Elementary divisors of Gram matrices of certain Specht modules

M. Künzer, G. Nebe

September 9, 2004

Abstract

The elementary divisors of the Gram matrices of Specht modules S^{λ} over the symmetric group are determined for two-row partitions and for two-column partitions λ. More precisely, the subquotients of the Jantzen filtration are calculated using Schaper’s formula. Moreover, considering a general partition λ of n at a prime p > n − λ_{1}, the only possible non trivial composition factor of S_{F_{p}}^{λ} is induced by the morphism of Carter and Payne, as shown by means of Kleshchev’s modular branching rule. This enables the Jantzen filtration to be calculated in this case as well.

0 Introduction  2
0.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.5 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 General considerations  6
1.1 Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Jantzen filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The reverse embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Block diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Elementary divisors for Specht modules  11
2.1 Specht modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Two-row partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Unimodular lattices in SQ^{(n−m,m)} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Some three- and four-part partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 At a large prime  22

AMS subject classification: 20C30.

4 Transposition  25
5 Explicit bases  26
5.1 A linear algebra lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Bases for hooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.3 Gram matrix entries for two-column partitions . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4 Bases for (2^{2}, 1^{n−4}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 Miscellanea  34
6.1 Symmetric partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2 A conjectural comparison of kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.3 A final remark on quasiblocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

0 Introduction

0.1 Problem

Specht modules S^{λ} are combinatorially defined ZS_{n}-modules, indexed by partitions λ of n, which yield a complete set of pairwise nonequivalent ordinary irreducible representations of the symmetric group S_{n} after scalar extension to C. A Specht module S^{λ} carries a nondegenerate S_{n}-invariant bilinear form, inducing an embedding into its Z-linear dual, S^{λ} [figure] S^{λ,∗}. The problem is to determine the structure of the quotient

S^{λ,∗}/S^{λ}

as an abelian group. Reformulated, we ask for the elementary divisors of the Gram matrix of this bilinear form on S^{λ}, or, for short, for the elementary divisors of S^{λ}.

0.2 Known Results

0.2.1 Simple modules

Let p be a prime. Denote by λ′ the transposed partition of λ. The number of elementary divisors of S^{λ} not divisible by p is either zero, or the dimension of a simple F_{p}S_{n}-module.

Theorem [5, 11.5]. Let D_{F_{p}}^{λ} be the image of S^{λ} \overline{}S^{λ,∗}/pS^{λ,∗}. If λ is p-regular, that is, λ′_{i} − λ′_{i+1} < p for all i ⩾ 1, then D_{F_{p}}^{λ} is a simple F_{p}S_{n}-module. Up to isomorphism, all simple F_{p}S_{n}-modules occur this way. If λ is p-singular, we have D_{F_{p}}^{λ} = 0.

Allowing for D_{F_{p}}^{λ} to denote the zero module if λ is not a p-regular partition turns out to be a convenient convention.

0.2.2 Schaper’s formula

First of all, the product of the elementary divisors of S^{λ} is known.

Theorem [9, p. 224]. There is an explicit combinatorial formula for the determinant of the Gram matrix of S^{λ}.

More precisely, given a prime p, the quotient S^{λ,∗}/S^{λ} is expressible as a linear combination of Specht modules in the Grothendieck group of ZS_{n}-modules of finite length by Schaper’s formula. Given a commutative ring A, we abbreviate S_{A}^{λ} := A ⊗_{Z} S^{λ}.

Theorem [28, p. 60], cf. [16, Cor. 5.33]. There are combinatorially determined integral coefficients α_{μ} such that

[S_{Z_{(p)}}^{λ,∗}/S_{Z_{(p)}}^{λ}] = ∑_{i⩾1}[S_{F_{p}}^{λ}(i)] = ∑_{μ} α_{μ}[S_{F_{p}}^{μ}] ∈ K_{0}(modf-Z_{(p)}S_{n}) ,

where S_{F_{p}}^{λ}(i) denotes the ith piece of the Jantzen filtration of S_{F_{p}}^{λ}.

Together with calculations of decomposition numbers due to James, Williams, To Law, Benson, Müller et al. [2, 5, 10, 18, 19, 20, 30], Schaper’s formula represents our principal tool.

0.2.3 Numerical results

There is an estimate for the first elementary divisor, found by James in the course of the construction of the simple modules D_{F_{p}}^{μ}.

Lemma [5, 10.4]. The product ∏_{i⩾1}(λ′_{i} − λ′_{i+1})! divides the first elementary divisor of S^{λ}. In turn, the first elementary divisor of S^{λ} divides the product ∏_{i⩾1}(λ′_{i} − λ′_{i+1})!^{i}.

For instance, 3! divides the first elementary divisor 12 of S^{(2^{3})}, which in turn divides 3!^{2}. The James factor ∏_{i⩾1}(λ′_{i} − λ′_{i+1})! will reappear constantly.

Numerically, the relation between the elementary divisors of S^{λ} and S^{λ′} has been known.

Proposition [13, 6.2.10]. Let n_{λ} := rk_{Z}S^{λ}, and let i ∈ [1, n_{λ}]. The product of the ith elementary divisor of S^{λ} and the (n_{λ} + 1 − i)th elementary divisor of S^{λ′} yields n!/n_{λ}.

In particular, the elementary divisors of S^{λ} and S^{λ′} mutually determine each other. We shall give a module version of this relation in terms of Jantzen subquotients (4.1).

0.2.4 Related work

Grabmeier used Schaper’s analoguous formula for the Weyl modules over the Schur algebra as an ingredient to determine the graduated hull of p-adic Schur algebras [4, 11.13].

Kleshchev and Sheth [12, 3.4], and independently, Reuter [26, 4.2.22], described the submodule structure of S_{F_{p}}^{(n−m,m)}.

0.3 Results

0.3.1 Two-row partitions

Let n ⩾ 1, let 0 ⩽ m ⩽ n/2 and let p be a prime. Since the decomposition numbers of S_{F_{p}}^{(n−m,m)} are in {0, 1} by James’ formula, the Jantzen filtration may be calculated by means of Schaper’s formula.

Theorem (2.5). The multiplicities of the simple modules in the subquotients of the Jantzen filtration of S_{F_{p}}^{(n−m,m)} are determined. In particular, the elementary divisors of S^{(n−m,m)} are calculated.

Moreover, combining arguments of Plesken [25] and Wirsing [31], we show that if m ⩾ 3, then S_{Q}^{(n−m,m)} does not contain a unimodular ZS_{n}-lattice, that is, a lattice X satisfying X ≃ X^{∗} (2.13). For m ∈ {1, 2}, unimodular lattices do occur and have been classified by Plesken [25, p. 98 and II.5].

0.3.2 At a large prime

Suppose given a partition λ of n and a prime p > n − λ_{1}. Using the theorem of Carter and Payne [3, p. 425], the direction of the Carter conjecture proven by James and Murphy [9, p. 222], as well as Kleshchev’s modular branching rule [11, 0.6], the Jantzen filtration of S_{F_{p}}^{λ} may be calculated.

Theorem (3.5). If p does not divide a first row hook length in the range [1, λ_{2}], then S_{F_{p}}^{λ} is simple. If p divides the first row hook length h_{t} of the node (1, t), t ∈ [1, λ_{2}], then [S_{F_{p}}^{λ}] = [D_{F_{p}}^{λ}] + [D_{F_{p}}^{λ[t]}], where λ[t] is the partition arising from λ by the according Carter-Payne box shift. The constituent [D_{F_{p}}^{λ[t]}] lies in the v_{p}(h_{t})th Jantzen subquotient.

0.3.3 Explicit diagonalization

The results mentioned so far are based on Schaper’s formula, so that no diagonalizing bases can be deduced. In general, an explicit diagonalization seems to be complicated. For hook partitions, however, it is easier to diagonalize directly (5.5) than to apply Schaper’s formula, as has already been remarked by James and Mathas [unpublished].

Moreover, for S^{(2^{2},1^{n−4})}, we give bases essentially diagonalizing the Gram matrix (5.14). In general, it might be worthwhile to employ modular morphisms in order to fully solve the diagonalization problem (cf. 6.8). But note that from such a complete solution, bases for the simple F_{p}S_{n}-modules would ensue.

0.3.4 The scope of Schaper’s formula

Some three- and four-part partitions are treated in §2.5, partly conjecturally. The problem that remains to be solved, once all decomposition numbers and Jantzen subquotients are known, is the following.

Let p be a prime, let e^{μ} be a primitive idempotent of Z_{(p)}S_{n} belonging to D_{F_{p}}^{μ} and let ε^{λ} be the central-primitive idempotent of QS_{n} belonging to S_{Q}^{λ}. The investigation of the elementary divisors of S^{λ} [figure] S^{λ,∗} can be reduced to the consideration of

S_{Z_{(p)}}^{λ}e^{μ} [figure] S_{Z_{(p)}}^{λ,∗}e^{μ} .

This is a ε^{λ}e^{μ}Z_{(p)}S_{n}e^{μ}-linear map, the determinant of which can be deduced from Schaper’s formula. So we are led to consider the Z_{(p)}-order ε^{λ}e^{μ}Z_{(p)}S_{n}e^{μ}, which is of rank [S_{F_{p}}^{λ} : D_{F_{p}}^{μ}]^{2} over Z_{(p)}.

In particular, if [S_{F_{p}}^{λ} : D_{F_{p}}^{μ}] = 1, we obtain ε^{λ}e^{μ}Z_{(p)}S_{n}e^{μ} ≃ Z_{(p)}, which enables us to calculate elementary divisors. If [S_{F_{p}}^{λ} : D_{F_{p}}^{μ}] > 1, however, Schaper’s formula alone is too coarse.

In another disguise, concerning the distribution of the multiplicity of a simple module D_{F_{p}}^{μ} over the Jantzen subquotients, Schaper’s formula gives the value of a certain sum of which one needs to know the summands (cf. 1.5 iii). If [S_{F_{p}}^{λ} : D_{F_{p}}^{μ}] = 1, then this sum consists of a single nonzero summand, that is, the Jantzen subquotient D_{F_{p}}^{μ} appears in is determined. But this purely numerical point of view hides the role of the ring ε^{λ}e^{μ}Z_{(p)}S_{n}e^{μ} in case [S_{F_{p}}^{λ} : D_{F_{p}}^{μ}] > 1.

0.4 Acknowledgements

We thank F. Lübeck for access to his immense computations [15], which served as our indispensable guideline. We thank A. Mathas for help with several details. We thank A. Kleshchev for help with modular branching (3.5). We thank E. Wirsing for help with Pell equations (2.13). We thank J. Müller for making [20] available to us. The first author would like to thank the IGD Lyon for kind hospitality during the first part of this project.

0.5 Conventions

(i) Composition of maps is written on the right, ^{\overline{}^{a}} ^{\overline{}^{b}} = ^{\overline{}^{ab}} . Exception is made for ‘standard’ maps, such as traces, characters … Unless mentioned otherwise, a module is a finitely generated right module.

(ii) For a, b ∈ Z, we denote by [a, b] := {c ∈ Z | a ⩽ c ⩽ b} the integral interval. (iii) If A is an assertion, which might be true or false, we let {A} equal 1 if A is true, and 0 if A is false.

If, in a sum, a summand has a factor {A} attached, this summand is zero if A is false, regardless whether it is well defined or not.

(iv) Let m be a positive integer, let ā denote the residue class of a modulo m. The assertion ā ∈

{b̄_{1}, …, b̄_{l}} will be written as a ≡_{m} b_{1}, …, b_{l}. In particular, a ≡_{m} b means a − b ∈ mZ.

(v) Let p be a prime, let n ⩾ 1. The F_{p}S_{n}-module DF_{p}^{μ} is defined for any composition μ of n, and we let it be zero if μ is not a p-regular partition. If μ is a p-regular partition, it is defined in [5, 11.2].

(vi) Given a partition λ, its transpose is denoted by λ′, i.e. j ⩽ λ_{i} ⟺ i ⩽ λ′_{j}. Given a λ-tableau [a], its transpose is a λ′-tableau denoted by [a′].

(vii) The sign of σ ∈ S_{n} is denoted by ε_{σ}.

(viii) Let Jac(B) denote the Jacobson radical of a ring B.

(ix) The binomial coefficient ( \begin{matrix}a \\ b \end{matrix} ) is zero unless a ⩾ b ⩾ 0.

(x) Given a ring A, we denote by modf-A the abelian category of A-modules of finite length, and by K_{0}(modf-A) its Grothendieck group. So K_{0}(modf-A) is the quotient of the free abelian group on the objects of modf-A modulo a relation X − (X′ + X′′) for each short exact sequence 0 \overline{}X′ \overline{}X \overline{}X′′ \overline{}0 in modf-A. The image of X ∈ Ob modf-A in K_{0}(modf-A) shall be denoted by [X]. Cf. [27, 3.1.6].

1 General considerations

1.1 Situation

1.1.1 An order Λ

Let R be a discrete valuation ring with maximal ideal generated by π, let K be its field of fractions. Given an R-module M, we denote its reduction by M̄ := M/πM.

Let A be a simple K-algebra (to which we may pass from a semisimple K-algebra, cf. 1.1). Let Λ ⊆ A be a full R-order in A, that is, an R-algebra that spans A as a vectorspace such that Λ is finitely generated free as a module over R. Let 1_{Λ} = e_{1} + ⋯ + e_{k} be an orthogonal decomposition into primitive idempotents of Λ, which correspond to indecomposable projective Λ-modules P_{j} := e_{j}Λ.

Remark 1.1 Suppose given a semisimple K-algebra B, a full R-order Ξ in B and a simple Ξ-lattice X, that is, a Ξ-module that is finitely generated free over R such that KX := K ⊗_{R}X is a simple B-module. Let ε ∈ B be the central-primitive idempotent that acts as identity on KX. Then X remains a simple lattice over the quasiblock Λ = Ξε, which is a full R-order in A = Bε, and to which we may reduce the situation for our purposes. Note that in general, the quasiblock Λ = Ξε will not be a direct summand of Ξ.

If Q is an indecomposable projective Ξ-module, then Qε (⊆ KQ) is either zero or an indecomposable projective Λ-module. All indecomposable projective Λ-modules are obtained this way.

For instance, if G is a finite group and K a field of characteristic 0, we may take B = KG, Ξ = RG and ε the rational central-primitive idempotent belonging to the simple KGmodule KX.

1.1.2 A suborder Δ ⊆ Λ

Let P_{j_{l}} be representatives of the isomorphism classes of the indecomposable projective Λ-modules, l ∈ [1, m], and let

f_{l} := ∑_{j′∈[1,k], P_{j′} ≃ P_{j_{l}}}e_{j′} .

Then the f_{l} + Jac(Λ) are the central-primitive idempotents of Λ/Jac(Λ). Letting D_{l} := e_{j_{l}}Λ/e_{j_{l}}Jac(Λ), a complete system of inequivalent simple Λ-modules is given by {D_{l} | l ∈ [1, m]}, and f_{l} operates as identity on D_{l}.

This allows to introduce an R-suborder

Δ := ∏_{l∈[1,m]} f_{l}Λf_{l} ⊆ Λ

that has the same Grothendieck group as Λ, but that allows to decompose modules into smaller pieces. Since Λ = ⨁_{l,l′∈[1,m]} f_{l}Λf_{l′} has Jacobson radical

Jac(Λ) = (⨁_{l} Jac(f_{l}Λf_{l})) ⊕ (⨁_{l≠l′} f_{l}Λf_{l′}) ,

the inclusion Δ ⊆ Λ induces an isomorphism

Δ/Jac(Δ) ^{\overline{}^{∼}} Λ/Jac(Λ) ,

whence restriction yields

K_{0}(modf-Δ) ^{\overline{}^{∼}} K_{0}(modf-Λ) .

1.1.3 Multiplicities

Let X be a simple Λ-lattice. There are various multiplicities to be considered.

By [X̄ : D_{l}] we denote the multiplicity of D_{l} in a composition series of X̄ in the sense of Jordan-Hölder. By [Λ : P_{j}] we denote the multiplicity of P_{j} as a direct summand of Λ, which is well defined since Krull-Schmidt holds for projective Λ-modules by Nakayama’s Lemma. Finally, by [KP_{j} : KX] we denote the multiplicity of KX in KP_{j} as a Jordan-Hölder constituent, or, equivalently, as a direct summand.

