation of (4.9), Carter and Payne [CP 80] have shown that

Hom_{KS_{n}}(K ⊗_{Z} S^{λ}, K ⊗_{Z} S^{μ}) ≠ 0,

K being an infinite field of characteristic p such that ilog_{p}d < v_{p}(m_{0}). This part of their result is recovered by (4.9).

Corollary 4.13 (to (4.9) via (3.16, 4.4)) The transpose in the sense of (3.16) of the morphism given in (4.9), interpreted as S^{λ}/m ^{\overline{}^{f}} S^{μ}/m, is given by

[figure: S^{μ′}/m; ft; S^{λ′}/m; ⟨b′⟩; −⟨b′⟩Θ_{φ},]

where the correspondoid {φ} ∈ {Φ^{λ′,μ′}} is given by

[μ′] ^{\overline{}^{\{φ\}}} N
j × i \overline{}j  for j × i ∈ [λ′] ∩ [μ′]
g × (λ′_{g} + i) \overline{}g + 1 for i ∈ [1, d]

All assertions on the Hom-groups in (4.9) have a counterpart, obtained by isomophic transport via the transposition isomorphism (3.16).

This follows by (4.9, 3.16) using (4.4). Concerning Θ_{φ}, we refer to (3.1).

Question 4.14 I do not know a formula for the transpose f^{t} in (4.13) in terms of λ′- polytabloids. Cf. (3.17, 4.15).

Example 4.15 (a fixed point, cf. [J 78, 24.4], [K 99, 4.4.1]) Consider the case n ⩾ 4, d = 2, λ = (2^{2}, 1^{n−4}) and μ = (1^{n}). The modulus becomes m = n − 1 if n − 1 is odd and m = (n − 1)/2 if n − 1 is even. We obtain

[figure: [μ′]; \{φ\}; N; 1 × i; \{ \begin{matrix}1 & for i ∈ [1, n − 2] \\ 2 & for i ∈ [n − 1, n], \end{matrix}]

which allows to calculate the transpose as mapping

[figure: S^{μ′}/m; ft; S^{λ′}/m; ⟨x⟩; − ∑_{i,j∈[1,n], i<j}{i, j},]

where [x] is the unique standard μ′-tableau and where we denote a λ′-tabloid by its second row. Similarly, for a λ′-polytabloid, we omit to denote the entries in the first row from column three onwards. I.e. we let a λ′-polytabloid be determined by the first two entries of the first and of the second row - which is not to be confused with a (2, 2)-polytabloid. Let

A := ∑_{j∈[4,n]}(j − 2) · ⟨\begin{matrix}1 & 3 \\ 2 & j \end{matrix}⟩ − ∑_{i,j∈[3,n], i<j} ⟨\begin{matrix}1 & 2 \\ i & j \end{matrix}⟩ ∈ S^{λ′}.

For k ∈ [3, n], l ∈ [4, n], k < l, we obtain the following list of values of the scalar product in M^{λ′}.

\begin{aligned}(A, \{k, l\}) & = & + & 0 & − & 1 \\ (A, \{1, l\}) & = & − & (l − 2) & + & (l − 3) \\ (A, \{2, l\}) & = & + & (l − 2) & + & (n − l) \\ (A, \{2, 3\}) & = & − & (n − 2)(n − 1)/2 + 1 & + & (n − 3) \\ (A, \{1, 2\}) & = & + & 0 & − & (n − 2)(n − 3)/2 \\ (A, \{1, 3\}) & = & + & (n − 2)(n − 1)/2 − 1 & + & 0. \end{aligned}

Hence, written in polytabloids, our map turns out to be

[figure: S^{μ′}/m; ft; S^{λ′}/m; ⟨x⟩; ∑_{j∈[4,n]}(j − 2) · ⟨\begin{matrix}1 & 3 \\ 2 & j \end{matrix}⟩ − ∑_{i,j∈[3,n], i<j} ⟨\begin{matrix}1 & 2 \\ i & j \end{matrix}⟩ .]

A Two lemmata

We present full versions of the two main lemmata of §2.2. For sake of completeness of the full versions, there is a certain overlap, concerning notation and conclusion, with their abridged versions (2.4, 2.10). We maintain the notation of §2.2.

Lemma A.1 (full version of (2.4)) Suppose g < p < k and s, t ⩾ 2. The map f′ annihilates G′_{[a],ξ,η}.

We fix a map

[figure: [1, λ_{1}]\{p, p + 1}; ẽ; [0, 2]; j; ẽ_{j}]

that maps g and k + 1 to e_{g} = e_{k+1} = 2, and that maps j ∈ [1, λ_{1}]\[g, k + 1] to e_{j} = 0. For α, β ∈ [0, 2], we denote by ẽαβ be the prolongation of ẽ to [1, λ_{1}] defined by ẽαβ|_{[1,λ_{1}]\{p,p+1}} := ẽ, (ẽαβ)_{p} := α and (ẽαβ)_{p+1} := β. We contend that

∑_{α,β∈[0,2]}X_{p}^{(2−α)}X_{p+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽαβ} = 0,

from which the lemma ensues.

Given α, β ∈ [0, 2], given a subset Ξ̃ ⊆ [1, 2] × {p, p + 1} with

\begin{aligned}#(Ξ̃ ∩ ([1, 2] × \{p\})) & = & α \\ #(Ξ̃ ∩ ([1, 2] × \{p + 1\})) & = & β, \end{aligned}

and given prescribed inverse images ξ^{−}, η^{−} ⊆ [1, 2] ‘within Ξ̃’, i.e. such that

(∗) \begin{matrix}ξ^{−} × \{p\} & ⊆ & Ξ̃ ∩ ([1, 2] × \{p\}) \\ η^{−} × \{p + 1\} & ⊆ & Ξ̃ ∩ ([1, 2] × \{p + 1\}), \end{matrix}

we let

\begin{aligned}Γ̇(ẽ, Ξ̃, ξ^{−}, η^{−}) & := & \{γ ∈ Γ̇(ẽαβ) & | & Ξ_{γ} ∩ ([1, 2] × \{p, p + 1\}) = Ξ̃, \\ & & & & γ^{−1}(ξ) = ξ^{−} × \{p\}, γ^{−1}(η) = η^{−} × \{p + 1\}\} \end{aligned}

and form the partial sum

Λ([a], ẽ, Ξ̃, ξ^{−}, η^{−}) := \frac{1}{s!t!} ∑_{γ∈Γ̇(ẽ,Ξ̃,ξ^{−},η^{−})}⟨a^{γ}⟩ε_{γ} ∘ (ξ ∪ η)

so that we can recover

(∗∗)  G′_{[a],ξ,η}f′_{ẽαβ} = ∑_{Ξ̃⊆[1,2]×{p,p+1}} ∑_{ξ^{−},η^{−}⊆[1,2] subject to (∗)}Λ([a], ẽ, Ξ̃, ξ^{−}, η^{−}).

