Morita equivalences of Ariki–Koike algebras

Richard Dipper and Andrew Mathas

Was lange währt, wird endlich gut.

Abstract We prove that every Ariki–Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki–Koike algebras which have q–connected parameter sets. A similar result is proved for the cyclotomic q–Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki–Koike algebras defined over fields of characteristic zero are now known in principle.

1 Introduction

Let R be a commutative ring with 1 and let q, Q_{1}, …, Q_{r} be elements of R with q invertible. Let Q = (Q_{1}, Q_{2}, …, Q_{r}). The Ariki–Koike algebra ℋ_{q,Q}(n) is the associative unital R–algebra with generators T_{0}, T_{1}, …, T_{n−1} subject to the following relations

(T_{0} − Q_{1}) … (T_{0} − Q_{r}) = 0
T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0}
(T_{i} + 1)(T_{i} − q) = 0  for 1 ≤ i ≤ n − 1
T_{i+1}T_{i}T_{i+1} = T_{i}T_{i+1}T_{i} for 1 ≤ i ≤ n − 2
T_{i}T_{j} = T_{j}T_{i}  for 0 ≤ i < j − 1 ≤ n − 2.

The main result of this paper states that up to Morita equivalence the Ariki– Koike algebras depend only on the q–orbits of the parameters in Q. More precisely we have the following.

1.1 Theorem Suppose that Q = Q_{1} ∐ Q_{2} ∐ ⋯ ∐ Q_{κ} (disjoint union) is a partitioning Π of the parameter set Q such that

f_{Π}(q, Q) = ∏_{1≤α<β≤κ} ∏_{{Q_{i}∈Qα \atop Q_{j} ∈Q_{β}}} ∏_{−n<a<n}(q^{a}Q_{i} − Q_{j})

A.M.S. subject classification (1991): 16G99, 20C20, 20G05.

This paper is a contribution to the DFG projects “Algorithmic number theory and algebra”

(DI-531/3-2) and “Quantisierte Koordinatenringe und verallgemeinerte q–Schur–Algebren”

(DI-531/5-1). Both authors acknowledge support from DFG. The second author was supported in part by the Sonderforschungsbereich 343 in Bielefeld and the University of Sydney research grant URG U0628.

is an invertible element of R. Then ℋ_{q,Q}(n) is Morita equivalent to the algebra

ℋ_{q,Π}(n) = ⨁_{{n_{1},...,nκ≥0 \atop n_{1}+···+nκ=n}} ℋ_{q,Q_{1}}(n_{1}) ⊗ ℋ_{q,Q_{2}}(n_{2}) ⊗ ⋯ ⊗ ℋ_{q,Q_{κ}}(n_{κ}).

Notice that the polynomial f_{Π}(q, Q) is invertible only if whenever there exist i and j with q^{a}Q_{i} = Q_{j} for some a with −n < a < n then Q_{i}, Q_{j} ∈ Q_{α} for some α. The appearance of these polynomials is not surprising because it was shown by Ariki [1] that over a field ℋ is semisimple if and only if

P_{ℋ} (q, Q) = ∏_{1≤i<j≤r} ∏_{−n<a<n}(q^{a}Q_{i} − Q_{j}) · ∏_{k=1}^{n}(1 + q + ⋯ + q^{k−1}) ≠ 0.

The polynomial P_{ℋ} (q, Q) is the analogue of the Poincaré polynomial for ℋ . The factors of P_{ℋ} (q, Q) in the right hand product determine whether or not the subalgebra ℋ_{q}(𝔖_{n}) is semisimple.

Permuting the parameters Q_{1}, …, Q_{r} does not affect ℋ up to isomorphism. In addition, if Q′ = cQ = (cQ_{1}, …, cQ_{r}), where c is any invertible element of R, then ℋ_{q,Q}(n) and ℋ_{q,Q′}(n) are isomorphic (replace T_{0} with c^{−1}T_{0}). We now rephrase Theorem 1.1 so that it says that up to Morita equivalence ℋ depends only on the q–orbits of Q_{1}, …, Q_{r}. To state this precisely, say that Q is q–connected if Q_{i} = q^{a_{i}} for some integer a_{i} for each i (for the purposes of Corollary 1.2 we could require that |a_{i}| < n, for each i, but this is not so important). Normally, Q will not be q–connected; however, up to a permutation of the Q_{α}’s, there is a unique partitioning Q = Q_{1} ∐ ⋯ ∐ Q_{κ} such that for α = 1, …, κ there exists an element c_{α} ∈ R such that Q_{i} ∈ Q_{α} if and only if Q_{i} = c_{α}q^{a_{i}} for some integer a_{i}; so, Q_{α} = c_{α}Q′_{α} for each α. By the above remarks, if c_{α} is invertible then Q′_{α} is q–connected and ℋ_{q,Q_{α}}(𝔖_{n}) is isomorphic to ℋ_{q,Q′_{α}}(𝔖_{n}); so Theorem 1.1 implies the following.