Denote E := dim_{K} End_{A}KX and E_{l} := dimR̄ End_{Λ}D_{l}. Brauer reciprocity holds,

E · [KP_{j_{l}} : KX] = E_{l} · [X̄ : D_{l}] ,

since the K-dimension of KXe_{j_{l}} and the R̄-dimension of X̄e_{j_{l}} coincide, and since the latter is calculated by the right hand side, as we see after passing to the case Λ = Δf_{l}, and subsequently reducing moritaequivalently to the case e_{j_{l}} = 1. Moreover,

E_{l} · [Λ : P_{j_{l}}] = dimR̄ D_{l} .

In particular, f_{l} is the sum of E_{l}^{−1} · dimR̄ D_{l} primitive idempotents.

More generally, if M is a Δ-module of finite length over R, then [M : D_{l}] denotes the Jordan-Hölder multiplicity of D_{l} in M as a Δ-module. If M happens to be the restriction of a Λ-module to Δ, this is the same as the multiplicity of D_{l} in M as a Λ-module. So we do not need to specify whether the multiplicity refers to Δ or to Λ.

Remark 1.2 If there exists a l ∈ [1, m] such that E_{l} = 1 and [X̄ : D_{l}] = 1, then E = 1. In other words, if some D_{l} is absolutely simple and appears with multiplicity 1 in X̄, then KX is absolutely simple, too.

Conversely, if KX is absolutely simple and [KP_{j_{l}} : KX] = 1, then D_{l} is absolutely simple.

1.2 Jantzen filtration

We recall the basic facts concerning Jantzen’s filtration arising from an embedding of simple Λ-lattices. This filtration is a tool to compare decomposition numbers with Jordan-Hölder multiplicities in the quotient of this embedding.

Let X ^{\overline{}^{φ}} Y be a nonzero Λ-linear map between simple lattices X and Y, thus necessarily injective. There exists an N ⩾ 0, which we choose and fix, such that π^{N}Y ⊆ Xφ.

Definition 1.3 For i ⩾ 0, we let

X̄(i) := ((π^{i}Y)φ^{−1} + πX)/πX ⊆ X̄

be the ith piece of the Jantzen filtration of X̄ with respect to X ^{\overline{}^{φ}} Y. In particular, X̄(0) = X̄, and X̄(N +1) = 0. Given l ∈ [1, m], the ith Jantzen multiplicity of D_{l} is given by

ϑ_{l,i} := [X̄(i)/X̄(i + 1) : D_{l}] .

Remark 1.4 Note that X(i) ≃ (π^{i}Y ∩ Xφ)/(π^{i}Y ∩ πXφ), and thus

\frac{X̄(i)}{X̄(i + 1)} ≃ \frac{π^{i}Y ∩ Xφ}{(π^{i+1}Y ∩ Xφ) + (π^{i}Y ∩ πXφ)} .

If we consider this subquotient as an R-module, by an elementary divisor decomposition we may assume that X = Y = R, and that φ is given by multiplication by π^{j} for some j ⩾ 0. Then X̄(i)/X̄(i + 1) is isomorphic to R̄ if i = j, and to zero otherwise. Returning to the general case, we obtain therefore

Y/Xφ ≃_{R} ⨁_{i⩾0}(R/π^{i}R)^{dimR̄ X̄(i)/X̄(i+1)} ≃_{R} ⨁_{l ∈ [1,m]} (⨁_{i⩾0}(R/π^{i}R)^{ϑ_{l,i}})^{dimR̄ D_{l}} .

Lemma 1.5 (Jantzen’s Lemma)

(i) In K_{0}(modf-Λ), we have [Y/Xφ] = ∑_{i⩾1}[X̄(i)] = ∑_{i⩾0} i [X̄(i)/X̄(i + 1)].

(ii) In K_{0}(modf-Δ), we have [Y f_{l}/Xφf_{l}] = ∑_{i⩾1}[X̄(i)f_{l}] = ∑_{i⩾0} i [X̄(i)f_{l}/X̄(i + 1)f_{l}] for any l ∈ [1, m].

(iii) Given l ∈ [1, m], we have

\begin{aligned}∑_{i⩾0} ϑ_{l,i} & = & [X̄ : D_{l}] & = & [X̄f_{l} : D_{l}] \\ ∑_{i⩾0} iϑ_{l,i} & = & [Y/Xφ : D_{l}] & = & [Y f_{l}/Xφf_{l} : D_{l}] . \end{aligned}

Assertion (ii) follows from (i) by decomposition in modf-Δ. The second formula in (iii) follows from (i, ii), the first follows from the definition of ϑ_{l,i}.

It remains to prove (i). There is a filtration of Λ-modules

π^{N}Xφ = π^{N−0}Xφ ∩ π^{N}Y ⊆ π^{N−1}Xφ ∩ π^{N}Y ⊆ ⋯ ⊆ π^{N−N}Xφ ∩ π^{N}Y = π^{N}Y ,

the ith subquotient of which is

\frac{π^{N−i}Xφ ∩ π^{N}Y}{π^{N−(i−1)}Xφ ∩ π^{N}Y} ≃ \frac{Xφ ∩ π^{i}Y}{πXφ ∩ π^{i}Y} = X̄(i)

for i ∈ [1, N].

Corollary 1.6 Suppose given l ∈ [1, m]. If [X̄ : D_{l}] = 1, then

ϑ_{l,i} = \{ \begin{matrix}1 & if i = [Y/Xφ : D_{l}] , \\ 0 & otherwise . \end{matrix}

Corollary 1.7 Suppose given l ∈ [1, m]. Let s ⩾ 1. If ϑ_{l,i} = 0 for i ∈ [0, s − 1] and if [Y/Xφ : D_{l}] ⩽ s[X̄ : D_{l}], then

ϑ_{l,i} = \{ \begin{matrix}[X̄ : D_{l}] & if i = s , \\ 0 & otherwise . \end{matrix}

In fact, the assumptions yield ∑_{i⩾s+1}(i − s)ϑ_{l,i} ⩽ 0.

1.3 The reverse embedding

Suppose given Λ-linear maps (X ^{\overline{}^{φ}} Y ^{\overline{}^{ψ}} X) = (X ^{\overline{}^{a}} X), where a ∈ R and v_{π}(a) = N. Note that given φ and N as above, such a map ψ exists. Consider the Jantzen filtration X̄(i) with respect to φ and the Jantzen filtration Ȳ (j) with respect to ψ.

Remark 1.8 Given i ∈ [0, N], we have \frac{X̄(i)}{X̄(i + 1)} ≃ \frac{Ȳ (N − i)}{Ȳ (N − i + 1)} . For i > N, we

have X̄(i)/X̄(i + 1) = 0.

In fact,

\begin{matrix}\frac{X̄(i)}{X̄(i + 1)} & ≃^{(1.4)} & \frac{π^{i}Y ∩ Xφ}{(π^{i+1}Y ∩ Xφ) + (π^{i}Y ∩ πXφ)} \\ ≃ \frac{Y ψ ∩ π^{N−i}X}{(πY ψ ∩ π^{N−i}X) + (Y ψ ∩ π^{N−i+1}X)} ≃^{(1.4)} \frac{Ȳ (N − i)}{Ȳ (N − i + 1)} . \end{matrix}

1.4 Block diagonalization

We shall describe, to a certain extent, the block matrices of the block diagonalization of X ^{\overline{}^{φ}} Y resulting from a decomposition of 1_{Λ} into orthogonal primitive idempotents. The problem that remains, once all decomposition numbers and Jantzen multiplicities determined, is to find the elementary divisors of the blocks of size > 1. This amounts to a study of the local and rationally simple R-algebras e_{j}Λe_{j}, which in concrete examples seems to be difficult to get a grip on.

Lemma 1.9 For l ∈ [1, m], we have

Y e_{j_{l}}/Xφe_{j_{l}} ≃_{R} ⨁_{i⩾0}(R/π^{i}R)^{ϑ_{l,i}·E_{l}} .

In fact,

Y e_{j_{l}}/Xφe_{j_{l}} ≃_{R} ⨁_{i⩾0}(R/π^{i}R)^{dimR̄ X̄(i)e_{j_{l}}/X̄(i+1)e_{j_{l}}}
= ⨁_{i⩾0}(R/π^{i}R)^{(dimR̄ X̄(i)f_{l}/X̄(i+1)f_{l})·E_{l}·(dimR̄ D_{l})^{−1}}
= ⨁_{i⩾0}(R/π^{i}R)^{[X̄(i)f_{l}/X̄(i+1)f_{l}:D_{l}]·E_{l}}
= ⨁_{i⩾0}(R/π^{i}R)^{ϑ_{l,i}·E_{l}} .

Corollary 1.10 Choosing R-linear bases in the direct summands Xe_{j} of X and Y e_{j} of Y, the matrix of φ appears in main diagonal block form. Given l ∈ [1, m], the block belonging to Xe_{j_{l}} ^{\overline{}^{φ}} Y e_{j_{l}} has edge length

dim_{K} KXe_{j_{l}} = E · [KP_{j_{l}} : KX] = E_{l} · [X̄ : D_{l}] ,

it appears with multiplicity

[Λ : P_{j_{l}}] = E_{l}^{−1} · dimR̄ D_{l}

and the valuation at π of its determinant is given by

E_{l} · [Y/Xφ : D_{l}] .

2 Elementary divisors for Specht modules

The calculation of the elementary divisors of the Gram matrix of a Specht lattice is equivalent to the determination of the quotient S^{λ,∗}/S^{λ} as an abelian group. Now, Schaper’s formula expresses SZ_{(p)}^{λ,∗}/SZ_{(p)}^{λ} as a linear combination of Specht modules in the Grothendieck group of Z_{(p)}S_{n}-modules of finite length. Provided the decomposition numbers of the occurring Specht modules are known, this allows, by means of Jantzen’s Lemma, to compare the decomposition of SZ_{(p)}^{λ,∗}/SZ_{(p)}^{λ} with the decomposition of SF_{p}^{λ}, and thus in simple cases to determine the distribution of the simple constituents of SF_{p}^{λ} over the subquotients of the Jantzen filtration. Together with the dimensions of these simple constituents, this yields the structure of SZ_{(p)}^{λ,∗}/SZ_{(p)}^{λ} as an abelian group by (1.4).

2.1 Specht modules

A λ-tabloid {a} is a λ-tableau [a] with unordered rows. Let M^{λ} be the free Z-module on the set of λ-tabloids, carrying a structure as a ZS_{n}-module by entrywise operation of S_{n}. That is, M^{λ} is isomorphic to the permutation module on (S_{λ_{1}} × ⋯ × S_{λ_{n}})\S_{n}. Let the S_{n}-invariant bilinear form (−, =) on M^{λ} be defined by

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

Let C_{a} ⩽ S_{n} denote the column stabilizer of [a]. A λ-polytabloid is given by

⟨a⟩ := ∑_{σ∈C_{a}}{a}σε_{σ} ∈ M^{λ} .

The Specht module S^{λ} is defined to be the Z-linear span of the λ-polytabloids in M^{λ}. It carries a ZS_{n}-module structure as a submodule of M^{λ}, and it carries an S_{n}-invariant bilinear form by restriction of (−, =) to S^{λ}, again denoted by (−, =). To simplify notation, we sometimes rescale by the James factor to

(−, =)_{0} := (∏_{i⩾1}(λ′_{i} − λ′_{i+1})!)^{−1} · (−, =) ,

cf. [5, 10.4].

A standard basis of S^{λ} over Z is given by the set of standard polytabloids, where a standard polytabloid ⟨a⟩ is attached to a tableau [a] with strictly increasing rows from left to right, and strictly increasing columns from top to bottom. For the Garnir relations between polytabloids we refer to [5, 7.2]. Let

[figure: S^{λ}; η; S^{λ,∗}; ξ; (ξ, −) ,]

whose cokernel shall be denoted by S^{λ,∗}/S^{λ}.

For a commutative ring A, we denote by S_{A}^{λ} the AS_{n}-module A ⊗_{Z} S^{λ}.

Given a prime p, the Jantzen filtration of S_{F_{p}}^{λ} with respect to S_{Z_{(p)}}^{λ} ^{\overline{}^{η}} S_{Z_{(p)}}^{λ,∗} has been defined in (1.3) as being given by

S_{F_{p}}^{λ}(i) = ((p^{i}S_{Z_{(p)}}^{λ,∗})η^{−1} + pS_{Z_{(p)}})/pS_{Z_{(p)}} ⊆ S_{F_{p}}^{λ} ,

yielding a decreasing filtration as i runs over Z_{⩾0}. Note that S_{F_{p}}^{λ}(0)/S_{F_{p}}^{λ}(1) ≃ D_{F_{p}}^{λ}, which is, according to the convention adopted here, nonzero if and only if λ is p-regular [5, 12.2, 11.1]. In this case, D_{F_{p}}^{λ} does not appear in S_{F_{p}}^{λ}(1) by [5, 12.2].

In a linear combination expressing [S_{F_{p}}^{λ}] in K_{0}(modf-Z_{(p)}S_{n}) in terms of simple modules, a lower index i ⩾ 0 indicates that this summand appears as a summand of the subquotient [S_{F_{p}}^{λ}(i)/S_{F_{p}}^{λ}(i + 1)]. Thus, this summand appears with multiplicity i in [S_{Z_{(p)}}^{λ,∗}/S_{Z_{(p)}}^{λ}] (1.5 iii). This is similar to the notation in [15], where one finds e.g.

[S_{F_{2}}^{(4,3,2^{2},1)}] = [D_{F_{2}}^{(8,3,1)}]_{2} + [D_{F_{2}}^{(6,5,1)}]_{3} + [D_{F_{2}}^{(10,2)}]_{4} + [D_{F_{2}}^{(10,2)}]_{7} + 3[D_{F_{2}}^{(12)}]_{4} + 2[D_{F_{2}}^{(12)}]_{7}
+[D_{F_{2}}^{(6,4,2)}]_{5} + [D_{F_{2}}^{(6,4,2)}]_{8} + [D_{F_{2}}^{(7,3,2)}]_{5} + [D_{F_{2}}^{(8,4)}]_{5} + [D_{F_{2}}^{(7,5)}]_{6} + [D_{F_{2}}^{(7,5)}]_{7}
+[D_{F_{2}}^{(11,1)}]_{6} + 4[D_{F_{2}}^{(11,1)}]_{9} + 2[D_{F_{2}}^{(9,3)}]_{7} + [D_{F_{2}}^{(9,3)}]_{9} + [D_{F_{2}}^{(5,4,2,1)}]_{10} .

(Presumably, there are also examples in which the multiplicity of a simple module is distributed over more than two Jantzen subquotients.) Thus for instance,

[S_{F_{2}}^{(4,3,2^{2},1)}(7)/S_{F_{2}}^{(4,3,2^{2},1)}(8)] = [D_{F_{2}}^{(10,2)}] + 2[D_{F_{2}}^{(12)}] + [D_{F_{2}}^{(7,5)}] + 2[D_{F_{2}}^{(9,3)}] .

Hence the multiplicity of Z/2^{7}Z as a summand of the abelian group S_{Z_{(2)}}^{(4,3,2^{2},1),∗}/S_{Z_{(2)}}^{(4,3,2^{2},1)}

is given by dim_{F_{2}} D_{F_{2}}^{(10,2)} + 2 dim_{F_{2}} D_{F_{2}}^{(12)} + dim_{F_{2}} D_{F_{2}}^{(7,5)} + 2 dim_{F_{2}} D_{F_{2}}^{(9,3)} (1.4).

2.2 Two-row partitions

We consider Specht modules indexed by two-row partitions, that is, partitions of the form (n − m, m). The basic ingredients that allow to use Jantzen’s Lemma are the theorem of James on the decomposition numbers of Specht modules of two-row partitions, which are contained in {0, 1}, and the theorem of Schaper, expressing [SZ_{(p)}^{λ,∗}/SZ_{(p)}^{λ}] as linear combination of Specht modules [SF_{p}^{μ}].

Let n ⩾ 1, let 0 ⩽ m ⩽ n/2 and let p be a prime number.

Lemma 2.1 ([5, 20.1]) We have rk_{Z} S^{(n−m,m)} = (\begin{matrix}n \\ m \end{matrix}) · \frac{n−2m+1}{n−m+1}.

Suppose given integers s and t. If s ⩾ 0 and t ⩾ 1, we write them p-adically as s + 1 = ∑_{i∈[0,k]} s_{i}p^{i} and t = ∑_{i∈[0,l]} t_{i}p^{i}, where s_{i}, t_{i} ∈ [0, p − 1] and where s_{k} ≠ 0 and t_{l} ≠ 0.