There exist elements x, y ∈ ξ, x ≠ y, x′, y′ ∈ η, x′ ≠ y′, z ∈ ξ̄, which we choose and fix. For γ ∈ Γ̇(ẽ00), we let x_{γ} := a_{j,γ̄(j,1)}, where j ∈ [p + 2, k + 1] is minimal with 1 × j ∈ Ξ_{γ}, and y_{γ} := a_{j,γ̄(j,2)}, where j ∈ [p + 2, k + 1] is minimal with 2 × j ∈ Ξ_{γ}. I.e. we pick the entries x_{γ}, y_{γ} that ‘cross the columns’ p and p + 1 under the operation of γ. We write

\begin{aligned}U_{γ} & := & (s + t − 2)!^{−1} · ⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (x_{γ}, x, x′) · (y_{γ}, y, y′) ∘ (ξ ∪ η) \\ V_{1,γ} & := & (s + t − 1)!^{−1} · ⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) ∘ (ξ ∪ η) \\ V_{2,γ} & := & (s + t − 1)!^{−1} · ⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (y_{γ}, y, y′) ∘ (ξ ∪ η), \end{aligned}

and let

A := ∑_{γ∈Γ̇(ẽ00)}U_{γ} · ∑_{w′∈η̄}(w′, z)
B := ∑_{γ∈Γ̇(ẽ00)}U_{γ} · (1 − ∑_{w∈ξ̄\z}(w, z))
C_{1} := ∑_{γ∈Γ̇(ẽ00)}U_{γ} · (z, x_{γ})
C_{2} := ∑_{γ∈Γ̇(ẽ00)}U_{γ} · (z, y_{γ})
D := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ} · ∑_{w∈ξ̄}(w, y_{γ})
= ∑_{γ∈Γ̇(ẽ00)}V_{2,γ} · ∑_{w∈ξ̄}(w, x_{γ})
H := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ} · ∑_{w′∈η̄}(w′, y_{γ})
= ∑_{γ∈Γ̇(ẽ00)}V_{2,γ} · ∑_{w′∈η̄}(w′, x_{γ})
F_{1} := ∑_{γ∈Γ̇(ẽ00)}V_{1,γ}
F_{2} := ∑_{γ∈Γ̇(ẽ00)}V_{2,γ}.

The equalities herein follow by the argument of (2.2 i) and by the independence of of the respective expressions of the choice of x, y, x′, y′.

Calculation of Λ-values. We shall distinguish subcases and subsubcases according to the summation in (∗∗).

To distinguish subcases, e.g. [\begin{matrix}+ & − \\ + & + \end{matrix}] designates the subset Ξ̃ = \{1 × p, 2 × p, 2 × (p + 1)\}; in general, the first factor of an element of [1, 2] × {p, p + 1} counts rows, the second counts columns, and a plus sign + denotes its appearance in Ξ̃.

To distinguish subsubcases, we denote e.g. by [\begin{matrix}⊕ & − \\ + & ⊕ \end{matrix}] the configuration Ξ̃ = [\begin{matrix}+ & − \\ + & + \end{matrix}], ξ^{−} = \{1\} and η^{−} = \{2\}; in general, the sign ⊕ in the left column means that its row number is an element of ξ^{−}, the sign ⊕ in the right column means that its row number is an element of η^{−}. Concerning the underlying set Ξ̃, the symbols ⊕ and + are synonymous.

Case f′_{ẽ22}.

Subcase Ξ̃ = [\begin{matrix}+ & + \\ + & + \end{matrix}].

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ ⊕ & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {1, 2}, {1, 2})
= \frac{1}{s!t!} ∑_{γ∈Γ̇(e,Ξ̃,\{1,2\},\{1,2\})}⟨a^{γ}⟩ε_{γ} ∘ (ξ ∪ η)
= \frac{1}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}≠y_{1}∈ξ, x′_{1}≠y′_{1}∈η}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}, x′_{1}) · (y_{γ}, y_{1}, y′_{1}) ∘ (ξ ∪ η)
= \frac{s(s−1)t(t−1)}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) ∘ (ξ ∪ η)

=^{(1.5)} \frac{s(s−1)t(t−1)·s!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (x_{γ}, x, x′) · (y_{γ}, y, y′) ∘ (ξ ∪ η) ∘ ξ̄
= s(s − 1)B.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ ⊕ & + \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {1, 2}, {1})

= \frac{s(s−1)t}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′_{1}) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s(s−1)t·(s−1)!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(x_{γ}, x, x′) · (y_{γ}, y, y′_{1}) · (ξ̄\z, η\{x′, y′}) · (y′_{1}, y′) ∘ (ξ ∪ η) ∘ (ξ̄∪ y′_{1})
= −\frac{s−1}{(t−1)!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(x_{γ}, x, x′) · (y_{γ}, y, y′_{1}) · (ξ̄\z, η\{x′, y′}) · (y′_{1}, y′) · (y′_{1}, y_{γ}) ∘ (ξ ∪ η) ∘ (ξ̄∪ y′_{1})
= −\frac{s−1}{(t−1)!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ (ξ̄∪ y′_{1})
= −(s − 1)(uB − A).