1.2 Corollary Suppose that each Q_{i} is invertible for 1 ≤ i ≤ r. Then the Ariki–Koike algebra ℋ_{q,Q}(n) is Morita equivalent to a direct sum of tensor products of Ariki–Koike algebras which have q–connected parameter sets.

The importance of this result stems from the work of Ariki [2] which showed that if R is a field of characteristic zero, q ≠ 1 and the parameter set Q is q–connected then the decomposition numbers of Ariki–Koike algebras can be computed in terms of the canonical basis of an associated integral highest weight module for an affine quantum group. In addition, Ariki [4] and Ariki and the second author [6] used the results of [2] to classify the irreducible representations of the Ariki–Koike algebras without any restrictions on the field, q or Q; this was done by first reducing to the case of q–connected parameter sets. Corollary 1.2 explains the reduction of [6] by showing that it comes from a Morita equivalence.

In fact, we can do more than this because Uglov [26] (extending the ideas of [23]), gave an algorithm for computing the decomposition matrices of the Ariki–Koike algebras which satisfy the restrictions of Ariki’s paper [2]. In the spirit of the Kazhdan–Lusztig conjectures, Uglov’s algorithm involves computing certain affine parabolic Kazhdan–Lusztig polynomials and evaluating them at 1. Combining [26] with Proposition 4.11(iii) below we obtain the following.

1.3 Corollary Suppose that R is a field of characteristic zero, q ≠ 1 and Q_{i} ≠ 0 for 1 ≤ i ≤ r. Then the decomposition matrix of ℋ_{q,Q}(n) is known.

Next consider the case where there exists an integer a with Q_{i} ≠ q^{a}Q_{j} if and only if i = j; then Theorem 1.1 says that ℋ_{q,Q}(n) is Morita equivalent to a direct sum of tensor products of Hecke algebras of type A. This special case is a result of Du and Rui [17, Theorem 4.14].

1.4 Corollary (Du–Rui) Suppose that ∏_{1≤i<j≤r} ∏_{−n<a<n}(q^{a}Q_{i} − Q_{j}) is an invertible element of R. Then ℋ_{q,Q}(n) is Morita equivalent to

⨁_{{n_{1},...,nr ≥0 \atop n_{1}+···+nr =n}} ℋ_{q}(𝔖_{n_{1}}) ⊗ ℋ_{q}(𝔖_{n_{2}}) ⊗ ⋯ ⊗ ℋ_{q}(𝔖_{n_{r}}).

In fact, when r = 2 this is a theorem of James and the first named author [15, Theorem 4.17]. Although this paper is largely motivated by [15], the techniques we use are very different. As in [15] we explicitly construct the projective generator of ℋ which induces the Morita equivalence of the main theorem; however, unlike [15, 17] we do this by adapting the standard basis of ℋ (from [16]) to give bases for the family of projective modules which describe the projective generator. This yields precise information, such as Specht series for these modules, which is new even in the special cases considered previously [15, 17]. Another consequence is that we are able to extend our results to the cyclotomic q–Schur algebras 𝒮_{q,Q}(n) of [16].

1.5 Theorem Suppose that Q = Q_{1} ∐ Q_{2} ∐ ⋯ ∐ Q_{κ} (disjoint union) is a partitioning Π of the parameter set Q such that f_{Π}(q, Q) is an invertible element of R. Then 𝒮_{q,Q}(n) is Morita equivalent to the algebra

𝒮_{q,Π}(n) = ⨁_{n_{1}+···+n_{κ}=n^{n_{1},...,nκ≥0}} 𝒮_{q,Q_{1}}(n_{1}) ⊗ 𝒮_{q,Q_{2}}(n_{2}) ⊗ ⋯ ⊗ 𝒮_{q,Q_{κ}}(n_{κ}).

Actually, a slightly more general statement is possible; see Theorem 5.2. Again some special cases of Theorem 1.5 were known previously. First, if

r = 2 then ℋ is an Iwahori–Hecke algebra of type B and in this case Theorem 1.5 can be deduced for special parameter sets from the results of [10, 20]. The case r = 2 is important because it has implications for representation theory of symplectic and unitary groups; see [8,9]. Secondly, Ariki [3] (see also Du and Rui [17]), have proved Theorem 1.5 under the assumptions of Corollary 1.4; here the cyclotomic q–Schur algebra arises as a quotient of a quantum group of type A acting on “q–tensor space” as the centralizing algebra of the Ariki–Koike algebra.

In order to prove Theorem 1.1 we first observe that it is enough to prove the following much simpler, but equivalent, result.