The integer s + 1 is said to contain t to base p if (s ⩾ 0, t ⩾ 1, k > l and t_{i} ∈ {0, s_{i}} for i ∈ [0, l]) or if (s ⩾ 0 and t = 0). Let

f_{p}(s, t) := \{ \begin{matrix}1 & if s + 1 contains t to base p \\ 0 & otherwise . \end{matrix}

Theorem 2.2 ([5, 24.15]) Let j ∈ [1, m]. We have

[S_{F_{p}}^{(n−m,m)} : D_{F_{p}}^{(n−j,j)}] = f_{p}(n − 2j, m − j) (∈ {0, 1}) .

Theorem 2.3 (particular case of Schaper’s formula [28, p. 60], see also [16, 5.33])

In K_{0}(modf-Z_{(p)}S_{n}), we have

[S_{Z_{(p)}}^{(n−m,m),∗}/S_{Z_{(p)}}^{(n−m,m)}] = ∑_{i∈[0,m−1]}v_{p} (\frac{n+1−m−i}{m−i}) [S_{F_{p}}^{(n−i,i)}] .

For j ∈ [0, m], we abbreviate

μ(n, m, p; j) := ∑_{i∈[j,m−1]} v_{p}(\frac{n + 1 − m − i}{m − i}) f_{p}(n − 2j, i − j) .

Corollary 2.4 Combining (2.2) and (2.3), we get

[S_{Z_{(p)}}^{(n−m,m),∗}/S_{Z_{(p)}}^{(n−m,m)}] = ∑_{j∈[0,m−1]}μ(n, m, p; j) [D^{(n−j,j)}] .

We denote by (b_{p; i,j})_{0⩽i,j⩽n/2} the inverse of the lower triangular unipotent integral matrix

(f_{p}(n − 2j, i − j))_{0⩽i,j⩽n/2} , and remark that dim_{F_{p}} D_{F_{p}}^{(n−j,j)} = ∑_{l∈[0,j]} b_{p; j,l} (\begin{matrix}n \\ l \end{matrix}) · \frac{n−2l+1}{n−l+1}.

Theorem 2.5 In K_{0}(modf-Z_{(p)}S_{n}), we have

(2.5.1)  [S_{F_{p}}^{(n−m,m)}] = ∑_{j∈[0,m]}f_{p}(n − 2j, m − j)[D_{F_{p}}^{(n−j,j)}]_{μ(n,m,p; j)} .

Thus as Z_{(p)}-modules, we have

(2.5.2)  S_{Z_{(p)}}^{(n−m,m),∗}/S_{Z_{(p)}}^{(n−m,m)} ≃ ⨁_{j ∈ [0,m−1]} (Z/p^{μ(n,m,p; j)}Z)^{∑_{l∈[0,j]} b_{p; j,l}( \begin{matrix}n \\ l \end{matrix} )· \frac{n−2l+1}{n−l+1}} .

In other words, the right hand side lists the p-part of the elementary divisors of the Gram matrix of the invariant bilinear form on the Specht module S^{(n−m,m)}, which is unique up to scalar, in an unordered manner.

Formula (2.5.1) follows by an application of (1.6) to (2.2) and (2.4), where R = Z_{(p)},

π = p, Λ = Z_{(p)}S_{n}, and where (X ^{\overline{}^{φ}} Y) = (S_{Z_{(p)}}^{(n−m,m)} ^{\overline{}^{η}} S_{Z_{(p)}}^{(n−m,m),∗}). Cf. [5, 4.12, 11.5].

Formula (2.5.2) now ensues by (1.4).

We draw a conclusion that will follow again from (3.5) below. Corollary 2.6 If p > m, we obtain

(2.6.1) [S_{F_{p}}^{(n−m,m)}] = [D_{F_{p}}^{(n−m,m)}]_{0} + ∑_{j∈[0,m−1]}{n ≡_{p} m + j − 1}[D_{F_{p}}^{(n−j,j)}]_{v_{p}(n+1−m−j)} .

Thus as Z_{(p)}-modules, we have

(2.6.2)  S_{Z_{(p)}}^{(n−m,m),∗}/S_{Z_{(p)}}^{(n−m,m)} ≃ ⨁_{j ∈ [0,m−1]}(Z_{(p)}/(n + 1 − m − j)Z_{(p)})^{( \begin{matrix}n \\ j \end{matrix} )· \frac{n−2j+1}{n−j+1}} .

For j ∈ [0, m − 1], we obtain μ_{n,m,p;j} = v_{p}(n + 1 − m − j), whence (2.6.1). In (2.6.2) and (2.5.2), we remark that the summands are zero unless j ≡_{p} n + 1 − m, which happens at most once. If j ≡_{p} n + 1 − m, we recall that the outer exponent in (2.5.2) is just dim_{F_{p}} D_{F_{p}}^{(n−j,j)}, which equals dim_{F_{p}} S_{F_{p}}^{(n−j,j)} by (2.2).

2.3 Numerical examples

Example 2.7 (case m = 1) Let n ⩾ 2. For p arbitrary, (2.6.2) yields

SZ_{(p)}^{(n−1,1),∗}/SZ_{(p)}^{(n−1,1)} ≃ Z_{(p)}/nZ_{(p)} .

This is also a particular case of (5.5) below. Example 2.8 (case m = 2) Let n ⩾ 4. For p > 2, (2.6.2) yields

SZ_{(p)}^{(n−2,2),∗}/SZ_{(p)}^{(n−2,2)} ≃ (Z_{(p)}/(n − 1)Z_{(p)}) ⊕ (Z_{(p)}/(n − 2)Z_{(p)})^{n−1} .

For p = 2, (2.5.2) yields

SZ_{(2)}^{(n−2,2),∗}/SZ_{(2)}^{(n−2,2)} ≃ (Z_{(2)}/ ( \begin{matrix}n−1 \\ 2 \end{matrix} ) Z_{(2)}) ⊕ (Z_{(2)}/(n − 2)Z_{(2)})^{n−2} .

Note that for n = 4, where the elementary divisors of S^{(2,2)} are 2 and 6, the result is formulated using redundant zero summands. In (5.14) below, we will give two bases essentially diagonalizing the Gram matrix of the Specht module to the transposed partition (cf. 4.3). Example 2.9 (case m = 3) Let n ⩾ 6. For p > 3, (2.6.2) yields

SZ_{(p)}^{(n−3,3),∗}/SZ_{(p)}^{(n−3,3)} ≃ (Z_{(p)}/(n−2)Z_{(p)})⊕(Z_{(p)}/(n−3)Z_{(p)})^{n−1}⊕(Z_{(p)}/(n−4)Z_{(p)})^{\frac{n(n−3)}{2}} .

For p = 3, (2.5.2) yields

SZ_{(3)}^{(n−3,3),∗}/SZ_{(3)}^{(n−3,3)} ≃ (Z_{(3)}/ ( \begin{matrix}n−2 \\ 3 \end{matrix} ) Z_{(3)}) ⊕ (Z_{(3)}/(n − 3)Z_{(3)})^{n−2}
⊕ (Z_{(3)}/(n − 4)Z_{(3)})^{\frac{n(n−3)}{2} −1} .

For p = 2, (2.5.2) yields

SZ_{(2)}^{(n−3,3),∗}/SZ_{(2)}^{(n−3,3)} ≃ (Z_{(2)}/\frac{n−2}{2^{\{n≡_{4}0\}}} Z_{(2)}) ⊕ (Z_{(2)}/ ( \begin{matrix}n−3 \\ 2 \end{matrix} ) Z_{(2)})^{(n−1)−\{n≡_{2}0\}}
⊕ (Z_{(2)}/(n − 4)Z_{(2)})^{\frac{n(n−3)}{2} −(n−1)+\{n≡_{4}0\}} .

Example 2.10 (case m = 4) Let n ⩾ 8. For p > 4, (2.6.2) yields

SZ_{(p)}^{(n−4,4),∗}/SZ_{(p)}^{(n−4,4)} ≃ (Z_{(p)}/(n − 3)Z_{(p)}) ⊕ (Z_{(p)}/(n − 4)Z_{(p)})^{n−1}

⊕ (Z_{(p)}/(n − 5)Z_{(p)})^{\frac{n(n−3)}{2}} ⊕ (Z_{(p)}/(n − 6)Z_{(p)})^{\frac{n(n−1)(n−5)}{6}} .

For p = 3, (2.5.2) yields

SZ_{(3)}^{(n−4,4),∗}/SZ_{(3)}^{(n−4,4)} ≃ (Z_{(3)}/ (3^{−\{n≡_{9}0,6\}}(n − 3)) Z_{(3)}) ⊕ (Z_{(3)}/ ( \begin{matrix}n−4 \\ 3 \end{matrix} ) Z_{(3)})^{(n−1)−\{n≡_{3}0\}}
⊕ (Z_{(3)}/(n − 5)Z_{(3)})^{\frac{n(n−3)}{2} −(n−1)}
⊕ (Z_{(3)}/(n − 6)Z_{(3)})^{\frac{n(n−1)(n−5)}{6} −(n−1)+\{n≡_{9}0,6\}} .

For p = 2, (2.5.2) yields

SZ_{(2)}^{(n−4,4),∗}/SZ_{(2)}^{(n−4,4)} ≃ (Z_{(2)}/ ( \begin{matrix}n−3 \\ 4 \end{matrix} ) Z_{(2)}) ⊕ (Z_{(2)}/ (2^{−\{n≡_{4}2\}}(n − 4)) Z_{(2)})^{n−2}

⊕ (Z_{(2)}/ ( \begin{matrix}n−5 \\ 2 \end{matrix} ) Z_{(2)})^{\frac{n(n−3)}{2} −\{n≡_{4}2\}(n−1)−\{n≡_{4}1\}}

⊕ (Z_{(2)}/(n − 6)Z_{(2)})^{\frac{n(n−1)(n−5)}{6} − \frac{n(n−3)}{2} +\{n≡_{4}2\}(n−2)} .

2.4 Unimodular lattices in S(n−m,m)_{Q}

Closely connected to the question for the elementary divisors of the Gram matrix is the question of the existence of unimodular lattices in a Specht module. For two-row partitions this investigation has been initiated by Plesken [25]. Our result now asserts the nonexistence of such lattices in the cases not treated in loc. cit. Reformulated, this amounts to the assertion that a certain system of two Pell equations is only trivially solvable, the proof of which we owe to E. Wirsing. The number of solutions of general Pellian systems has been studied extensively (cf. [1], where also further references may be found). The method employed in the particular case here does not seem to generalize.

Let R be a localization of Z at a maximal ideal, or Z itself, let λ be a partition of n, and let X ⊆ S_{Q}^{λ} be a full RS_{n}-lattice, i.e. an RS_{n}-submodule that is finitely generated free over R of rank dim_{Q} S_{Q}^{λ}. We denote by

X^{#} := {v ∈ S_{Q}^{λ} | (v, X) ⊆ R} ⊆ S_{Q}^{λ}

its dual lattice. Note that X^{#} ≃ X^{∗}, the latter denoting the abstract R-dual.

If X ≃ X^{∗}, the RS_{n}-lattice X is called unimodular. Under our assumptions, this is equivalent to the existence of a scalar a ∈ Q such that X = aX^{#}.

Given an inclusion X ⊆ Y ⊆ S_{Q}^{λ} of full RS_{n}-lattices, by self duality of simple F_{p}S_{n}- modules for each prime p, we have [Y/X] = [X^{#}/Y^{#}] in the Grothendieck group [5, 11.5].

We include a proof of a corollary of Plesken, restricting his argument to this corollary as well.

Proposition 2.11 ([25, Cor. II.4]) Let p be a prime and assume λ to be p-regular. If S_{Q}^{λ} contains a unimodular Z_{(p)}S_{n}-lattice, then

[S_{Z_{(p)}}^{λ,∗}/S_{Z_{(p)}}^{λ}] ∈ 2K_{0}(modf-Z_{(p)}S_{n}) .

Let X ⊆ S_{Q}^{λ} be a unimodular Z_{(p)}S_{n}-lattice, let a ∈ Q such that X = aX^{#}. Let m ∈ Z_{(p)} be such that mS_{Z_{(p)}}^{λ} ⊆ X. The filtration

mS_{Z_{(p)}}^{λ} ⊆ X = aX^{#} ⊆ am^{−1}S_{Z_{(p)}}^{λ,#}

shows that [am^{−1}S_{Z_{(p)}}^{λ,#}/mS_{Z_{(p)}}^{λ}] ∈ 2K_{0}(modf-Z_{(p)}S_{n}). Therefore, there exists s ∈ Z such that

[S_{Z_{(p)}}^{λ,∗}/S_{Z_{(p)}}^{λ}] + s[S_{F_{p}}^{λ}] ∈ 2K_{0}(modf-Z_{(p)}S_{n})

Since λ is p-regular, counting multiplicities of D_{F_{p}}^{λ} yields s ∈ 2Z by (1.5 i) and [5, 12.2].

Remark 2.12 The converse to (2.11) holds as well, provided the decomposition numbers of S_{F_{p}}^{λ} are in {0, 1}. For this direction, λ need not be p-regular.

In fact, suppose [S_{Z_{(p)}}^{λ,∗}/S_{Z_{(p)}}^{λ}] ∈ 2K_{0}(modf-Z_{(p)}S_{n}). Let S_{Z_{(p)}}^{λ} ⊆ M ⊆ S_{Q}^{λ,∗} be a full Z_{(p)}S_{n}-lattice that is maximal with respect to the property that (M, M) ⊆ Z_{(p)}. Let a ⩾ 0 such that p^{2^{a}}M^{#} ⊆ M.

If p^{2^{b}}M^{#} ⊆ M for some b ⩾ 1, then (p^{2^{b−1}}M^{#}, p^{2^{b−1}}M^{#}) = (M^{#}, p^{2^{b}}M^{#}) ⊆ (M^{#}, M) ⊆ Z_{(p)}, whence M = M + p^{2^{b−1}}M^{#} by maximality of M, i.e. p^{2^{b−1}}M^{#} ⊆ M. By induction, starting with b = a, we conclude that pM^{#} ⊆ M ⊆ M^{#}.

Now by the filtration S_{Z_{(p)}}^{λ} ⊆ M ⊆ M^{#} ⊆ S_{Z_{(p)}}^{λ,#}, the decomposition numbers of M^{#}/M are even. But since M^{#}/M is a quotient of M^{#}/pM^{#}, the decomposition numbers of which are in {0, 1} by assumption, we infer that M = M^{#}.

Now let λ = (n − m, m). The cases m = 1 and m = 2 have been treated in [25, p. 98 and II.5].

Theorem 2.13 Let 3 ⩽ m ⩽ n/2. The module S_{Q}^{(n−m,m)} does not contain a unimodular ZS_{n}-lattice.

Given a prime p, Schaper’s formula reads

[S_{Z_{(p)}}^{(n−m,m),∗}/S_{Z_{(p)}}^{(n−m,m)}] = ∑_{i∈[1,m]}v_{p}(\frac{n − 2m + 1 + i}{i}) [S_{F_{p}}^{(n−m+i,m−i)}] .

First, let us consider the case 2m = n. If there was a unimodular ZS_{n}-lattice, there would be a unimodular Z_{(3)}S_{n}-lattice. However, D_{F_{3}}^{(m+2,m−2)} appears in [S_{Z_{(3)}}^{(n−m,m),∗}/S_{Z_{(3)}}^{(n−m,m)}] with multiplicity 1, so this is impossible by (2.11).

Now, let us consider the case 2m < n and assume the existence of a unimodular ZS_{n}- lattice. At a prime p, existence of a unimodular Z_{(p)}S_{n}-lattice implies by (2.11) that all multiplicities of simple modules in [S_{Z_{(p)}}^{(n−m,m),∗}/S_{Z_{(p)}}^{(n−m,m)}] are even. Considering the

multiplicity of D_{F_{p}}^{(n−m+i,m−i)} for i ∈ [1, m], beginning with i = 1, we obtain the condition that 2 divides v_{p}(\frac{n−2m+1+i}{i}) for all i ∈ [1, m]. Since this holds for all primes p, and at least for i ∈ [1, 3], we conclude that there exist positive integers x, y, z such that

\begin{aligned}n − 2m + 2 & = & x^{2} \\ n − 2m + 3 & = & 2y^{2} \\ n − 2m + 4 & = & 3z^{2} , \end{aligned}

whence

(∗)  \begin{aligned}2y^{2} − x^{2} & = & 1 \\ 3z^{2} − x^{2} & = & 2 . \end{aligned}

E. Wirsing [31] proved that (x, y, z) = (1, 1, 1) is the only solution of (∗) in positive integers, as reproduced below. Since x = 1 would correspond to n = 2m−1, this assertion contradicts the assumption on the existence of a unimodular ZS_{n}-lattice and proves the theorem.