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ ⊕ & ⊕ \end{matrix}]. By symmetry, we obtain from subsubcase [\begin{matrix}⊕ & ⊕ \\ ⊕ & + \end{matrix}]

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {1, 2}, {2}) = −(s − 1)(uB − A).

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ ⊕ & + \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {1, 2}, {})

= \frac{s(s−1)}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y, y′_{1}) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s(s−1)·(s−2)!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(x_{γ}, x, x′_{1}) · (y_{γ}, y, y′_{1}) · (ξ̄\z, η\{x′, y′}) · (x′_{1}, x′) · (y′_{1}, y′) ∘ (ξ ∪ η) ∘ (ξ̄∪ x′_{1} ∪ y′_{1})
= \frac{1}{t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(x_{γ}, x, x′_{1}) · (y_{γ}, y, y′_{1}) · (ξ̄\z, η\{x′, y′}) · (x′_{1}, x′) · (y′_{1}, y′) · (x′_{1}, x_{γ}) · (y′_{1}, y_{γ})∘
∘(ξ ∪ η) ∘ (ξ̄∪ x′_{1} ∪ y′_{1})
= \frac{1}{t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ (ξ̄∪ x′_{1} ∪ y′_{1})
= (u − 1)(uB − 2A).

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ + & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {1}, {1, 2})

= \frac{st(t−1)}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y_{1}, y′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{st(t−1)·(s+1)!(t−2)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y_{1}, y′) · (ξ̄\y_{1}, η\\{x′, y′\}) ∘ (ξ ∪ η)
= \frac{s(s+1)}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)
= (s + 1)sD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ + & + \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {1}, {1})
= \frac{st}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y_{1}, y′_{1}) ∘ (ξ ∪ η)

=^{(1.5)} \frac{st·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y_{1}, y′_{1}) · (ξ̄\y_{1}, η\\{x′, y′\}) · (y′_{1}, y′) ∘ (ξ ∪ η)
= −\frac{s}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) · (y_{γ}, y_{1}, y′, y′_{1}) · (y′_{1}, y′) · (y_{γ}, y′_{1}) ∘ (ξ ∪ η)
= −\frac{su}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)
= −suD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ + & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {1}, {2})
= \frac{st}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}, y′) ∘ (ξ ∪ η)

=^{(1.5)} \frac{st·s!(t−1)!}{s!t!(s+1−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}, y′) · (ξ̄\y_{1}, η\\{x′, y′\}) · (x′_{1}, x′) ∘ (ξ ∪ η)
= −\frac{s}{(s+1−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}) · (x′_{1}, x′) · (x_{γ}, x′_{1}) ∘ (ξ ∪ η)
= −\frac{su}{(s+1−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)
= −suD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ + & + \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {1}, {})
= \frac{s}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}, y′_{1}) ∘ (ξ ∪ η)

=^{(1.5)} \frac{s·(s−1)!t!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}, y′_{1}) · (ξ̄\y_{1}, η\{x′, y′}) · (x′_{1}, x′) · (y′_{1}, y′) ∘ (ξ ∪ η)
= \frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(ξ̄, η\x′) · (x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}, y′, y′_{1}) · (x′_{1}, x′) · (y′_{1}, y′) · (x_{γ}, x′_{1}) · (y_{γ}, y′_{1}) ∘ (ξ ∪ η)
= \frac{u(u−1)}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)
= u(u − 1)D.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & ⊕ \\ ⊕ & ⊕ \end{matrix}]. By symmetry, we obtain from subsubcase [\begin{matrix}⊕ & ⊕ \\ + & ⊕ \end{matrix}]

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {2}, {1, 2}) = (s + 1)sD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & ⊕ \\ ⊕ & + \end{matrix}]. By symmetry, we obtain from subsubcase [\begin{matrix}⊕ & + \\ + & ⊕ \end{matrix}]

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {2}, {1}) = −suD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & + \\ ⊕ & ⊕ \end{matrix}]. By symmetry, we obtain from subsubcase [\begin{matrix}⊕ & ⊕ \\ + & + \end{matrix}]

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {2}, {2}) = −suD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & + \\ ⊕ & + \end{matrix}]. By symmetry, we obtain from subsubcase [\begin{matrix}⊕ & + \\ + & + \end{matrix}]

Λ([a], ẽ,^{+}^{+}_{+}_{+}, {2}, {}) = u(u − 1)D.

Subsubcases [\begin{matrix}+ & ⊕ \\ + & ⊕ \end{matrix}], [\begin{matrix}+ & ⊕ \\ + & + \end{matrix}], [\begin{matrix}+ & + \\ + & ⊕ \end{matrix}], [\begin{matrix}+ & + \\ + & + \end{matrix}]. The Λ-value is zero by the Garnir relations. Summing up Λ-values over subcases and subsubcases, we obtain

G′_{[a],ξ,η}f′_{ẽ22} = 2(s − u)A + (s − u)(s − u − 1)B + 2(s − u)(s − u + 1)D

Case f′_{ẽ12}. Subcase Ξ̃ = [\begin{matrix}+ & + \\ − & + \end{matrix}]. Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ − & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{−}_{+}, {1}, {1, 2})
= \frac{st(t−1)}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{st(t−1)·s!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ ξ̄
= −\frac{s}{(t−2)!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ ξ̄
= −sB.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ − & + \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{−}_{+}, {1}, {1})
= \frac{st}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y′_{1}) ∘ (ξ ∪ η)
=^{(1.5)} \frac{st·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y′_{1}) · (ξ̄, η\x′) ∘ (ξ ∪ η)
= sH.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ − & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{−}_{+}, {1}, {2})
= \frac{st}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{st·(s−1)!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y′) · (ξ̄\z, η\\{x′, y′\}) · (x′_{1}, x′) ∘ (ξ ∪ η) ∘ (ξ̄∪ x′_{1})
= \frac{1}{(t−1)!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(x_{γ}, x, x′_{1}) · (y_{γ}, y, y′) · (ξ̄\z, η\{x′, y′}) · (x′_{1}, x′) · (x′_{1}, x_{γ}) ∘ (ξ ∪ η) ∘ (ξ̄∪ x′_{1})
= \frac{1}{(t−1)!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ (ξ̄∪ x′_{1})
= uB − A.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ − & + \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{−}_{+}, {1}, {})
= \frac{s}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y′_{1}) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s·(s−1)!t!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y′_{1}) · (ξ̄, η\x′) · (x′_{1}, x′) ∘ (ξ ∪ η)
= −\frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y′_{1}) · (ξ̄, η\x′) · (x′_{1}, x′) · (x′_{1}, x_{γ}) ∘ (ξ ∪ η)
= −\frac{u−1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y′_{1}) · (ξ̄, η\x′) ∘ (ξ ∪ η)
= −(u − 1)H.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & ⊕ \\ − & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{−}_{+}, {}, {1, 2})
= \frac{t(t−1)}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}, x′) · (y_{γ}, y′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{t(t−1)·(s+1)!(t−2)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}, x′) · (y_{γ}, y′) · (ξ̄\x_{1}, η\\{x′, y′\}) ∘ (ξ ∪ η)
= −\frac{s+1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (x_{γ}, x_{1}) · (y_{γ}, y, y′) ∘ (ξ ∪ η)
= −(s + 1)D.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & + \\ − & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{−}_{+}, {}, {2})
= \frac{t}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}, x′_{1}) · (y_{γ}, y′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{t·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}, x′_{1}) · (y_{γ}, y′) · (ξ̄\x_{1}, η\\{x′, y′\}) · (x′_{1}, x′) ∘ (ξ ∪ η)
= \frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (x_{γ}, x_{1}, x′, x′_{1}) · (y_{γ}, y, y′) · (x′_{1}, x′) · (x′_{1}, x_{γ}) ∘ (ξ ∪ η)
= \frac{u}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (x_{γ}, x_{1}) · (y_{γ}, y, y′) ∘ (ξ ∪ η)
= uD.

Subsubcases [\begin{matrix}+ & ⊕ \\ − & + \end{matrix}], [\begin{matrix}+ & + \\ − & + \end{matrix}]. The Λ-value is zero by the Garnir relations. Subcase Ξ̃ = [\begin{matrix}− & + \\ + & + \end{matrix}]. Obtained, by symmetry, from subcase Ξ̃ = [\begin{matrix}+ & + \\ − & + \end{matrix}]. Summing up Λ-values over subcases and subsubcases, we obtain

G′_{[a],ξ,η}f′_{ẽ12} = −2A − 2(s − u)B − 2(s − u + 1)D + 2(s − u + 1)H.

Case f′_{ẽ02}. Subcase Ξ̃ = [\begin{matrix}− & + \\ − & + \end{matrix}].

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}− & ⊕ \\ − & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{−}^{+}_{−}_{+}, {}, {1, 2})
= \frac{t(t−1)}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{t(t−1)·s!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ ξ̄
= \frac{1}{(t−2)!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ ξ̄
= B.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}− & ⊕ \\ − & + \end{matrix}]. We calculate

Λ([a], ẽ,^{−}^{+}_{−}_{+}, {}, {1})
= \frac{t}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′_{1}) ∘ (ξ ∪ η)
=^{(1.5)} \frac{t·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′_{1}) · (ξ̄, η\x′) ∘ (ξ ∪ η)
= −\frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y′_{1}) · (ξ̄, η\x′) ∘ (ξ ∪ η)
= −H.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}− & + \\ − & ⊕ \end{matrix}]. By symmetry, we obtain from subsubcase [\begin{matrix}− & ⊕ \\ − & + \end{matrix}] Λ([a], ẽ,^{−}^{+}_{−}_{+}, {}, {2}) = −H.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}− & + \\ − & + \end{matrix}]. The Λ-value is zero by the Garnir relations. Summing up Λ-values over subcases and subsubcases, we obtain

G′_{[a],ξ,η}f′_{ẽ02} = B − 2H.

Case f′_{ẽ21}. Subcase Ξ̃ = [\begin{matrix}+ & + \\ + & − \end{matrix}]. Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ ⊕ & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{−}, {1, 2}, {1})
= \frac{s(s−1)t}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s(s−1)t·(s−1)!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y) · (ξ̄\z, η\\{x′, y′\}) · (y_{γ}, y′) ∘ (ξ ∪ η) ∘ (ξ̄∪ y_{γ})
= \frac{(s−1)}{(t−1)!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ (ξ̄∪ y_{γ})
= (s − 1)(B − C_{2}).

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ ⊕ & − \end{matrix}]. We calculate

\begin{aligned}& Λ([a], ẽ,^{+}^{+}_{+}_{−}, \{1, 2\}, \{\}) \\ = & \frac{s(s−1)}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y) ∘ (ξ ∪ η) \end{aligned}

=^{(1.5)} \frac{s(s−1)·(s−2)!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(x_{γ}, x, x′_{1}) · (y_{γ}, y) · (ξ̄\z, η\{x′, y′}) · (x′_{1}, x′) · (y_{γ}, y′) ∘ (ξ ∪ η) ∘ (ξ̄∪ x′_{1} ∪ y_{γ})
= −\frac{1}{t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y)·
·(ξ̄\z, η\{x′, y′}) · (x′_{1}, x′) · (y_{γ}, y′) · (x′_{1}, x_{γ}) ∘ (ξ ∪ η) ∘ (ξ̄∪ x′_{1} ∪ y_{γ})
= −\frac{1}{t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ (ξ̄∪ x′_{1} ∪ y_{γ})
= −(uB − uC_{2} − A).