First, we remark that a solution (x, y, z) consists of pairwise coprime integers, and that x ≡_{2} 1.

The rational points (ξ, ζ) on the ellipse ξ^{2} + 3ζ^{2} = 4 are parametrized by

(ξ, ζ) = (2\frac{1 − 3t^{2}}{1 + 3t^{2}} , \frac{4t}{1 + 3t^{2}} ) ,

where t ∈ Q ∪ \{∞\}. Letting (ξ, ζ) = (\frac{x}{y}, \frac{z}{y}), a solution (x, y, z) yields such a rational point. We may exclude t = ∞, corresponding to (−2, 0), since z ≠ 0. Writing t = \frac{r}{s} with coprime positive integers r and s, we obtain r ≡_{2} s ≡_{2} 1 since r ≢_{2} s would imply x ≡_{2} 0. If s ≢_{3} 0, then rs and (s^{2} + 3r^{2})/4 are coprime, whence

(x, y, z) = ((s^{2} − 3r^{2})/2, (s^{2} + 3r^{2})/4, rs) .

Now x^{2} = 3z^{2} − 2 yields

(∗∗_{1})  s^{4} − 1 = 2 · (\frac{3}{4}(r^{2} − s^{2}))^{2} .

If s ≡_{3} 0, then rs/3 and (s^{2} + 3r^{2})/12 are coprime, whence

(x, y, z) = \frac{1}{3} ((s^{2} − 3r^{2})/2, (s^{2} + 3r^{2})/4, rs) .

Now x^{2} = 3z^{2} − 2 yields

(∗∗_{2})  r^{4} − 1 = 2 · (\frac{1}{12}(s^{2} − 9r^{2}))^{2} .

By a result of Euler, however, the integral equation

(∗∗)  u^{4} − v^{4} = 2w^{2}

is unsolvable if w ≠ 0 [32, p. 82]. Therefore, the only solution to (∗) in positive integers results from (∗∗_{2}) as being (x, y, z) = (1, 1, 1).

2.5 Some three- and four-part partitions

The partial results and conjectures that follow we could lift from the table of F. Lübeck [15] and from calculations of decomposition numbers due to James, Williams, To Law, Benson, Müller et al. [2, 5, 10, 18, 19, 20, 30]. See also [5, p. 113]. A general result for p large that covers the respective first cases listed in this section, is given below (3.5).

Recall that we stipulated the module D_{F_{p}}^{μ} to be zero whenever μ is not a p-regular partition.

Proposition 2.14 (n − 3, 2, 1) Let n ⩾ 5. For p > 3, we have

[S_{F_{p}}^{(n−3,2,1)}] = [D_{F_{p}}^{(n−3,2,1)}]_{0} + {n ≡_{p} 3}[D_{F_{p}}^{(n−2,1^{2})}]_{v_{p}(n−3)} + {n ≡_{p} 1}[D_{F_{p}}^{(n−2,2)}]_{v_{p}(n−1)} .

For p = 2, we have

\begin{aligned}[S_{F_{2}}^{(n−3,2,1)}] & = & [D_{F_{2}}^{(n−3,2,1)}]_{0} + \{n ≡_{2} 1\}[D_{F_{2}}^{(n−2,2)}]_{v_{2}(n−1)+v_{2}(n−3)} \\ & & +\{n ≡_{2} 1\}[D_{F_{2}}^{(n)}]_{2\{n≡_{4}1\}+(v_{2}(n−3)−1)\{n≡_{4}3\}} . \end{aligned}

For p = 3, we have

[S_{F_{3}}^{(n−3,2,1)}] = [D_{F_{3}}^{(n−3,2,1)}]_{0} + [D_{F_{3}}^{(n−3,3)}]_{1} + {n ≡_{3} 1}[D_{F_{3}}^{(n−2,2)}]_{1+v_{3}(n−1)}
+{n ≡_{3} 0}[D_{F_{3}}^{(n−1,1)}]_{1+v_{3}(n−3)} + {n ≡_{3} 0}[D_{F_{3}}^{(n−2,1^{2})}]_{v_{3}(n−3)}
+(1 + {n ≡_{9} 2, 3, 4})[D_{F_{3}}^{(n)}]_{1} .

Schaper’s formula (see e.g. [16, 5.33]) reads

\begin{aligned}[S_{Z_{(p)}}^{(n−3,2,1),∗}/S_{Z_{(p)}}^{(n−3,2,1)}] & = & v_{p}(3)[S_{F_{p}}^{(n−3,3)}] + v_{p}(n − 1)[S_{F_{p}}^{(n−2,2)}] − v_{p}(\frac{n−1}{3}) [S_{F_{p}}^{(n)}] \\ & & +v_{p}(n − 3)[S_{F_{p}}^{(n−2,1^{2})}] , \end{aligned}

where p is an arbitrary prime, and where n ⩾ 6. We expand this formula into simple modules using [5, 24.15] and [10, App.]. Since by loc. cit. the decomposition numbers of S_{F_{p}}^{(n−3,2,1)} itself are contained in {0, 1, 2}, we may apply (1.6, 1.7) to obtain the distribution of the occurring simple modules over the Jantzen subquotients as stated above.

Lemma 2.15 Let n ⩾ 6. We have [S_{F_{2}}^{(n−4,2^{2})} : D_{F_{2}}^{(n−2,2)}] = {n ≡_{2} 0} .

This follows by [10, 3.13] if n is odd, and by [10, 4.5] if n is even. We state this explicitly since in this case we do not have λ_{3} < p, cf. [10, Introduction; more precisely, 4.13].

Conjecture 2.16 Let n ⩾ 6. We have [S_{F_{2}}^{(n−4,2^{2})} : D_{F_{2}}^{(n)}] = 1 + {n ≡_{4} 0, 1} . Proposition 2.17 (n − 4, 2^{2}) Let n ⩾ 6. For p > 3, we have

[S_{F_{p}}^{(n−4,2^{2})}] = [D_{F_{p}}^{(n−4,2^{2})}]_{0} + {n ≡_{p} 2}[D_{F_{p}}^{(n−2,2)}]_{v_{p}(n−2)} + {n ≡_{p} 3}[D_{F_{p}}^{(n−3,2,1)}]_{v_{p}(n−3)} . Suppose p = 2. If (2.16) holds true, then

\begin{aligned}[S_{F_{2}}^{(n−4,2^{2})}] & = & [D_{F_{2}}^{(n−4,3,1)}]_{1} + (1 + \{n ≡_{4} 0, 1\})[D_{F_{2}}^{(n)}]_{1+\{n≡_{4}2,3\}} \\ & & +\{n ≡_{2} 0\}[D_{F_{2}}^{(n−2,2)}]_{v_{2}(n−2)+1} + \{n ≡_{2} 1\}[D_{F_{2}}^{(n−3,2,1)}]_{v_{2}(n−3)+1} . \end{aligned}

Suppose p = 3. For n = 6, we have

[S_{F_{3}}^{(2^{3})}] = [D_{F_{3}}^{(3,2,1)}]_{1} + [D_{F_{3}}^{(6)}]_{2} .

If n ⩾ 7, there exist a_{n}, b_{n} ⩾ 1 with a_{n} + b_{n} = (1 + v_{3}(n − 3)) + (1 + v_{3}(n − 3)) such that

[S_{F_{3}}^{(n−4,2^{2})}] = [D_{F_{3}}^{(n−4,2^{2})}]_{0} + [D_{F_{3}}^{(n−4,4)}]_{1} + {n ≡_{3} 0}[D_{F_{3}}^{(n−3,2,1)}]_{v_{3}(n−3)}
+{n ≡_{3} 0}[D_{F_{3}}^{(n−3,3)}]_{v_{3}(n−3)+1} + {n ≡_{3} 2}[D_{F_{3}}^{(n−2,2)}]_{v_{3}(n−2)+1}
+(1 + {n ≡_{9} 4, 5, 6})[D_{F_{3}}^{(n−1,1)}]_{1} + {n ≡_{9} 0, 6}[D_{F_{3}}^{(n)}]_{2}
+{n ≡_{9} 3}([D_{F_{3}}^{(n)}]_{a_{n}} + [D_{F_{3}}^{(n)}]_{b_{n}}) .

Schaper’s formula reads

[S_{Z_{(p)}}^{(n−4,2^{2}),∗}/S_{Z_{(p)}}^{(n−4,2^{2})}] = v_{p}(\frac{3}{2}) [S_{F_{p}}^{(n−4,4)}] + v_{p}(\frac{n−2}{2}) [S_{F_{p}}^{(n−2,2)}] − v_{p}(\frac{n−2}{3}) [S_{F_{p}}^{(n−1,1)}]

+v_{p}(2)[S_{F_{p}}^{(n−4,3,1)}] + v_{p}(n − 3)[S_{F_{p}}^{(n−3,2,1)}] − v_{p}(\frac{n−3}{2}) [S_{F_{p}}^{(n−2,1^{2})}] ,

where p is an arbitrary prime and where n ⩾ 8. The result now follows by [5, 24.15] and [10], using (1.6, 1.7, 2.15). Proposition 2.18 (n − 4, 3, 1) Suppose n ⩾ 7. For p > 4, we have

\begin{aligned}[S_{F_{p}}^{(n−4,3,1)}] & = & [D_{F_{p}}^{(n−4,3,1)}]_{0} + \{n ≡_{p} 2\}[D_{F_{p}}^{(n−3,3)}]_{v_{p}(n−2)} + \{n ≡_{p} 4\}[D_{F_{p}}^{(n−2,1^{2})}]_{v_{p}(n−4)} \\ & & +\{n ≡_{p} 5\}[D_{F_{p}}^{(n−3,2,1)}]_{v_{p}(n−5)} . \end{aligned}

Suppose p = 2. There exist a_{n}, b_{n}, c_{n}, d_{n}, e_{n}, f_{n}, g_{n}, h_{n}, i_{n}, j_{n} ⩾ 1 with

\begin{aligned}a_{n} + b_{n} & = & 1 + (1 + v_{2}(n − 2)) \\ c_{n} + d_{n} & = & 2 + 2 \\ e_{n} + f_{n} + g_{n} & = & 1 + 1 + v_{2}(n − 4) \\ h_{n} + i_{n} + j_{n} & = & 1 + 1 + v_{2}(n − 5) \end{aligned}

such that

[S_{F_{2}}^{(n−4,3,1)}] = [D_{F_{2}}^{(n−4,3,1)}]_{0} + [D_{F_{2}}^{(n−4,4)}]_{2} + {n ≡_{2} 0}[D_{F_{2}}^{(n−3,3)}]_{2+v_{2}(n−2)}
+{n ≡_{2} 1}[D_{F_{2}}^{(n−3,2,1)}]_{v_{2}(n−5)} + {n ≡_{4} 0}[D_{F_{2}}^{(n−2,2)}]_{v_{2}(n−4)}
+{n ≡_{4} 1}[D_{F_{2}}^{(n−2,2)}]_{1+v_{2}(n−5)} + {n ≡_{4} 2}([D_{F_{2}}^{(n−2,2)}]_{a_{n}} + [D_{F_{2}}^{(n−2,2)}]_{b_{n}})
+{n ≡_{4} 0}[D_{F_{2}}^{(n−1,1)}]_{2+v_{2}(n−4)} + {n ≡_{8} 2, 7}[D_{F_{2}}^{(n)}]_{2}
+{n ≡_{8} 3, 6}([D_{F_{2}}^{(n)}]_{c_{n}} + [D_{F_{2}}^{(n)}]_{d_{n}}) + 2{n ≡_{8} 0, 1}[D_{F_{2}}^{(n)}]_{1}
+{n ≡_{8} 4}([D_{F_{2}}^{(n)}]_{e_{n}} + [D_{F_{2}}^{(n)}]_{f_{n}} + [D_{F_{2}}^{(n)}]_{g_{n}})
+{n ≡_{8} 5}([D_{F_{2}}^{(n)}]_{h_{n}} + [D_{F_{2}}^{(n)}]_{i_{n}} + [D_{F_{2}}^{(n)}]_{j_{n}}) .

Suppose p = 3. We have

\begin{aligned}[S_{F_{3}}^{(n−4,3,1)}] & = & [D_{F_{3}}^{(n−4,3,1)}]_{0} + \{n ≡_{3} 2\}[D_{F_{3}}^{(n−3,3)}]_{v_{3}(n−2)+v_{3}(n−5)} + \{n ≡_{3} 2\}[D_{F_{3}}^{(n−3,2,1)}]_{v_{3}(n−5)} \\ & & +\{n ≡_{3} 1\}[D_{F_{3}}^{(n−2,1^{2})}]_{v_{3}(n−4)} + \{n ≡_{9} 2\}[D_{F_{3}}^{(n)}]_{2} + \{n ≡_{9} 5\}[D_{F_{3}}^{(n)}]_{v_{3}(n−5)−1} . \end{aligned}

Schaper’s formula reads

\begin{aligned}[S_{Z_{(p)}}^{(n−4,3,1),∗}/S_{Z_{(p)}}^{(n−4,3,1)}] & = & −v_{p}(\frac{n−2}{4}) [S_{F_{p}}^{(n)}] + v_{p}(n − 2)[S_{F_{p}}^{(n−3,3)}] + v_{p}(4)[S_{F_{p}}^{(n−4,4)}] \\ & & +v_{p}(\frac{n−4}{2}) [S_{F_{p}}^{(n−2,1^{2})}] + v_{p}(n − 5)[S_{F_{p}}^{(n−3,2,1)}] , \end{aligned}

where p is an arbitrary prime, and where n ⩾ 8. The result follows by [5, 24.15] and [10, App.], using (1.6). Lemma 2.19 Suppose given n ⩾ 6.

(i) If n ⩾ 8, then [S_{F_{2}}^{(n−4,2,1^{2})} : D_{F_{2}}^{(n−4,3,1)}] = 1. (ii) For p an arbitrary prime, we have [S_{F_{p}}^{(n−4,2,1^{2})} : D_{F_{p}}^{(n−3,2,1)}] = {n ≡_{p} 1}. (iii) For p > 4 prime, we have [S_{F_{p}}^{(n−4,2,1^{2})} : D_{F_{p}}^{(n−3,1^{3})}] = {n ≡_{p} 4}. This follows from [10, 3.6, 3.13]. Again, we state this explicitly, this time since (n − 4, 2, 1^{2}) is a four-part-partition. Assertion (i) also follows by [6, Th. A]. Conjecture 2.20 Suppose given n ⩾ 6.

(i) If n ⩾ 9, then [S_{F_{2}}^{(n−4,2,1^{2})} : D_{F_{2}}^{(n−4,4)}] = 1. (ii) If n ⩾ 7, then [S_{F_{2}}^{(n−4,2,1^{2})} : D_{F_{2}}^{(n−3,3)}] = {n ≡_{2} 0}. (iii) We have [S_{F_{2}}^{(n−4,2,1^{2})} : D_{F_{2}}^{(n−2,2)}] = 2{n ≡_{4} 0, 1} + 3{n ≡_{4} 2} + {n ≡_{4} 3}. (iv) We have [S_{F_{2}}^{(n−4,2,1^{2})} : D_{F_{2}}^{(n−1,1)}] = 2{n ≡_{4} 0} + {n ≡_{4} 2}.

(v) If n ⩾ 7, then [S_{F_{2}}^{(n−4,2,1^{2})} : D_{F_{2}}^{(n)}] = {n ≡_{8} 7} + 2{n ≡_{8} 0, 2, 3} + 3{n ≡_{8} 1, 4, 6} + 4{n ≡_{8} 5}.

(vi) We have [S_{F_{3}}^{(n−4,2,1^{2})} : D_{F_{3}}^{(n−3,3)}] = {n ≡_{3} 1}. (vii) We have [S_{F_{3}}^{(n−4,2,1^{2})} : D_{F_{3}}^{(n)}] = {n ≡_{3} 1} + {n ≡_{9} 4}. Proposition 2.21 (n − 4, 2, 1^{2}) Suppose n ⩾ 6. For p > 4, we have

[S_{F_{p}}^{(n−4,2,1^{2})}] = [D_{F_{p}}^{(n−4,2,1^{2})}]_{0} + {n ≡_{p} 1}[D_{F_{p}}^{(n−3,2,1)}]_{v_{p}(n−1)} + {n ≡_{p} 4}[D_{F_{p}}^{(n−3,1^{3})}]_{v_{p}(n−4)} . Suppose p = 2. For n = 6, we obtain

[S_{F_{2}}^{(2^{2},1^{2})}] = [D_{F_{2}}^{(4,2)}]_{3} + [D_{F_{2}}^{(5,1)}]_{2} + [D_{F_{2}}^{(6)}]_{2} .

If n ⩾ 7 and if (2.20 i-v) and (2.16) hold true, then there exist a_{n}, …, z_{n} ⩾ 1 with

\begin{aligned}a_{n} + b_{n} & = & 1 + (3 + v_{2}(n − 4)) \\ c_{n} + d_{n} + e_{n} & = & 1 + 1 + 4 \\ f_{n} + g_{n} & = & 3 + 3 \\ h_{n} + i_{n} + j_{n} & = & 2 + 2 + (1 + v_{2}(n − 1)) \\ k_{n} + l_{n} & = & 2 + 2 \\ m_{n} + o_{n} + q_{n} & = & 2 + 2 + (1 + v_{2}(n − 4)) \\ r_{n} + s_{n} + t_{n} + u_{n} & = & 1 + 1 + 3 + 3 \\ v_{n} + w_{n} + x_{n} & = & 1 + 1 + 3 \\ y_{n} + z_{n} & = & (1 + v_{2}(n − 4)) + (1 + v_{2}(n − 4)) \end{aligned}

such that

[S_{F_{2}}^{(n−4,2,1^{2})}] = [D_{F_{2}}^{(n−4,3,1)}]_{3} + [D_{F_{2}}^{(n−4,4)}]_{1} + {n ≡_{2} 1}[D_{F_{2}}^{(n−3,2,1)}]_{3+v_{2}(n−1)}

+{n ≡_{2} 0}[D_{F_{2}}^{(n−3,3)}]_{1+v_{2}(n−4)} + (2{n ≡_{4} 1} + {n ≡_{4} 3})[D_{F_{2}}^{(n−2,2)}]_{1} +{n ≡_{4} 0}([D_{F_{2}}^{(n−2,2)}]_{a_{n}} + [D_{F_{2}}^{(n−2,2)}]_{b_{n}}) +{n ≡_{4} 2}([D_{F_{2}}^{(n−2,2)}]_{c_{n}} + [D_{F_{2}}^{(n−2,2)}]_{d_{n}} + [D_{F_{2}}^{(n−2,2)}]_{e_{n}}) +{n ≡_{8} 0}([D_{F_{2}}^{(n)}]_{f_{n}} + [D_{F_{2}}^{(n)}]_{g_{n}}) +{n ≡_{8} 1}([D_{F_{2}}^{(n)}]_{h_{n}} + [D_{F_{2}}^{(n)}]_{i_{n}} + [D_{F_{2}}^{(n)}]_{j_{n}}) +{n ≡_{8} 2, 3}([D_{F_{2}}^{(n)}]_{k_{n}} + [D_{F_{2}}^{(n)}]_{l_{n}}) +{n ≡_{8} 4}([D_{F_{2}}^{(n)}]_{m_{n}} + [D_{F_{2}}^{(n)}]_{o_{n}} + [D_{F_{2}}^{(n)}]_{q_{n}}) +{n ≡_{8} 5}([D_{F_{2}}^{(n)}]_{r_{n}} + [D_{F_{2}}^{(n)}]_{s_{n}} + [D_{F_{2}}^{(n)}]_{t_{n}} + [D_{F_{2}}^{(n)}]_{u_{n}}) +{n ≡_{8} 6}([D_{F_{2}}^{(n)}]_{v_{n}} + [D_{F_{2}}^{(n)}]_{w_{n}} + [D_{F_{2}}^{(n)}]_{x_{n}}) + {n ≡_{8} 7}[D_{F_{2}}^{(n)}]_{3} +{n ≡_{4} 2}[D_{F_{2}}^{(n−1,1)}]_{2} + {n ≡_{4} 0}([D_{F_{2}}^{(n−1,1)}]_{y_{n}} + [D_{F_{2}}^{(n−1,1)}]_{z_{n}}) .

Suppose p = 3. If (2.20 vi, vii) hold true, then

\begin{aligned}[S_{F_{3}}^{(n−4,2,1^{2})}] & = & [D_{F_{3}}^{(n−4,2,1^{2})}]_{0} + \{n ≡_{3} 1\}[D_{F_{3}}^{(n−3,3)}]_{v_{3}(n−1)} + (\{n ≡_{3} 1\} + \{n ≡_{9} 4\})[D_{F_{3}}^{(n)}]_{v_{3}(n−1)} \\ & & +\{n ≡_{3} 1\}[D_{F_{3}}^{(n−3,2,1)}]_{v_{3}(n−1)+v_{3}(n−4)} . \end{aligned}

Schaper’s formula reads

[S_{Z_{(p)}}^{(n−4,2,1^{2}),∗}/S_{Z_{(p)}}^{(n−4,2,1^{2})}] = v_{p}(2)[S_{F_{p}}^{(n−4,2^{2})}] + v_{p}(4)[S_{F_{p}}^{(n−4,3,1)}] + v_{p}(n − 1)[S_{F_{p}}^{(n−3,2,1)}]
−v_{p}(2)[S_{F_{p}}^{(n−4,4)}] − v_{p}(\frac{n−1}{2}) [S_{F_{p}}^{(n−2,2)}] + v_{p}(\frac{n−1}{4}) [S_{F_{p}}^{(n)}]
+v_{p}(n − 4)[S_{F_{p}}^{(n−3,1^{3})}] ,

where p is an arbitrary prime, and where n ⩾ 8. The result follows by [5, 24.15] and [10, App.], using (1.6, 1.7).

As a general pattern, one seems to observe the following.

Conjecture 2.22 Let p be a prime, let n ⩾ 1. Given a partition λ of n and an integer k ⩾ 0, we denote by λ + k the partition (λ_{1} + k, λ_{2}, λ_{3}, …). Suppose λ to be a partition of n, and μ to be a p-regular partition of n.

(i) There exists integers a, J ⩾ 0 such that for j ⩾ J, we have

[S_{F_{p}}^{λ} : D_{F_{p}}^{μ}] = [S_{F_{p}}^{λ+jp^{a}} : D_{F_{p}}^{μ+jp^{a}}] .

(ii) Let λ^{r} denote the p-regularization of λ in the sense of [6], and suppose (λ^{r})_{1} = λ_{1}. The multiplicity [S_{Z_{(p)}}^{λ+k,∗}/S_{Z_{(p)}}^{λ+k} : D_{F_{p}}^{(λ+k)^{r}}] does not depend on k ⩾ 0. (Cf. [6, Th. A] and (1.6).)

3 At a large prime

Notation 3.1 Let n ⩾ 1. Given a partition μ of n and j ∈ [1, μ_{2}], we denote by μ[j] the partition of n defined by

μ[j]_{i} := \{ \begin{matrix}μ_{1} + μ_{μ′_{j}} − j + 1 & if i = 1 \\ j − 1 & if i = μ′_{j} \\ μ_{i} & otherwise \end{matrix}

for i ⩾ 0. I.e. we cut the last row that meets the jth column in the jth column and append that piece in the first row.

Let R(μ) = {i ⩾ 1 | μ_{i} > μ_{i+1}} be the set of row numbers of removable nodes. Given k ∈ R(μ), we denote by μ(k) the partition of n − 1 defined by

μ(k)_{i} := \{ \begin{matrix}μ_{i} − 1 & if i = k \\ μ_{i} & else \end{matrix}

for i ⩾ 0. I.e. the diagram of μ(k) is obtained from the diagram of μ by removing the last node in row k.

Let λ be a partition of n, and suppose λ ≠ (n). Suppose p to be a prime strictly bigger than n − λ_{1} = ∑_{i⩾2} λ_{i}. Given j ∈ [1, λ_{2}], we denote by h_{j} := λ_{1} − j + λ′_{j} the hook length of the node (1, j) of the diagram of λ. Note that p divides at most one of the numbers h_{j} for j ∈ [1, λ_{2}] since h_{1} − h_{λ_{2}} < p. If p divides h_{t}, we denote s = λ′_{t}. Note that t ∈ [λ_{s+1} + 1, λ_{s}].

We shall need a particular case of the direction of the Carter Conjecture that has been proven by James and Murphy.

Theorem 3.2 ([9, p. 222]) If p does not divide h_{j} for all j ∈ [1, λ_{2}], then S_{F_{p}}^{λ} is irreducible.

The following is a particular case of a result of Carter and Payne.

Theorem 3.3 ([3, p. 425]) Suppose that p divides h_{t}. Then Hom_{F_{p}S_{n}}(S_{F_{p}}^{λ[t]}, S_{F_{p}}^{λ}) ≠ 0. In particular, [S_{F_{p}}^{λ} : D_{F_{p}}^{λ[t]}] ⩾ 1.

The idea of proof and the main ingredient of (3.4), hence of (3.5), are due to Kleshchev.

Proposition 3.4 Suppose that p divides h_{t}. Then [S_{F_{p}}^{λ}] = [D_{F_{p}}^{λ}] + [D_{F_{p}}^{λ[t]}].

Consider the case λ_{1} > λ_{2}. By (3.3), it suffices to show equality of the restrictions to S_{n−1} in the Grothendieck group. To prove this equality, by induction, we may assume the assertion to hold for S_{n−1}, yielding

[S_{F_{p}}^{λ}]|_{S_{n−1}} =^{[5, 9.2]} ∑_{i∈R(λ)}[S_{F_{p}}^{λ(i)}]
=^{induction} ∑_{i∈R(λ)} ([D_{F_{p}}^{λ(i)}] + {i ≠ 1 and λ_{i} ≠ t}[D_{F_{p}}^{λ(i)[t]}]
+ {i = 1 and λ_{s+1} + 1 < t}[D_{F_{p}}^{λ(1)[t−1]}]) .

On the other hand, since all removable nodes, except for (s, t) if λ_{s} = t, are normal, and, using p | h_{t}, all normal nodes are good, modular branching [11, 0.6] enables us to calculate

[D_{F_{p}}^{λ}]|_{S_{n−1}} = ∑_{i∈R(λ)}{λ_{i} ≠ t}[D_{F_{p}}^{λ(i)}] ,

as well as

[D_{F_{p}}^{λ[t]}]|_{S_{n−1}} = {λ_{s+1} + 1 < t}[D_{F_{p}}^{λ(s)[t−1]}] + ∑_{i∈R(λ)\{s}}[D_{F_{p}}^{λ(i)[t]}] .

We are done by remarking that if λ_{s} > t, then λ(s)[t] = λ(1)[t]; that if λ_{s} = t, then λ(s) = λ(1)[t]; and that if λ_{s+1} + 1 < t, then λ(s)[t − 1] = λ(1)[t − 1].

Consider the case λ_{1} = λ_{2}. From p | h_{t} we conclude that λ_{2} + λ′_{1} ⩾ p + 1, and from p > n − λ_{1} we infer that λ_{2} + λ′_{1} < p + 2. Now, λ_{2} + λ′_{1} = p + 1 leads to λ = (k^{2}, 1^{l}), with k ⩾ 1, l ⩾ 0 and k + l = p − 1, so in particular t = 1.

If k = 1, then [S_{F_{p}}^{λ}] = [D_{F_{p}}^{λ}] + [D_{F_{p}}^{λ[t]}], since [D_{F_{p}}^{(1^{p})}] = 0 by convention, and since [S_{F_{p}}^{(1^{p})}] =

[D_{F_{p}}^{(2,1^{p−2})}] by [6, Th. A].

If k > 1, we obtain one the one hand, by induction,

[S_{F_{p}}^{λ}]|_{S_{n−1}} = [D_{F_{p}}^{(k,k−1,1^{l})}] + {l ⩾ 1} ([D_{F_{p}}^{(k+1,k−1,1^{l−1})}] + [D_{F_{p}}^{(k^{2},1^{l−1})}]) + {l = 0}[D_{F_{p}}^{(n−1)}] .

On the other hand, modular branching [11, 0.6] yields

\begin{aligned}[D_{F_{p}}^{(k^{2},1^{l})}]|_{S_{n−1}} & = & [D_{F_{p}}^{(k,k−1,1^{l})}] \\ \{l ⩾ 1\}[D_{F_{p}}^{(k+1,k,1^{l−1})}]|_{S_{n−1}} & = & \{l ⩾ 1\} ([D_{F_{p}}^{(k+1,k−1,1^{l−1})}] + [D_{F_{p}}^{(k^{2},1^{l−1})}]) \\ \{l = 0\}[D_{F_{p}}^{(n)}]|_{S_{n−1}} & = & \{l = 0\}[D_{F_{p}}^{(n−1)}] . \end{aligned}

Thus

[S_{F_{p}}^{λ}] = [S_{F_{p}}^{(k^{2},1^{l})}]
= [D_{F_{p}}^{(k^{2},1^{l})}] + {l ⩾ 1}[D_{F_{p}}^{(k+1,k,1^{l−1})}] + {l = 0}[D_{F_{p}}^{(n)}]
= [D_{F_{p}}^{λ}] + [D_{F_{p}}^{λ[t]}] .

Theorem 3.5 Let λ be a partition of n, let p be a prime such that p > n − λ_{1}. Let h_{j} and, if existent, t be as in (3.1).

If p does not divide h_{j} for any j ∈ [1, λ_{2}], then

[S_{F_{p}}^{λ}] = [D_{F_{p}}^{λ}]_{0} .

If p divides h_{t}, then

[S_{F_{p}}^{λ}] = [D_{F_{p}}^{λ}]_{0} + [D_{F_{p}}^{λ[t]}]_{v_{p}(h_{t})} .

If p divides h_{t}, Schaper’s formula (see e.g. [16, 5.33]) reads

[S_{Z_{(p)}}^{λ,∗}/S_{Z_{(p)}}^{λ}] = ∑_{i∈[2,s]}(−1)^{s−i}v_{p}(h_{t})[S_{F_{p}}^{λ⟨i⟩}] ,

where the diagram of the partition of λ⟨i⟩ is obtained by removing the skew hook belonging to the node (i, t) from the diagram of λ, and by attaching as many nodes to the first row as there are in this skew hook (cf. [9, p. 223]).

Now, if h⟨i⟩_{j} denotes the hook length of the node (1, j) of the diagram of λ⟨i⟩, where j ∈ [1, λ_{2}], and if p divides h⟨i⟩_{t⟨i⟩}, then λ⟨i⟩[t⟨i⟩] = λ⟨i − 1⟩ for i ∈ [3, s]. Moreover, λ⟨s⟩ = λ[t], and S_{F_{p}}^{λ⟨2⟩} = D_{F_{p}}^{λ⟨2⟩} by (3.2). Thus by (3.4), we have

[S_{Z_{(p)}}^{λ,∗}/S_{Z_{(p)}}^{λ}] = v_{p}(h_{t})[D_{F_{p}}^{λ[t]}] ,

and we are done by (1.6).

Corollary 3.6 Suppose p divides h_{t}. As modules over Z_{(p)}, we have

S_{Z_{(p)}}^{λ,∗}/S_{Z_{(p)}}^{λ} ≃ (Z_{(p)}/h_{t}Z_{(p)})^{dim_{^{F}^{p}} D_{Fp}^{λ[t]}} ,

where, using the notation of the proof of (3.5), we have

dim_{F_{p}} D_{F_{p}}^{λ[t]} = ∑_{i∈[2,s]}(−1)^{s−i} dim_{F_{p}} S_{F_{p}}^{λ⟨i⟩} .

This follows using (3.5), its proof and (1.4).

4 Transposition

Let n ⩾ 1. Given a ZS_{n}-module X, we denote X^{−} := X ⊗_{Z} S^{(1^{n})}. Given a partition λ, we denote n_{λ} := rk_{Z} S^{λ}.