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ + & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{−}, {1}, {1})
= \frac{st}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)

=^{(1.5)} \frac{st·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y_{1}) · (ξ̄\y_{1}, η\\{x′, y′\}) · (y_{γ}, y′) ∘ (ξ ∪ η)
= \frac{s}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)
= sD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ + & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{−}, {1}, {})
= \frac{s}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)

=^{(1.5)} \frac{s·(s−1)!t!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·
·(x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}) · (ξ̄\y_{1}, η\{x′, y′}) · (y_{γ}, y′) · (x′_{1}, x′) ∘ (ξ ∪ η)
= −\frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′_{1}) · (y_{γ}, y_{1}) · (x′_{1}, x′) · (x_{γ}, x′_{1}) ∘ (ξ ∪ η)
= −\frac{u}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)
= −uD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & ⊕ \\ ⊕ & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{−}, {2}, {1})
= \frac{st}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}, x′) · (y_{γ}, y) ∘ (ξ ∪ η)

=^{(1.5)} \frac{st·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}, x′) · (y_{γ}, y) · (ξ̄\x_{1}, η\\{x′, y′\}) · (y_{γ}, y′) ∘ (ξ ∪ η)
= \frac{s}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (x_{γ}, x_{1}) · (y_{γ}, y, y′) ∘ (ξ ∪ η)
= sD.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & + \\ ⊕ & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{+}_{−}, {2}, {})
= \frac{s}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}, x′_{1}) · (y_{γ}, y) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s·(s−1)!t!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·

·(x_{γ}, x_{1}, x′_{1}) · (y_{γ}, y) · (ξ̄\x_{1}, η\{x′, y′}) · (y_{γ}, y′) · (x′_{1}, x′) ∘ (ξ ∪ η)

= −\frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄, x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (x_{γ}, x_{1}, x′, x′_{1}) · (y_{γ}, y, y′) · (x′_{1}, x′) · (x_{γ}, x′_{1}) ∘ (ξ ∪ η)
= −\frac{u}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (x_{γ}, x_{1}) · (y_{γ}, y, y′) ∘ (ξ ∪ η)
= −uD.

Subsubcases [\begin{matrix}+ & ⊕ \\ + & − \end{matrix}], [\begin{matrix}+ & + \\ + & − \end{matrix}]. The Λ-value is zero by the Garnir relations. Subcase Ξ̃ = [\begin{matrix}+ & − \\ + & + \end{matrix}]. Obtained, by symmetry, from subcase Ξ̃ = [\begin{matrix}+ & + \\ + & − \end{matrix}]. Summing up Λ-values over subcases and subsubcases, we obtain

G′_{[a],ξ,η}f′_{ẽ21} = 2A + 2(s − u − 1)B − (s − u − 1)(C_{1} + C_{2}) + 4(s − u)D.

Case f′_{ẽ11}. Subcase Ξ̃ = [\begin{matrix}+ & + \\ − & − \end{matrix}]. Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & ⊕ \\ − & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{−}_{−}, {1}, {1})
= \frac{st}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{st·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (ξ̄, η\x′) ∘ (ξ ∪ η)
= sF_{1}.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & + \\ − & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{+}_{−}_{−}, {1}, {})
= \frac{s}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s·(s−1)!t!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (ξ̄, η\x′) · (x′_{1}, x′) ∘ (ξ ∪ η)
= −\frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′_{1}) · (ξ̄, η\x′) · (x′_{1}, x′) · (x′_{1}, x_{γ}) ∘ (ξ ∪ η)
= −\frac{u}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) ∘ (ξ ∪ η)
= −uF_{1}.

Subsubcases [\begin{matrix}+ & ⊕ \\ − & − \end{matrix}], [\begin{matrix}+ & + \\ − & − \end{matrix}]. The Λ-value is zero by the Garnir relations. Subcase Ξ̃ = [\begin{matrix}− & − \\ + & + \end{matrix}]. Obtained, by symmetry, from subcase Ξ̃ = [\begin{matrix}+ & + \\ − & − \end{matrix}]. Subcase Ξ̃ = [\begin{matrix}+ & − \\ − & + \end{matrix}].

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & − \\ − & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{−}_{−}_{+}, {1}, {2})
= \frac{st}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) · (y_{γ}, y′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{st·(s−1)!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) · (y_{γ}, y′) · (ξ̄\z, η\\{x′, y′\}) · (x_{γ}, x′) ∘ (ξ ∪ η) ∘ (ξ̄∪ x_{γ})
= −\frac{1}{(t−1)!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ (ξ̄∪ x_{γ})
= −(B − C_{1}).

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & − \\ − & ⊕ \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{−}_{−}_{+}, {}, {2})
= \frac{t}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}) · (y_{γ}, y′) ∘ (ξ ∪ η)
=^{(1.5)} \frac{t·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x_{1}) · (y_{γ}, y′) · (ξ̄\x_{1}, η\\{x′, y′\}) · (x_{γ}, x′) ∘ (ξ ∪ η)
= −\frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{x_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\y′) · (x_{γ}, x_{1}) · (y_{γ}, y, y′) ∘ (ξ ∪ η)
= −D.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & − \\ − & + \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{−}_{−}_{+}, {1}, {})
= \frac{s}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) · (y_{γ}, y′_{1}) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s·(s−1)!t!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) · (y_{γ}, y′_{1}) · (ξ̄, η\x′) · (x_{γ}, x′) ∘ (ξ ∪ η)
= \frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y′_{1}) · (ξ̄, η\x′) ∘ (ξ ∪ η)
= H.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & − \\ − & + \end{matrix}]. The Λ-value is zero by the Garnir relations. Subcase Ξ̃ = [\begin{matrix}− & + \\ + & − \end{matrix}]. Obtained, by symmetry, from subcase Ξ̃ = [\begin{matrix}+ & − \\ − & + \end{matrix}]. Summing up Λ-values over subcases and subsubcases, we obtain

G′_{[a],ξ,η}f′_{ẽ11} = −2B + (C_{1} + C_{2}) − 2D + 2H + (s − u)(F_{1} + F_{2}).