Proposition 4.1 Suppose given n ⩾ 1, p a prime and λ a partition of n. For i ∈ [0, v_{p}(n!/n_{λ})], we have

\frac{S_{F_{p}}^{λ}(i)}{S_{F_{p}}^{λ}(i + 1)} ≃ (\frac{S_{F_{p}}^{λ′}(v_{p}(n!/n_{λ}) − i)}{S_{F_{p}}^{λ′}(v_{p}(n!/n_{λ}) − i + 1)})^{−} .

For i > v_{p}(n!/n_{λ}), we have S_{F_{p}}^{λ}(i)/S_{F_{p}}^{λ}(i + 1) = 0.

By [5, 6.7] (and [13, 6.2.5]), we have an isomorphism

[figure: S^{λ}; ∼; S^{λ′,∗,−}; ⟨a⟩; ({a′}, −) .]

Given a λ-tableau [a], we let C_{a} denote the column stabilizer of [a] in S_{n}, and κ_{a}^{−} := ∑_{σ∈C_{a}} σε_{σ}; we let R_{a} denote the row stabilizer of [a] in S_{n} and ρ_{a}^{+} := ∑_{σ∈R_{a}} σ. Consider the composition

[figure: S^{λ}; η; S^{λ,∗}; ∼; S^{λ′,−}; η^{−}; S^{λ′,∗,−}; ∼; S^{λ}; ⟨a⟩; (\{a\}, −)κ_{a}^{−}; ⟨a′⟩κ_{a}^{−}; (\{a′\}, −)ρ_{a}^{+}κ_{a}^{−}; ⟨a⟩ρ_{a}^{+}κ_{a}^{−} = \frac{n!}{n_{λ}} ⟨a⟩ ,]

the last equality following from [5, 23.2 ii]. We localize at p and apply (1.8) to X = S_{Z_{(p)}}^{λ},

Y = S_{Z_{(p)}}^{λ′,−} and N = v_{p}(n!/n_{λ}).

Corollary 4.2 We have S_{F_{p}}^{λ}(v_{p}(n!/n_{λ}))/S_{F_{p}}^{λ}(v_{p}(n!/n_{λ}) + 1) ≃ D_{F_{p}}^{λ′,−}.

Corollary 4.3 For i ∈ [1, n_{λ}], the product of the ith elementary divisor of the Gram matrix of S^{λ} and the (n_{λ} + 1 − i)th elementary divisor of the Gram matrix of S^{λ′} yields n!/n_{λ}.

In particular, the largest elementary divisor of the Gram matrix of S^{λ} divides n!/n_{λ}.

The first argument is to localize at a prime p and to apply (1.4) to (4.1). In this sense, (4.1) is the module version of (4.3).

The second argument is to consider the composition used in the proof of (4.1) directly, which yields the result without reverting to localization.

Remark 4.4 A Brauer-Nesbitt type argument (cf. §6.3) shows that if G is a finite group, R a discrete valuation ring of characteristic zero with fraction field K splitting G, and X a simple RG-lattice carrying a nondegenerate G-invariant R-bilinear form, then the quotient of the last elementary divisor by the first elementary divisor of the Gram matrix of this bilinear form divides |G|/rk_{R}X in R. In this generality, there is no substitute for transposition of partitions. Nonetheless, in our particular case this fact can be derived using transposition of partitions (4.3).

The following has been discovered by C. D. Gay and rederived by James and Murphy (cf. [9, p. 234]).

Corollary 4.5 The product of the determinants of the Gram matrices of S^{λ} and of S^{λ′} is given by (n!/n_{λ})^{n_{λ}}.

For later use, we record the related

Lemma 4.6 We have (⟨a⟩ρ_{a}^{+}, −) = \frac{n!}{n_{λ}} (\{a\}, −).

To this end, we compose the string of morphisms displayed in the proof of (4.1) from S^{λ,∗} to S^{λ,∗}.

Sometimes, we shall abbreviate ⟨a⟩ρ_{a}^{+} =: ⟨a⟩ρ^{+}.

5 Explicit bases

So far, we have been calculating elementary divisors of Gram matrices for Specht modules without reference to any explicit diagonalization. Instead, we made use of known decomposition numbers as well as of Schaper’s formula. It would be preferable to construct two bases diagonalizing the Gram matrix, but this seems to be difficult in general. For hook partitions, however, explicit diagonalization is a simple (and precise) way to get the elementary divisors (5.5). We also treat the partition (2^{2}, 1^{n−4}) in this manner (5.14), but already this modest case indicates how complicated such bases seem to look like in general, at least in terms of polytabloids. We remark that even for simple modules whose dimension is known (such as two-row partitions, three-row partitions with third row < p, or those treated in [17]), explicit bases are largely unknown (or at least unpublished). Such a basis, lifted to the Specht module of the same partition, could locally be used as part of a diagonalizing basis. The bases found so far diagonalize globally. To simplify, we spare ourselves the construction of the second basis by giving the first basis in a convenient manner with respect to the standard choice of the second, see §5.1. By a matrix inversion of a unipotent upper triangular integral matrix, a diagonalizing second basis ensues.

5.1 A linear algebra lemma

Lemma 5.1 Let X be a finitely generated free Z-module of rank m ⩾ 1, let (−, =) be a nondegenerate bilinear form on X. Let G be the Gram matrix of that bilinear form with respect to some Z-linear basis. Let (x_{1}, …, x_{m}), (y_{1}, …, y_{m}) be two tuples of elements of X such that the following hold.

(i) We have (x_{i}, y_{i}) ≠ 0 for all i ∈ [1, m].

(ii) We have (x_{i}, y_{j}) = 0 for all i, j ∈ [1, m] such that i > j.

(iii) (x_{i}, y_{i}) divides (x_{i}, y_{j}) for all i, j ∈ [1, m] such that i < j.

(iv) (x_{i}, y_{i}) divides (x_{i+1}, y_{i+1}) for all i ∈ [1, m − 1].

(v) ∏_{i∈[1,m]}(x_{i}, y_{i}) = ± det G.

Then the tuples (x_{1}, …, x_{m}) and (y_{1}, …, y_{m}) are Z-linear bases of X. The elementary divisors of G are given by the tuple of integers

((x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{n}, y_{n})) .

Let g : X \overline{}X^{∗} send ξ ∈ X to (ξ, −). The tuple (x_{1}, …, x_{m}) is linear independent over Z, as its image in X^{∗} via g shows, using (i, ii). Likewise, the tuple (y_{1}, …, y_{m}) is linearly independent. In the composite embedding

Z⟨x_{1}, …, x_{m}⟩ [figure: f] X [figure: g] X^{∗} [figure: h] Z⟨y_{1}, …, y_{m}⟩^{∗} ,

both fgh and g have determinant ± det G, by (v). Thus f and h are equalities. Now (i, ii, iii, iv) ensure that the elementary divisors result as claimed.

The element y_{i} of the second basis is called the diagonal correspondent of the element x_{i} of the first basis.

5.2 Bases for hooks

The result of this section has independently been observed by Mathas and James [unpublished]. As ingredient for (5.1 v), we shall make use of the determinant of the Gram matrix, but not of Schaper’s formula itself.

Let n ⩾ 2, let l ∈ [0, n − 1], and let λ := (n − l, 1^{l}) be a hook partition of n. Note that a λ-polytabloid is determined by the tuple of entries in the first column, read from top to bottom. So given a tuple b = (b_{0}, b_{1}, …, b_{l}), the entries b_{i} ∈ [1, n] pairwise distinct, we shall write ⟨b⟩ for the λ-polytabloid it determines. In the sequel, by a tuple we understand a tuple with pairwise distinct entries. To the underlying set of a tuple we refer without further comment.

Counting ordered tuples, we obtain rk_{Z} S^{λ} = (\begin{matrix}n−1 \\ l \end{matrix}).

If b is a tuple and x an element, then (x, b) denotes the tuple b with an entry x appended on the left, (b, x) denotes the tuple b with an entry x appended on the right, etc. If tuple brackets appear within tabloid or polytabloid brackets, we omit the tuple brackets. For

instance, if n = 5, l = 2 and b = (5, 3), then ⟨2, b⟩ = ⟨\begin{matrix}2 & 1 & 4 \\ 5 \\ 3 \end{matrix}⟩.

Lemma 5.2 We have #(S^{λ,∗}/S^{λ}) = l!^{( \begin{matrix}n−1 \\ l \end{matrix} )} · n^{( \begin{matrix}n−2 \\ l−1 \end{matrix} )} .

This follows by induction on n using the Branching Theorem for Determinants [9, p. 225].

Lemma 5.3 Suppose given increasingly ordered tuples b, c ⊆ [2, n − 1] of length #b = #c = l. We have

(⟨n, b⟩, ⟨1, c⟩)_{0} = \{ \begin{matrix}1 & if & b = c , \\ 0 & if & b ≠ c . \end{matrix}

Lemma 5.4 Suppose given increasingly ordered tuples d ⊆ [2, n−1] of length #d = l −1, and c ⊆ [2, n] of length #c = l. We have

( ∑_{s∈[1,n−1]\d}⟨s, d, n⟩, ⟨1, c⟩)_{0} = \{ \begin{matrix}n & if & (d, n) = c , \\ 0 & if & (d, n) ≠ c . \end{matrix}

In fact,

\begin{aligned}( ∑_{s∈[1,n−1]\d}⟨s, d, n⟩, −)_{0} & = & (n − l − 1)!^{−1}(⟨1, d, n⟩ρ^{+}, −)_{0} \\ & =^{(4.6)} & n · (\{1, d, n\}, −) . \end{aligned}

Theorem 5.5 As abelian groups, we have

S^{λ,∗}/S^{λ} = S^{(n−l,1^{l}),∗}/S^{(n−l,1^{l})} ≃ (Z/l! Z)^{( \begin{matrix}n−2 \\ l \end{matrix} )} ⊕ (Z/n · l! Z)^{( \begin{matrix}n−2 \\ l−1 \end{matrix} )} .

Bases x and y that trigonalize the Gram matrix in the sense of (5.1) are given by a tuple y consisting of the standard basis, ordered in such a manner that all elements without entry n in the first column are before all elements with entry n in the first column, and by a tuple x = (x′, x′′), with x′ consisting of the elements of the form ⟨n, b⟩ as in (5.3), with respective diagonal correspondent ⟨1, b⟩ in y, and x′′ consisting of elements of the form ∑_{s∈[1,n]\d}⟨s, d, n⟩ as in (5.4), with respective diagonal correspondent ⟨1, d, n⟩ in y.

Example 5.6 Let n = 4 and λ = (2, 1^{2}). With respect to

x = (⟨ \begin{matrix}4 & 1 \\ 2 \\ 3 \end{matrix} ⟩ , ⟨ \begin{matrix}1 & 3 \\ 2 \\ 4 \end{matrix} ⟩ + ⟨ \begin{matrix}3 & 1 \\ 2 \\ 4 \end{matrix} ⟩ , ⟨ \begin{matrix}1 & 2 \\ 3 \\ 4 \end{matrix} ⟩ + ⟨ \begin{matrix}2 & 1 \\ 3 \\ 4 \end{matrix} ⟩) , y = (⟨ \begin{matrix}1 & 4 \\ 2 \\ 3 \end{matrix} ⟩ , ⟨ \begin{matrix}1 & 3 \\ 2 \\ 4 \end{matrix} ⟩ , ⟨ \begin{matrix}1 & 2 \\ 3 \\ 4 \end{matrix} ⟩) ,

the Gram matrix ( (x_{i}, y_{j}) )_{i,j} takes the form ( \begin{matrix}2 & −2 & 2 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{matrix} ).

Remark 5.7 (Jantzen subquotients) Let p be a prime. To consider the Jantzen subquotients, we have to distinguish two cases.

Case p ⩾ 3. Given a partition ν, we denote by ν^{r} the p-regularized partition in the sense of [6, p. 46], where the corresponding diagram construction is explained – the diagram of ν being given by [ν] := {(i, j) ∈ Z_{⩾1} × Z_{⩾1} | j ⩽ ν_{i}}.

Now, if p does not divide n, then S_{F_{p}}^{λ} ≃ D_{F_{p}}^{λ^{r}} is simple [6, p. 52]. By (5.5), we obtain

[S_{F_{p}}^{λ}] = [D_{F_{p}}^{λ^{r}}]_{v_{p}(l!)} .

So suppose that p divides n.

If l = 0, we obtain

[S_{F_{p}}^{λ}] = [S_{F_{p}}^{(n)}] = [D_{F_{p}}^{(n)}]_{0} .

If l = n − 1, we obtain by [6, Th. A]

[S_{F_{p}}^{λ}] = [S_{F_{p}}^{(1^{n})}] = [D_{F_{p}}^{(1^{n})^{r}}]_{v_{p}(n!)} .

If l ∈ [1, n−2], the decomposition numbers of [6, p. 52] and the long exact hook sequence of [22, Lem. 2] (cf. [13, 4.2.3, 4.2.4]) give

[S_{F_{p}}^{λ}] = [S_{F_{p}}^{(n−l,1^{l})}] = [D_{F_{p}}^{(n−l−{ l ⩾ n−n/p},1^{∗})^{r}}]_{v_{p}(l!)} + [D_{F_{p}}^{(n−(l−1)−{ (l−1) ⩾ n−n/p},1^{∗})^{r}}]_{v_{p}(l! n)} ,

where we abbreviated (n − j, 1^{∗}) := (n − j, 1^{j}). In fact, the image in S_{F_{p}}^{λ} of the differential of the long exact hook sequence equals S_{F_{p}}^{λ}(l! n), as results from a comparison of [13, 4.2.3] with (5.4).

Case p = 2. Suppose given k, m ⩾ 0, written 2-adically as k + 1 = ∑_{s∈[0,K]} a_{s}2^{s} and m = ∑_{s∈[0,M]} b_{s}2^{s}, where a_{s}, b_{s} ∈ {0, 1}, where a_{K} = 1 and where b_{M} = 1 if m ⩾ 1, and M = −1 if m = 0. Assume K > M. Let t := max ({s ∈ [1, M] | a_{s} < b_{s}} ∪ {0}), and define

F_{2}(k, m) := ∑_{i⩾0}f_{2}(k, m−2i) = 2^{∑_{u∈[1,t]} a_{u}}·(1 + ∑_{s∈[t+1,M]}a_{s}b_{s} · 2^{∑_{u∈[t+1,s−1]} a_{u}} )·{km ≡_{2} 0} ,

the notation being as for (2.2).

First, suppose 2l ⩽ n − 1. For j ∈ [0, l], [5, p. 93] and [5, 24.15] yield [S_{F_{2}}^{λ} : D_{F_{2}}^{(n−j,j)}] = F_{2}(n − 2j, l − j) (cf. [29]). If n is even, we use the same argument as in the case p ⩾ 3, if n is odd, there is no further argument necessary, to conclude

[S_{F_{2}}^{λ}] = [S_{F_{2}}^{(n−l,1^{l})}] = ∑_{j∈[0,l]} F_{2}(n − 2j, l − j) [D_{F_{2}}^{(n−j,j)}]_{v_{2}(l! n^{{l≢_{2}j}})} .

Next, suppose 2l ⩾ n − 1. Since [S_{F_{2}}^{λ}] = [S_{F_{2}}^{λ′,∗,−}] = [S_{F_{2}}^{λ′}], we obtain in the same manner

[S_{F_{2}}^{λ}] = [S_{F_{2}}^{(n−l,1^{l})}] = [S_{F_{2}}^{(l+1,1^{n−l−1})}] = ∑_{j∈[0,n−l−1]} F_{2}(n−2j, n−l−1−j) [D_{F_{2}}^{(n−j,j)}]_{v_{2}(l! n^{{l≢_{2}j}})} .

Example 5.8 We have

[SF_{2}^{(6,1^{8})}] = 3[DF_{2}^{(14,0)}]_{7} + 2[DF_{2}^{(13,1)}]_{8} + 2[DF_{2}^{(12,2)}]_{7} + [DF_{2}^{(11,3)}]_{8} + [DF_{2}^{(10,4)}]_{7} + [DF_{2}^{(9,5)}]_{8} .

5.3 Gram matrix entries for two-column partitions

Let n ⩾ 1, 1 ⩽ h ⩽ n/2 and λ = (2^{h}, 1^{n−2h}). Recall that tuples have pairwise distinct entries by convention.