Case f′_{ẽ01}. Subcase Ξ̃ = [\begin{matrix}− & + \\ − & − \end{matrix}]. Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}− & ⊕ \\ − & − \end{matrix}]. We calculate

\begin{aligned}& Λ([a], ẽ,^{−}^{+}_{−}_{−}, \{\}, \{1\}) \\ = & \frac{t}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) ∘ (ξ ∪ η) \end{aligned}

\begin{matrix}=^{(1.5)} & \frac{t·s!(t−1)!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (ξ̄, η\x′) ∘ (ξ ∪ η) \\ = & −\frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (ξ̄, η\x′) ∘ (ξ ∪ η) \\ = & −F_{1}. \end{matrix}

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}− & + \\ − & − \end{matrix}]. The Λ-value is zero by the Garnir relations. Subcase Ξ̃ = [\begin{matrix}− & − \\ − & + \end{matrix}]. Obtained, by symmetry, from subcase Ξ̃ = [\begin{matrix}− & + \\ − & − \end{matrix}]. Summing up Λ-values over subcases and subsubcases, we obtain

G′_{[a],ξ,η}f′_{ẽ01} = −(F_{1} + F_{2}).

Case f′_{ẽ20}. Subcase Ξ̃ = [\begin{matrix}+ & − \\ + & − \end{matrix}]. Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & − \\ ⊕ & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{−}_{+}_{−}, {1, 2}, {})
= \frac{s(s−1)}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) · (y_{γ}, y) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s(s−1)(s−2)!}{s!t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ}·

·(x_{γ}, x) · (y_{γ}, y) · (ξ̄\z, η\{x′, y′}) · (x_{γ}, x′) · (y_{γ}, y′) ∘ (ξ ∪ η) ∘ (ξ̄∪ x_{γ} ∪ y_{γ})

= \frac{1}{t!(s+t−2)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (y_{γ}, y, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ (ξ ∪ η) ∘ (ξ̄∪ x_{γ} ∪ y_{γ})
= B − C_{1} − C_{2}.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & − \\ + & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{−}_{+}_{−}, {1}, {})
= \frac{s}{s!t!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s(s−1)!t!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) · (y_{γ}, y_{1}) · (ξ̄\y_{1}, η\\{x′, y′\}) · (x_{γ}, x′) · (y_{γ}, y′) ∘ (ξ ∪ η)
= \frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)} ∑_{y_{1}∈ξ̄}⟨a^{γ}⟩ε_{γ} · (ξ̄, η\x′) · (x_{γ}, x, x′) · (y_{γ}, y_{1}) ∘ (ξ ∪ η)
= D.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & − \\ ⊕ & − \end{matrix}]. By symmetry, we obtain from subsubcase [\begin{matrix}⊕ & − \\ + & − \end{matrix}] Λ([a], ẽ,^{+}^{−}_{+}_{−}, {2}, {}) = D.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & − \\ + & − \end{matrix}]. The Λ-value is zero by the Garnir relations. Summing up Λ-values over subcases and subsubcases, we obtain

G′_{[a],ξ,η}f′_{ẽ20} = B − (C_{1} + C_{2}) + 2D.

Case f′_{ẽ10}. Subcase Ξ̃ = [\begin{matrix}+ & − \\ − & − \end{matrix}].

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}⊕ & − \\ − & − \end{matrix}]. We calculate

Λ([a], ẽ,^{+}^{−}_{−}_{−}, {1}, {})

= \frac{s}{s!t!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) ∘ (ξ ∪ η)
=^{(1.5)} \frac{s(s−1)!t!}{s!t!(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x) · (ξ̄, η\x′) · (x_{γ}, x′) ∘ (ξ ∪ η)
= \frac{1}{(s+t−1)!} ∑_{γ∈Γ̇(ẽ00)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x, x′) · (ξ̄, η\x′) ∘ (ξ ∪ η)
= F_{1}.

Subsubcase (Ξ̃, ξ^{−}, η^{−}) = [\begin{matrix}+ & − \\ − & − \end{matrix}]. The Λ-value is zero by the Garnir relations. Subcase Ξ̃ = [\begin{matrix}− & − \\ + & − \end{matrix}]. Obtained, by symmetry, from subcase Ξ̃ = [\begin{matrix}+ & − \\ − & − \end{matrix}]. Summing up Λ-values over subcases and subsubcases, we obtain

G′_{[a],ξ,η}f′_{ẽ10} = (F_{1} + F_{2}).

Case f′_{ẽ00}. The Garnir relations yield

G′_{[a],ξ,η}f′_{ẽ00} = 0.

We note that s − u = X_{p} − X_{p+1} and evaluate the linear combination

∑_{α,β∈[0,2]}X_{p}^{(2−α)}X_{p+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽαβ} ; \begin{aligned}= & 1 & ·1 & ·(2(X_{p} − X_{p+1})A + (X_{p} − X_{p+1})(X_{p} − X_{p+1} − 1)B \\ & & & +2(X_{p} − X_{p+1})(X_{p} − X_{p+1} + 1)D) \\ + & X_{p} & ·1 & ·( − 2A − 2(X_{p} − X_{p+1})B − 2(X_{p} − X_{p+1} + 1)D \\ & & & +2(X_{p} − X_{p+1} + 1)H) \\ + & X_{p}(X_{p} + 1) & ·1 & ·(B − 2H) \\ + & 1 & ·X_{p+1} & ·(2A + 2(X_{p} − X_{p+1} − 1)B \\ & & & −(X_{p} − X_{p+1} − 1)(C_{1} + C_{2}) + 4(X_{p} − X_{p+1})D) \\ + & X_{p} & ·X_{p+1} & ·( − 2B + (C_{1} + C_{2}) − 2D + 2H \\ & & & +(X_{p} − X_{p+1})(F_{1} + F_{2})) \\ + & X_{p}(X_{p} + 1) & ·X_{p+1} & ·( − (F_{1} + F_{2})) \\ + & 1 & ·X_{p+1}(X_{p+1} + 1) & ·(B − (C_{1} + C_{2}) + 2D) \\ + & X_{p} & ·X_{p+1}(X_{p+1} + 1) & ·((F_{1} + F_{2})) \\ + & X_{p}(X_{p} + 1) & ·X_{p+1}(X_{p+1} + 1) & ·(0) \\ = & 0. \end{aligned}