Lemma 5.9 Let i ∈ [0, h]. Suppose given pairwise disjoint tuples φ, ψ, ϑ, ζ ⊆ [1, n] of length

\begin{aligned}#φ & = & n − 2h + i , \\ #ψ & = & i , \\ #ϑ & = & h − i , \\ #ζ & = & h − i . \end{aligned}

Let [a] be the tableau with first column (ϑ, φ) and second column (ζ, ψ). Let [b] be the tableau with first column (ζ, φ) and second column (ϑ, ψ). Then

(⟨a⟩, ⟨b⟩) = h! · (h − i)! · (n − 2h + i)! .

We have to calculate

[figure: (; ϑ; φ; ζ; ψ; ,; ζ; φ; ϑ; ψ; ); = ∑_{τ ∈ C_{a}} ∑_{σ ∈ C_{b}} {{a}τ = {b}σ}ε_{τ}ε_{σ} .]

First, assume that ζ and ψ remain fixed in [a], i.e. that τ restricts to the identity on ζ ∪ψ. In order to obtain {a}τ = {b}σ, the permutation σ has to restrict to the identity on ζ ∪ ψ as well. Moreover, σ and τ have to restrict to the same automorphism of ϑ, whence the factor (h−i)!, and to the same automorphism on φ, whence the factor (n−2h+i)!. Next, removing the assumption, we remark that we may simultaneously permute the rows 1 to h of {a}τ and of {b}σ, whence the factor h!.

Lemma 5.10 Suppose given increasingly ordered tuples ξ, η ⊆ [1, n] of length h with intersection of cardinality i. We write ⟨ξ⟩ for the polytabloid with second column ξ, and first column [1, n]\ξ, increasingly ordered from top to bottom. Let ∑ ξ denote the sum of the entries of ξ. Then

(⟨ξ⟩, ⟨η⟩)_{0} = \frac{(h − i)!(n − 2h + i)!}{(n − 2h)!} · (−1)^{h−i+∑ ξ+∑ η} .

This follows from (5.9) by reordering the columns of the polytabloids involved.

5.4 Bases for (2^{2}, 1^{n−4})

Let n ⩾ 6, let λ = (2^{2}, 1^{n−4}). Given a, b ∈ [1, n], a ≠ b, we denote by ⟨\begin{matrix}a \\ b \end{matrix}⟩ the λpolytabloid with second column (a, b), and whose first column is increasingly ordered from top to bottom. That is, we use the notation that has already been employed in (5.10).

Lemma 5.11 We have

#(S^{λ,∗}/S^{λ}) = (2(n − 4)!)^{(n^{2}−3n)/2} · (n − 1)^{(n^{2}−3n−2)/2} · (n − 2)^{(n^{2}−5n+2)/2} · 2 .

The transposed partition has #(S^{(n−2,2),∗}/S^{(n−2,2)}) = (n − 1) · (n − 2)^{n−1}/2, as we take from the first example on [9, p. 224]. Since rk_{Z} S^{λ} = n(n−3)/2 by [5, 20.1], the assertion ensues from (4.5).

Lemma 5.12 We obtain the following table for values of (−, =)_{0} = \frac{1}{2(n−4)!}(−, =), in which the columns are indexed by the standard polytabloid basis of S^{λ}.

(−, =)_{0}  ⟨ \begin{matrix}b \\ n \end{matrix} ⟩  ⟨ \begin{matrix}b \\ n−1 \end{matrix} ⟩  ⟨ \begin{matrix}b \\ c \end{matrix} ⟩
b ∈ [2, n − 1]  b ∈ [2, n − 2]  b ∈ [2, n − 2],

c ∈ [4, n − 2], b < c

⟨ \begin{matrix}3 \\ n \end{matrix} ⟩  \begin{matrix}(n − 2)(n − 3) \\ if b = 3 \\ (−1)^{b} · (n − 3) \\ if b ≠ 3 \end{matrix}  \begin{matrix}(n − 3) \\ if b = 3 \\ (−1)^{b} · 2 \\ if b ≠ 3 \end{matrix}  \begin{matrix}(−1)^{c+n+1} · (n − 3) \\ if b = 3 \\ (−1)^{b+c+n+1} · 2 \\ if b ≠ 3 \end{matrix}

⟨ \begin{matrix}3 \\ 4 \end{matrix} ⟩

\begin{matrix}(−1)^{n+1} · (n − 3) \\ if b = 3 \\ (−1)^{n} · (n − 3) \\ if b = 4 \\ (−1)^{b+n+1} · 2 \\ if b ∉ \{3, 4\} \end{matrix} \begin{matrix}(−1)^{n} · (n − 3) \\ if b = 3 \\ (−1)^{n+1} · (n − 3) \\ if b = 4 \\ (−1)^{b+n} · 2 \\ if b ∉ \{3, 4\} \end{matrix}

(n − 2)(n − 3) if b = 3, c = 4 (n − 3) if b = 2, c = 4 (−1)^{c+1} · (n − 3) if b = 3, c ≠ 4 (−1)^{c} · (n − 3) if b = 4, c ≠ 4 (−1)^{b+c+1} · 2 if b, c ∉ {3, 4}

\begin{matrix}⟨ \begin{matrix}k \\ n−1 \end{matrix} ⟩ + (−1)^{k} ⟨ \begin{matrix}1 \\ n−1 \end{matrix} ⟩ \\ k ∈ [2, n − 2] \end{matrix}  \begin{matrix}(n − 1) \\ if b = k \\ 0 \\ if b ≠ k \end{matrix}  \begin{matrix}(n − 1)(n − 3) \\ if b = k \\ 0 \\ if b ≠ k \end{matrix}  \begin{matrix}(−1)^{n+c} · (n − 1) \\ if k = b \\ (−1)^{n+b} · (n − 1) \\ if k = c \\ 0 \\ if k ∉ \{b, c\} \end{matrix}

\begin{matrix}⟨ & \begin{matrix}3 \\ n \end{matrix} & ⟩ \\ +(−1)^{n} ⟨ \begin{matrix}n−1 \\ n \end{matrix} ⟩ \\ −(n − 3) ⟨ \begin{matrix}3 \\ n−1 \end{matrix} ⟩ \end{matrix}  0  \begin{matrix}−(n − 1)(n − 3)(n − 4) \\ if b = 3 \\ (−1)^{b+1} · (n − 1)(n − 4) \\ if b ≠ 3 \end{matrix} \begin{matrix}(−1)^{c+n+1} · (n − 1)(n − 4) \\ if b = 3 \\ (−1)^{b+c+n+1} · 2(n − 1) \\ if b ≠ 3 \end{matrix}

\begin{matrix}⟨ \begin{matrix}k \\ l \end{matrix} ⟩ ρ^{+} \\ k ∈ [2, n − 1], \\ l ∈ [4, n − 1], k < l \end{matrix}

0

\begin{matrix}(n − 1)(n − 2) \\ if k = b, l = n − 1 \\ 0 \\ else \end{matrix}

(n − 1)(n − 2) if k = b, l = c (−1)^{l+1}(n − 1)(n − 2) if k = c, b = 2 0 else

This table is to be verified using (5.10), except for the last row, for which we may most conveniently use (4.6). Explicitly, we have for example ⟨\begin{matrix}3 \\ 4 \end{matrix}⟩ ρ^{+} = ⟨\begin{matrix}3 \\ 4 \end{matrix}⟩−⟨\begin{matrix}1 \\ 4 \end{matrix}⟩+⟨\begin{matrix}3 \\ 2 \end{matrix}⟩+⟨\begin{matrix}1 \\ 2 \end{matrix}⟩.

Example 5.13 If n = 6, we obtain the following table.

\begin{matrix}(−, =)_{0} & ⟨ \begin{matrix}5 \\ 6 \end{matrix} ⟩ & ⟨ \begin{matrix}4 \\ 6 \end{matrix} ⟩ & ⟨ \begin{matrix}3 \\ 6 \end{matrix} ⟩ & ⟨ \begin{matrix}2 \\ 6 \end{matrix} ⟩ & ⟨ \begin{matrix}4 \\ 5 \end{matrix} ⟩ & ⟨ \begin{matrix}3 \\ 5 \end{matrix} ⟩ & ⟨ \begin{matrix}2 \\ 5 \end{matrix} ⟩ & ⟨ \begin{matrix}3 \\ 4 \end{matrix} ⟩ & ⟨ \begin{matrix}2 \\ 4 \end{matrix} ⟩ \\ ⟨ \begin{matrix}3 \\ 6 \end{matrix} ⟩ & −3 & 3 & 12 & 3 & 2 & 3 & 2 & −3 & −2 \\ ⟨ \begin{matrix}3 \\ 4 \end{matrix} ⟩ & 2 & 3 & −3 & −2 & −3 & 3 & 2 & 12 & 3 \\ ⟨ \begin{matrix}4 \\ 5 \end{matrix} ⟩ + ⟨ \begin{matrix}1 \\ 5 \end{matrix} ⟩ & 0 & 5 & 0 & 0 & 15 & 0 & 0 & −5 & 5 \\ ⟨ \begin{matrix}3 \\ 5 \end{matrix} ⟩ − ⟨ \begin{matrix}1 \\ 5 \end{matrix} ⟩ & 0 & 0 & 5 & 0 & 0 & 15 & 0 & 5 & 0 \\ ⟨ \begin{matrix}2 \\ 5 \end{matrix} ⟩ + ⟨ \begin{matrix}1 \\ 5 \end{matrix} ⟩ & 0 & 0 & 0 & 5 & 0 & 0 & 15 & 0 & 5 \\ ⟨ \begin{matrix}3 \\ 6 \end{matrix} ⟩ + ⟨ \begin{matrix}5 \\ 6 \end{matrix} ⟩ − 3 · ⟨ \begin{matrix}3 \\ 5 \end{matrix} ⟩ & 0 & 0 & 0 & 0 & −10 & −30 & −10 & −10 & −10 \\ ⟨ \begin{matrix}4 \\ 5 \end{matrix} ⟩ ρ^{+} & 0 & 0 & 0 & 0 & 20 & 0 & 0 & 0 & 20 \\ ⟨ \begin{matrix}3 \\ 5 \end{matrix} ⟩ ρ^{+} & 0 & 0 & 0 & 0 & 0 & 20 & 0 & 0 & 0 \\ ⟨ \begin{matrix}2 \\ 5 \end{matrix} ⟩ ρ^{+} & 0 & 0 & 0 & 0 & 0 & 0 & 20 & 0 & 0 \\ ⟨ \begin{matrix}3 \\ 4 \end{matrix} ⟩ ρ^{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & 0 \\ ⟨ \begin{matrix}2 \\ 4 \end{matrix} ⟩ ρ^{+} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 \end{matrix}

Using (5.1, 5.11), this yields

S^{(2^{2},1^{2}),∗}/S^{(2^{2},1^{2})} ≃ (Z/4Z)^{1} ⊕ (Z/20Z)^{3} ⊕ (Z/40Z)^{1} ⊕ (Z/80Z)^{4}

as abelian groups.

Theorem 5.14 (cf. (2.8)) Suppose n ⩾ 6. As abelian groups, we obtain

S^{(2^{2},1^{n−4}),∗}/S^{(2^{2},1^{n−4})} ≃ (Z/2^{{n≡_{2}1}} · 2(n − 4)! Z)^{1}
⊕ (Z/(n − 1) · 2(n − 4)! Z)^{n−3}
⊕ (Z/2^{{n≡_{2}0}}(n − 1) · 2(n − 4)! Z)^{1}
⊕ (Z/(n − 2)(n − 1) · 2(n − 4)! Z)^{(n^{2}−5n+2)/2} .

Trigonalizing bases x and y in the sense of (5.1) are given as follows. Let

y_{i} := ⟨\begin{matrix}n−i \\ n \end{matrix}⟩ for i ∈ [1, n − 2]
y_{(n−2)+i} := ⟨\begin{matrix}n−1−i \\ n−1 \end{matrix}⟩ for i ∈ [1, n − 3]
y_{(2n−5)+i} := ⟨\begin{matrix}b_{i} \\ c_{i} \end{matrix} ⟩  for i ∈ [1, (\begin{matrix}n−3 \\ 2 \end{matrix})− 1] ,

where b_{i}, c_{i} ∈ [2, n − 2], b_{i} < c_{i}, (b_{i}, c_{i}) ≠ (2, 3), and where i < j implies that c_{i} > c_{j}, or that c_{i} = c_{j} and b_{i} > b_{j} (i.e. we choose a reverse lexicographical ordering, read from bottom to top). Let

x_{i} := (−1)^{n−i} ⟨\begin{matrix}1 \\ n−1 \end{matrix}⟩ + ⟨\begin{matrix}n−i \\ n−1 \end{matrix}⟩ for i ∈ [2, n − 2]
x_{(n−2)+i} := ⟨\begin{matrix}n−1−i \\ n−1 \end{matrix}⟩ ρ^{+}  for i ∈ [2, n − 3]
x_{(2n−5)+i} := ⟨\begin{matrix}b_{i} \\ c_{i} \end{matrix} ⟩ ρ^{+}  for i ∈ [1, (\begin{matrix}n−3 \\ 2 \end{matrix})− 1] .

Moreover, let

x_{1} := \{ \begin{matrix}⟨\begin{matrix}3 \\ 4 \end{matrix}⟩ & if n is odd, \\ ⟨\begin{matrix}3 \\ n \end{matrix}⟩ + \frac{n−2}{2} ⟨\begin{matrix}3 \\ 4 \end{matrix}⟩ & if n is even, \end{matrix}

and

x_{(n−2)+1} := \{ \begin{matrix}\frac{n−3}{2} ⟨\begin{matrix}n−2 \\ n−1 \end{matrix}⟩ ρ^{+} & − & \frac{n−1}{2} (⟨\begin{matrix}3 \\ n \end{matrix}⟩ + (−1)^{n} ⟨\begin{matrix}n−1 \\ n \end{matrix}⟩ − (n − 3) ⟨\begin{matrix}3 \\ n−1 \end{matrix}⟩) & if n is odd, \\ ⟨\begin{matrix}n−2 \\ n−1 \end{matrix}⟩ ρ^{+} & + & (⟨\begin{matrix}3 \\ n \end{matrix}⟩ + (−1)^{n} ⟨\begin{matrix}n−1 \\ n \end{matrix}⟩ − (n − 3) ⟨\begin{matrix}3 \\ n−1 \end{matrix}⟩) & if n is even. \end{matrix}

Conditions (i, ii, iii, iv) of (5.1) follow by (5.12), condition (v) follows by (5.11).

6 Miscellanea

6.1 Symmetric partitions

A partition λ is called symmetric if λ = λ′, otherwise, it is called asymmetric. We remark that by induction, ordinary branching [5, 9.2 ii] shows that if λ ≠ (1) is symmetric, then n_{λ} := rk_{Z} S^{λ} is even.

Suppose given a symmetric partition λ ≠ (1). Let m be the middle jump factor, i.e. the quotient of the (n_{λ}/2 + 1)st and the (n_{λ}/2)th elementary divisor of the Gram matrix of S^{λ}. Let H := ∏_{i∈[1,s]}(2λ_{i} − 2i + 1) be the product of the main diagonal hook lengths, where s = λ_{s}.

Proposition 6.1 The quotient H/m is a square in Q.

By (4.3), the quotient (n!/n_{λ})/m is the square of the (n_{λ}/2)th elementary divisor of S^{λ}. But since n!/n_{λ} is the product of all hook lengths of λ, and since λ is symmetric, we conclude that H/m is a square in Q.

Conjecture 6.2 The quotient H/m is an integer.

Here is the list of the symmetric non-hook partitions whose elementary divisors are known so far [15]. By (4.3), it is sufficient to list the respective first half of the elementary divisors. Thus, the last elementary divisor we give is the (n_{λ}/2 + 1)st, together with its multiplicity.