Lemma A.2 (full version of (2.10)) Suppose g = p < k and s, t ⩾ 2. There exist elements x′, y′ ∈ η, x′ ≠ y′, z ∈ ξ̄, which we choose and fix. For γ ∈ Γ̇(ẽ0), we let x_{γ} := a_{j,γ̄(j,1)}, where j ∈ [g + 2, k + 1] is minimal with 1 × j ∈ Ξ_{γ}, and y_{γ} := a_{j,γ̄(j,2)}, where j ∈ [g + 2, k + 1] is minimal with 2 × j ∈ Ξ_{γ}. I.e. we pick the entries x_{γ}, y_{γ} that ‘cross the column’ g + 1 under the operation of γ. The set of maps

[figure: [1, λ_{1}]\{g + 1}; ẽ; [0, 2]; j; ẽ_{j}]

that send g and k + 1 to ẽ_{g} = ẽ_{k+1} = 2, and that map j ∈ [1, λ_{1}]\[g, k + 1] to e_{j} = 0, is denoted by Ẽ. For ẽ ∈ Ẽ and β ∈ [0, 2], we denote by ẽβ the prolongation of ẽ to [g, k + 1] by (ẽβ)_{g+1} = β. For γ ∈ Γ̇(ẽ0), we write

U_{γ} := ⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\{x′, y′}) · (x_{γ}, x′) · (y_{γ}, y′)

and let

\begin{aligned}A_{[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · ∑_{w′∈η̄}(w′, z) \\ B_{[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (1 − ∑_{w∈ξ̄\z}(w, z)) \\ C_{1,[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (z, x_{γ}) \\ C_{2,[a],ξ,η,ẽ} & := & ∑_{γ∈Γ̇(ẽ0)}U_{γ} · (z, y_{γ}). \end{aligned}

We obtain

\begin{matrix}G′_{[a],ξ,η}f′ = (X_{g} + 2)∑_{ẽ∈Ẽ}( ∏_{j∈[g+2,k]}X_{j}^{(2−ẽ_{j})})(2A_{[a],ξ,η,ẽ} + (X_{g} + 1)B_{[a],ξ,η,ẽ} \\ −X_{g+1}(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ})). \end{matrix}

Having fixed a map ẽ ∈ Ẽ, we need to evaluate the expression

∑_{β∈[0,2]}X_{g+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽβ}.

Given β ∈ [0, 2], given a subset Ξ̃ ⊆ [1, 2] × {g + 1} of cardinality #Ξ̃ = β and given a subset η^{−} ⊆ [1, 2] such that

(∗) η^{−} × (g + 1) ⊆ Ξ̃,

we let

Γ̇(ẽ, Ξ̃, η^{−}) := {γ ∈ Γ̇(ẽβ) | Ξ_{γ} ∩ ([1, 2] × {g + 1}) = Ξ̃, γ^{−1}(η) = η^{−} × {g + 1}}

and form the partial sum

Λ([a], ẽ, Ξ̃, η^{−}) := \frac{1}{s!t!} ∑_{γ∈Γ̇(ẽ,Ξ̃,η^{−})}⟨a^{γ}⟩ε_{γ} ∘ (ξ ∪ η)

so that we can recover

G′_{[a],ξ,η}f′_{ẽβ} = ∑_{Ξ̃⊆[1,2]×{g+1}} ∑_{η^{−}⊆[1,2] subject to (∗)}Λ([a], ẽ, Ξ̃, η^{−}).

Calculation of Λ-values. To distinguish subcases and subsubcases, we adapt the according notation of (A.1).

Case f′_{ẽ2}.

Subcase Ξ̃ = [\begin{matrix}+ \\ + \end{matrix}].

Subsubcase (Ξ̃, η^{−}) = [\begin{matrix}⊕ \\ ⊕ \end{matrix}]. We calculate

Λ([a], ẽ, {+ \atop +}, \{1, 2\})
= \frac{t(t−1)}{s!t!} ∑_{γ∈Γ̇(ẽ0)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′) ∘ (ξ ∪ η)

=^{(1.4)} \frac{t(t−1)(s+2)!}{s!t!} ∑_{γ∈Γ̇(ẽ0)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′) · (ξ̄\z, η\\{x′, y′\}) ∘ ξ̄
= (s + 2)(s + 1)B_{[a],ξ,η,ẽ}.

Subsubcase (Ξ̃, η^{−}) = [\begin{matrix}⊕ \\ + \end{matrix}]. We calculate

Λ([a], ẽ, {+ \atop +}, \{1\})
= \frac{t}{s!t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′_{1}) ∘ (ξ ∪ η)

=^{(1.4)} \frac{t(s+1)!}{s!t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′_{1}) · (ξ̄\z, η\\{x′, y′\}) · (y′_{1}, y′) ∘ (ξ̄∪ y′_{1})
= −\frac{(s+1)}{(t−1)!} ∑_{γ∈Γ̇(ẽ0)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (y_{γ}, y′_{1}) · (ξ̄\z, η\\{x′, y′\}) · (y′_{1}, y′) · (y′_{1}, y_{γ}) ∘ (ξ̄∪ y′_{1})
= −\frac{(s+1)}{(t−1)!} ∑_{γ∈Γ̇(ẽ0)} ∑_{y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (ξ̄\z, η\\{x′, y′\}) · (y_{γ}, y′) ∘ (ξ̄∪ y′_{1})
= −(s + 1)(uB_{[a],ξ,η,ẽ} − A_{[a],ξ,η,ẽ}).

Subsubcase (Ξ̃, η^{−}) = [\begin{matrix}+ \\ ⊕ \end{matrix}]. By symmetry, we obtain from subsubcase [\begin{matrix}⊕ \\ + \end{matrix}]

Λ([a], ẽ, {+ \atop +}, \{2\}) = −(s + 1)(uB_{[a],ξ,η,ẽ} − A_{[a],ξ,η,ẽ}).