λ m H elementary divisors

(2, 2)  3 3 2^{1} · 6^{1}
(3, 2, 1) 5 5 1^{4} · 3^{4} · 15^{4} ⋯
(3^{2}, 2)  15 15 8^{21} · 120^{21}
(3^{3})  15 15 24^{21} · 360^{21}
(4, 2, 1^{2}) 7 7 2^{14} · 8^{31} · 56^{31} ⋯
(4, 3, 2, 1) 21 21 1^{41} · 3^{176} · 15^{167} · 315^{167} ⋯
(4, 3^{2}, 1) 21 21 8^{372} · 40^{222} · 840^{222} ⋯
(4^{2}, 2^{2})  35 35 8^{131} · 24^{353} · 72^{836} · 2520^{836} ⋯
(4^{2}, 3, 2) 35 35 16^{428} · 48^{2081} · 144^{1781} · 5040^{1781} ⋯
(5, 2, 1^{3}) 9 9 6^{56} · 30^{168} · 270^{168} ⋯
(5, 3, 2, 1^{2}) 3 27 2^{1046} · 4^{460} · 8^{44} · 16^{386} · 48^{1560} · 144^{354} · 432^{354} ⋯
(5, 3^{2}, 1^{2}) 3 27 4^{2329} · 20^{181} · 40^{857} · 120^{2559} · 360^{2082} · 1080^{2082} ⋯
(6, 2, 1^{4}) 11 11 24^{208} · 48^{2} · 144^{840} · 1584^{840} ⋯
(7, 2, 1^{5}) 13 13 120^{792} · 840^{3960} · 10920^{3960} ⋯

Let α be the product of the strictly upper diagonal hook lengths of λ. Let γ be the first elementary divisor of the Gram matrix of S^{λ}. We denote E := End_{ZA_{n}}S^{λ} and E_{Q} := Q ⊗_{Z} E. Symmetry of λ gives

S^{λ} ≃_{ZA_{n}} S^{λ,−} ≃^{[5, 6.7]}_{ZS_{n}} S^{λ′,∗} = S^{λ,∗} ≃ S^{λ,#},

and so

(S^{λ} ^{\overline{}^{β}} S^{λ}) := (S^{λ} [figure] S^{λ,#} ^{\overline{}^{∼}} S^{λ}) ∈ E

(cf. §2.4). By definition of β, we have ρ^{λ}(σ)β = βρ^{λ}(σ)ε_{σ}, where ρ^{λ}(σ) is the operation of σ ∈ S_{n} on S^{λ}.

Proposition 6.3 Suppose (−1)^{(n−s)/2}H not to be a square in Z and denote χ := \sqrt{(−1)^{(n−s)/2}H}. We have an isomorphism E_{Q} ≃ Q(χ) (since a splitting field for E_{Q} contains the ordinary character values given by [8, 2.5.12, 2.15.13]), which we fix and use as an identification. Let h ∈ Z_{>0} be maximal such that h^{2} divides H, and let χ_{0} := \sqrt{(−1)^{(n−s)/2}H/h^{2}}. The following hold.

(i) We have γ | αh.

(ii) If the ring of algebraic integers in E_{Q} is given by Z[χ_{0}], then E = Z[αγ^{−1}χ]. If the ring of algebraic integers in E_{Q} is given by Z[(1 + χ_{0})/2], then the following holds. If αγ^{−1}h is even, then E = Z[αγ^{−1}χ]; if αγ^{−1}h is odd, then E = Z[αγ^{−1}h(1+χ_{0})/2] or E = Z[αγ^{−1}χ].

(iii) The elementary divisors of the Gram matrix of S^{λ} and the elementary divisors of the operation of αχ ∈ E on the E-module S^{λ} coincide.

Using the map S_{n} \overline{}Gal(E_{Q}/Q) given by conjugation by means of ρ^{λ}, we conclude that β has trace zero, i.e. that β = ±α̃χ for some α̃ ∈ Q_{>0}.

On the other hand, β^{2} = (−1)^{(n−s)/2}α̃^{2}H shows that the ith and the (n_{λ} + 1 − i)th elementary divisor of S^{λ} multiply to give α̃^{2}H for i ∈ [1, n_{λ}]. But by (4.3), they multiply to give n!/n_{λ}, which is the product of the hook lengths of λ [5, 20.1]. Therefore, α̃ = α, which proves (iii).

Suppose Z[x] to be the ring of algebraic integers in a quadratic number field, and suppose F ⊆ Z[x] to be a subring of Z-rank 2. Choosing a Z-linear basis (u + vx, w + yx) of F, where u, v, w, y ∈ Z, the existence of 1 ∈ F shows that there are a, b ∈ Z such that au + bw = 1, av + by = 0. Base change by (\begin{matrix}a & b \\ −w & u \end{matrix}) yields a Z-linear basis of the form (1, dx), d ⩾ 1, i.e. E = Z[dx].

Now, γ^{−1}αχ is still contained in E. More precisely, α is minimal with this property, for otherwise γ^{−1}αχ would not have 1 as first elementary divisor.

Therefore, if the ring of algebraic integers in E_{Q} is given by Z[χ_{0}], minimality of α yields d = αγ^{−1}h. If the ring of algebraic integers in E_{Q} is given by Z[(1 + χ_{0})/2], minimality of α yields d = αγ^{−1}h if d is odd, and d = 2αγ^{−1}h if d is even. This proves (ii). Moreover, in all three cases αγ^{−1}h is integral, as asserted in (i).

Proposition 6.4 Suppose (−1)^{\frac{n−s}{2}} H to be a square in Z, so in particular n ≡_{4} s. We have an isomorphism E_{Q} ≃ Q × Q (since the central-primitive idempotents of CA_{n} belonging to the summands of S_{C}^{λ}|_{A_{n}} already lie in QA_{n}, as the character values given in [8, 2.5.12, 2.15.13] show), which we fix and use as an identification. The following hold.

(i) We have γ | α\sqrt{H} (= \sqrt{n!/n_{λ}}).

(ii) We have E = \{(a, b) ∈ Z × Z | a ≡_{d} b\} with d = 2αγ^{−1}\sqrt{H} if αγ^{−1} is even, and either d = 2αγ^{−1}\sqrt{H} or d = αγ^{−1}\sqrt{H} if αγ^{−1} is odd.

(iii) The elementary divisors of the Gram matrix of S^{λ} and those of the operation of α\sqrt{H}(1, −1) on the E-module S^{λ} coincide.

The nontrivial automorphism of E_{Q} turns β into −β, whence β = ±α̃(1, −1) for some α̃ ∈ Q_{>0}. Since β^{2} = α̃^{2}, comparison with (4.3) yields α̃ = α\sqrt{H}, which proves (iii).

We have E = \{(a, b) ∈ Z × Z | a ≡_{d} b\} for some d ⩾ 1. But αγ^{−1}\sqrt{H} is minimal in Q_{>0} with αγ^{−1}\sqrt{H}(1, −1) ∈ E, that is, αγ^{−1}\sqrt{H} is integral – as claimed in (i) – and minimal with 2αγ^{−1}\sqrt{H} ≡_{d} 0. Thus, d = αγ^{−1}\sqrt{H} if d is odd, and d = 2αγ^{−1}\sqrt{H} if d is even, whence (ii).

Example 6.5 Here are some examples of the subring E ⊆ E_{Q}.

\begin{matrix}λ & E & α & γ \\ (3, 2, 1) & Z[3(1 + \sqrt{5})/2] & 3 & 1 \\ (4, 1^{3}) & Z[\sqrt{−7}] & 6 & 6 \\ (4, 2, 1^{2}) & Z[4\sqrt{−7}] & 8 & 2 \\ (3^{2}, 2) & Z[\sqrt{−15}] & 8 & 8 \\ (3^{3}) & Z[\sqrt{−15}] & 24 & 24 \\ (5, 1^{4}) & \{(a, b) ∈ Z × Z | a ≡_{6} b\} & 24 (\sqrt{H} = 3) & 24 \end{matrix}

According to (6.3 iii, 6.4 iii), the main problem that remains to be solved is to determine the structure of S^{λ} as an E-module. For example, the elementary divisors of S^{(3,2,1)} are given by 1^{4} · 3^{4} · 15^{4} · 45^{4}, whereas on the free E-module E^{8}, multiplication by αχ = 3\sqrt{5} has the elementary divisors 1^{8} · 45^{8}. Thus by (6.3 iii), S^{λ} is not free over E, and not locally free at 3 either.

Example 6.6 The asymmetric example S^{μ} of smallest dimension for which two successive elementary divisors have quotient bigger than the outer hook length μ_{1}+μ′_{1}−1, is given by μ = (9, 2^{2}, 1). In fact, we obtain the elementary divisors 4^{792} · 8^{3421} · 120^{493} · 960^{714} · 2880^{586} [15], whereas the outer hook length is 12.

The following remark on partitions of rectangular shape pertains in particular to partitions of quadratic shape.

Remark 6.7 Let μ be a partition such that μ_{1} = μ_{h}, where h = μ′_{1}. Let ν be the partition obtained from μ by removing the lower right node in its diagram, i.e. ν_{h} := μ_{h}−1, ν_{j} := μ_{j} for j ≠ h. Then the elementary divisors of S^{μ} are given by the elementary divisors of S^{ν}, multiplied by the constant factor h.

We have S^{μ}|_{S_{n−1}} ^{\overline{}^{∼}} S^{ν} [5, 9.3], by sending a ν-polytabloid to the μ-polytabloid obtained by adding the entry n in the lower right corner. Thus the invariant bilinear form on S^{ν} induces an invariant bilinear form on S^{μ}, whence the Gram matrix of S^{μ} is a scalar multiple of the Gram matrix of S^{ν}. This scalar is calculated to be h by the Branching Theorem for Determinants [9, p. 225].

6.2 A conjectural comparison of kernels

There seems to be a connection between elementary divisors and modular morphisms. Apart from the fact that both yield necessary conditions on the shape of the quasiblock (cf. §6.3), we do not know of any a priori reason for such a connection to exist.

Let h ⩾ 1, let k ∈ [1, h]. Abbreviate ν(k) := (2^{h−k}, 1^{n−2h+k}) and b := n − 2h + 1. Let F^{ν(0)} be the free Z-module on the ν(0)-tableaux, endowed with the natural operation of S_{n}, yielding F^{ν(0)} ≃ ZS_{n}. Consider the morphism

[figure: F^{ν(0)}; f_{k}; S^{ν(k)}/(b + k)S^{ν(k)}; [a]; ∑_{ζ∈Z(a,k)}(−1)^{Σ(ζ)}⟨a^{ζ}⟩ ,]

where Z(a, k) is the set of subtuples of length k of the second column of [a], where [a^{ζ}] is the μ-tableau given by removing ζ from the second column of [a] and appending it at the bottom of the first column, and where Σ(ζ) is the sum of the positions of the entries of ζ. So for example, if the second column is given by (3, 5, 6, 7, 9), and if ζ = (5, 7, 9), then Σ(ζ) = 2 + 4 + 5.

Let p be a prime. Given m ⩾ 1, we denote ilog_{p}(m) := max{i ∈ Z_{⩾0} | p^{i} ⩽ m}. Let m? := ∏_{p prime} p^{ilog_{p}(m)} (‘m quasi-factorial’, e.g. 4? = 12). From [14, 4.9] we take the morphism

[figure: S^{ν(0)}; s_{k}; S^{ν(k)}/(b + k)S^{ν(k)}; ⟨a⟩; k? · ([a]f_{k}) .]

We will form an intersection of the kernels of a slight variation of these morphisms – the factor k? can sometimes be lowered on a smaller domain.

Let S^{ν(0), ∩ 0} := S^{ν(0)}, and s̃_{1} := s_{1}.

Assume submodules S^{ν(0), ∩ l−1} ⊆ S^{ν(0)} and morphisms S^{ν(0), ∩ l−1} ^{\overline{}^{s̃_{l}}} S^{ν(l)}/(b + l)S^{ν(l)} to be constructed for l ∈ [1, k − 1].

Let S^{ν(0), ∩ k−1} := ⋂_{l∈[1,k−1]} Kern s̃_{l}. Let

[figure: S^{ν(0), ∩ k−1}; s̃_{k}; S^{ν(k)}/(b + k)S^{ν(k)}; ∑_{⟨a⟩} t_{⟨a⟩}⟨a⟩; \frac{k?}{gcd(k?,γ_{k−1})} ∑_{⟨a⟩} t_{⟨a⟩}([a]f_{k}) ,]

where the t_{⟨a⟩} are coefficients in Z, and where γ_{k−1} is the first elementary divisor of S^{ν(0), ∩ k−1} [figure] S^{ν(0)}.

Conjecture 6.8 The kernel of

S^{ν(0)} ^{\overline{}^{η}} S^{ν(0),∗}/\frac{h! (n−h+1)!}{n−2h+1} S^{ν(0),∗} = S^{ν(0),∗}/\frac{n!}{rk_{Z}S^{ν(0)}} S^{ν(0),∗}

is given by the intersection

S^{ν(0), ∩ h} = ⋂_{k∈[1,h]} Kern s̃_{k} .

Conjecture (6.8) holds if n ⩽ 8 (direct computation), or if h = 1 (5.5). If h = 2, the kernel is contained in the intersection (5.14).

6.3 A final remark on quasiblocks

Let p be a prime, let R = Z_{(p)}, and let ε^{λ} denote the central-primitive idempotent of QS_{n} belonging to S_{Q}^{λ}, for λ a partition of n. The quasiblocks ε^{λ}RS_{n} of RS_{n} may be viewed as the ‘building blocks’ of the locally integral representation theory of the symmetric group. The Gram matrix determines a certain part of the structure of such a quasiblock, sometimes it even describes it in its entirety.

Wedderburn’s isomorphism QS_{n} ^{\overline{}^{∼}} ∏_{λ} End_{Q}S_{Q}^{λ} restricts to RS_{n} [figure] ∏_{λ} End_{R}S_{R}^{λ}, which

in turn induces ε^{λ}RS_{n} [figure: ρ^{λ}] End_{R}S_{R}^{λ}, given by sending ε^{λ}σ to the operation ρ^{λ}(σ) of σ ∈ S_{n} on S_{R}^{λ}. A necessary condition for an R-linear endomorphism of S_{R}^{λ} to lie in the image of this embedding is its compatibility with the RS_{n}-linear embedding S_{R}^{λ} [figure: η] S_{R}^{λ,∗}. Namely, if S_{R}^{λ} ^{\overline{}^{φ}} S_{R}^{λ} is in the image of ρ^{λ}, then there exists an R-linear operation S_{R}^{λ,∗} ^{\overline{}^{ψ}} S_{R}^{λ,∗} such that φη = ηψ. In terms of matrices, this means that the image ρ^{λ}(RS_{n}) = ρ^{λ}(ε^{λ}RS_{n}) is contained in Γ_{R}^{λ} := (R)_{n_{λ}} ∩ (G^{λ}(R)_{n_{λ}}(G^{λ})^{−1}), where G^{λ} denotes the Gram matrix. For instance, we obtain

ε^{(2,1)}Z_{(3)}S_{3} \overline{}^{ρ^{(2,1)}}_{∼} Γ_{Z_{(3)}}^{(2,1)} = (Z_{(3)})_{2} ∩ ((\begin{matrix}1 & 0 \\ 0 & 3 \end{matrix}) (Z_{(3)})_{2} (\begin{matrix}1 & 0 \\ 0 & 3 \end{matrix})^{−1}) = (\begin{matrix}Z_{(3)} & Z_{(3)} \\ (3) & Z_{(3)} \end{matrix} ) .

But in general, the inclusion

ε^{λ}RS_{n} [figure: ρ^{λ}] Γ_{R}^{λ}

is not surjective – as an example we may take λ = (3, 2, 1) over R = Z_{(3)}.

If all indecomposable projective ε^{λ}RS_{n}-lattices are simple, that is, if the quasiblock is a tiled order, we do not know whether

(i) Jac(ε^{λ}RS_{n}) [figure: ρ^{λ}] Jac(Γ_{R}^{λ}) ,

and if so, whether

(ii) there exists a K ⩾ 0 such that for all k ⩾ K

Jac^{k}(ε^{λ}RS_{n}) \overline{}^{ρ^{λ}}_{∼} Jac^{k}(Γ_{R}^{λ}) .

Cf. [13, 6.1.26]. Several examples of tiled quasiblocks are given in [23, 24]. An example of a tiled quasiblock that requires K = 2 is given in [21, 5.6.12 ii].

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M. Künzer, G. Nebe Abt. Reine Mathematik Universität Ulm D-89069 Ulm kuenzer@mathematik.uni-ulm.de nebe@mathematik.uni-ulm.de

Note added in proof. Matthew Fayers independently obtained (2.5) and (4.1) using different methods. See M. Fayers, On the structure of Specht modules, J. Lond. Math. Soc. (2) 67, p. 85–102, 2003.

Note added in proof. G. E. Murphy obtained (2.5) in an explicit form that avoids matrix inversion (Thesis, Queen Mary College, London, 1987). It is stated in loc. cit. (6.27) as being dependent of the conjecture loc. cit. (6.24), which in turn has been proven by Schaper [28, p. 60].