Subsubcase (Ξ̃, η^{−}) = [\begin{matrix}+ \\ + \end{matrix}]. We calculate

Λ([a], ẽ, {+ \atop +}, \{\})
= \frac{1}{s!t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′_{1}) · (y_{γ}, y′_{1}) ∘ (ξ ∪ η)

=^{(1.4)} \frac{s!}{s!t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′_{1}) · (y_{γ}, y′_{1}) · (ξ̄\z, η\\{x′, y′\}) · (x′_{1}, x′) · (y′_{1}, y′) ∘ (ξ̄∪ x′_{1} ∪ y′_{1})
= \frac{1}{t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ}·

·(x_{γ}, x′_{1}) · (y_{γ}, y′_{1}) · (ξ̄\z, η\{x′, y′}) · (x′_{1}, x′) · (y′_{1}, y′) · (x′_{1}, x_{γ}) · (y′_{1}, y_{γ}) ∘ (ξ̄∪ x′_{1} ∪ y′_{1})

= \frac{1}{t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{x′_{1}≠y′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (x_{γ}, x′) · (y_{γ}, y′) ∘ (ξ̄∪ x′_{1} ∪ y′_{1})
= (u − 1)uB_{[a],ξ,η,ẽ} − 2(u − 1)A_{[a],ξ,η,ẽ}.

Case f′_{ẽ1}.

Subcase Ξ̃ = [\begin{matrix}+ \\ − \end{matrix}].

Subsubcase (Ξ̃, η^{−}) = [\begin{matrix}⊕ \\ − \end{matrix}]. We calculate

Λ([a], ẽ, {+ \atop −}, \{1\})
= \frac{t}{s!t!} ∑_{γ∈Γ̇(ẽ0)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) ∘ (ξ ∪ η)

=^{(1.4)} \frac{t(s+1)!}{s!t!} ∑_{γ∈Γ̇(ẽ0)}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′) · (ξ̄\z, η\\{x′, y′\}) · (y_{γ}, y′) ∘ (ξ̄∪ y_{γ})
= (s + 1)(B_{[a],ξ,η,ẽ} − C_{2,[a],ξ,η,ẽ}).

Subsubcase (Ξ̃, η^{−}) = [\begin{matrix}+ \\ − \end{matrix}]. We calculate

Λ([a], ẽ, {+ \atop −}, \{\})
= \frac{1}{s!t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′_{1}) ∘ (ξ ∪ η)
=^{(1.4)} \frac{s!}{s!t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′_{1}) · (ξ̄\z, η\\{x′, y′\}) · (x′_{1}, x′) · (y_{γ}, y′) ∘ (ξ̄∪ x′_{1} ∪ y_{γ})
= −\frac{1}{t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (x_{γ}, x′_{1}) · (ξ̄\z, η\\{x′, y′\}) · (x′_{1}, x′) · (y_{γ}, y′) · (x_{γ}, x′_{1}) ∘ (ξ̄∪ x′_{1} ∪ y_{γ})
= −\frac{1}{t!} ∑_{γ∈Γ̇(ẽ0)} ∑_{x′_{1}∈η̄}⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (x_{γ}, x′) · (y_{γ}, y′) ∘ (ξ̄∪ x′_{1} ∪ y_{γ})
= −(uB_{[a],ξ,η,ẽ} − uC_{2,[a],ξ,η,ẽ} − A_{[a],ξ,η,ẽ}).

Subcase Ξ̃ = [\begin{matrix}− \\ + \end{matrix}]. Obtained, by symmetry, from subcase Ξ̃ = [\begin{matrix}+ \\ − \end{matrix}]. Case f′_{ẽ0}. Subcase Ξ̃ = [\begin{matrix}− \\ − \end{matrix}]. Subsubcase (Ξ̃, η^{−}) = [\begin{matrix}− \\ − \end{matrix}]. We calculate

Λ([a], ẽ, {− \atop −}, \{\})
= \frac{1}{s!t!} ∑_{γ∈Γ̇(ẽ0)}⟨a^{γ}⟩ε_{γ} ∘ (ξ ∪ η)
=^{(1.4)} \frac{s!}{s!t!} ∑_{γ∈Γ̇(ẽ0)}⟨a^{γ}⟩ε_{γ} · (ξ̄\z, η\\{x′, y′\}) · (x_{γ}, x′) · (y_{γ}, y′) ∘ (ξ̄∪ x_{γ} ∪ y_{γ})
= B_{[a],ξ,η,ẽ} − C_{1,[a],ξ,η,ẽ} − C_{2,[a],ξ,η,ẽ}.

We note that s − u = X_{g} − X_{g+1} and evaluate the linear combination

∑_{β∈[0,2]}X_{g+1}^{(2−β)}G′_{[a],ξ,η}f′_{ẽβ}

= 1  ·(2(X_{g} − X_{g+1} + 2)A_{[a],ξ,η,ẽ}
+(X_{g} − X_{g+1} + 2)(X_{g} − X_{g+1} + 1)B_{[a],ξ,η,ẽ})
+ X_{g+1}  ·(2A_{[a],ξ,η,ẽ} + 2(X_{g} − X_{g+1} + 1)B_{[a],ξ,η,ẽ}
−(X_{g} − X_{g+1} + 1)(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}))
+ X_{g+1}(X_{g+1} + 1) ·(B_{[a],ξ,η,ẽ} − (C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}))

= 2(X_{g} + 2)A_{[a],ξ,η,ẽ} + (X_{g} + 2)(X_{g} + 1)B_{[a],ξ,η,ẽ} − X_{g+1}(X_{g} + 2)(C_{1,[a],ξ,η,ẽ} + C_{2,[a],ξ,η,ẽ}).

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[G 50] Théorie de la répresentation linéaire des groupes symétriques, Thèse, Mém. Soc. Roy. Sc. Liège, (4), 10, 1950.

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[J 78] The Representation Theory of the Symmetric Groups, SLN 682, 1978.

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[K 99] Ties for the ZS_{n}, thesis, http://www.mathematik.uni-bielefeld.de/∼kuenzer, Bielefeld, 1999.

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[Sp 35] Die irreduziblen Darstellungen der Symmetrischen Gruppe, Math. Z. 39, p. 696-711, 1935.

Matthias Künzer Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld

kuenzer@mathematik.uni-bielefeld.de