1.6 Theorem Fix an integer s with 1 ≤ s ≤ r and suppose that

f_{s}(q, Q) = ∏_{1≤i≤s<j≤r} ∏_{−n<a<n}(q^{a}Q_{i} − Q_{j})

is an invertible element of R. Then ℋ_{q,Q} is Morita equivalent to

ℋ_{q,s,Q}(n) = ⨁_{b=0}^{n} ℋ_{q,(Q_{s+1},...,Q_{r})}(𝔖_{b}) ⊗ ℋ_{q,(Q_{1},...,Q_{s})}(𝔖_{n−b}).

The general case follows by iterating Theorem 1.6, using the remarks before Corollary 1.2.

The proof of Theorem 1.6 is based on an explicit decomposition of ℋ = ℋ_{q,Q}(n) into a direct sum of projective right ideals V^{b} using the standard basis of ℋ . The different projective modules V^{b} have no direct summand in common and V = ⨁_{b=0}^{n} V^{b} is a progenerator for ℋ . In Theorem 3.20 we give an explicit formula for the multiplicity of each V^{b} in the regular representation of ℋ and in Theorem 4.7 we show that the endomorphism ring of V is ℋ_{q,s,Q}(n); this proves Theorem 1.6 and that the Morita equivalences of Theorem 1.6 are given by the functors −⊗_{ℋ_{q,s,Q}} V and Hom_{ℋ} (V, −). The explicit description of these functors enables us to extend our results to the cyclotomic q-Schur algebras in section 5.

2 The standard basis theorem

Let 𝔖_{n} be the symmetric group on {1, 2, …, n}, acting from the right, and let s_{1}, …, s_{n−1} be the standard Coxeter generators of 𝔖_{n}; that is, s_{i} = (i, i + 1) for all i. If w ∈ 𝔖_{n} write w = s_{i_{1}}s_{i_{2}} … s_{i_{k}} and say this expression is reduced if k is minimal; in this case, k is the length of w and we write ℓ(w) = k and define T_{w} = T_{i_{1}}T_{i_{2}} … T_{i_{k}}. Let ℋ (𝔖_{n}) be the R–span of { T_{w} | w ∈ W }; then ℋ (𝔖_{n}) is a free subalgebra of ℋ of rank n! which is isomorphic to the Iwahori–Hecke algebra of 𝔖_{n}. The Iwahori–Hecke algebra ℋ (𝔖_{n}) is described in detail in [24].

Let L_{1} = T_{0} and for 1 ≤ i < n set L_{i+1} = q^{−1}T_{i}L_{i}T_{i}. These elements satisfy the following fundamental relations (see [5, (3.3)] and [16, (2.1)]).

2.1 Suppose that 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n. Then (i) L_{i} and L_{j} commute. (ii) T_{i} and L_{j} commute if i ≠ j − 1, j. (iii) T_{i} commutes with L_{i}L_{i+1} and with L_{i} + L_{i+1}. (iv) If a ∈ R and i ≠ j then T_{i} commutes with (L_{1} − a)(L_{2} − a) … (L_{j} − a).

The importance of these elements derives from the following result.

2.2 (Ariki–Koike [5, Theorem 3.10]) The algebra ℋ is free as an R–module with basis

{ L_{1}^{d_{1}}L_{2}^{d_{2}} … L_{n}^{d_{n}}T_{w} | w ∈ 𝔖_{n} and 0 ≤ d_{m} ≤ r − 1 for m = 1, 2, …, n } .

In particular, ℋ is free of rank r^{n}n!

Note that because q is invertible so are the elements T_{i}, for i = 1, …, n − 1; explicitly, T_{i}^{−1} = q^{−1}(T_{i} − q + 1). Consequently, T_{w} is invertible for all w ∈ 𝔖_{n}.

Let ∗: ℋ ⟶ ℋ be the anti–automorphism of ℋ determined by T_{i}^{∗} = T_{i} for i = 0, 1, …, n − 1. Then T_{w}^{∗} = T_{w^{−1}} for all w ∈ 𝔖_{n} and L_{i}^{∗} = L_{i} for i = 1, 2, …, n.

As we next recall, the Ariki–Koike algebra has another basis which is better adapted to the study of its representation theory; this basis is cellular, in the sense of Graham and Lehrer [18]. (Graham and Lehrer were the first to construct a cellular basis of ℋ ; the basis we use is due to Gordon James and the authors [16].)

A composition of an integer m ≥ 0 is an ordered sequence of non–negative integers τ = (τ_{1}, τ_{2} …) such that |τ| = ∑_{i≥1} τ_{i} = m; if the sequence is non–increasing then τ is a partition of m. A multicomposition of n (with r– components) is an ordered r–tuple μ = (μ^{(1)}, …, μ^{(r)}) of compositions such that |μ^{(1)}| + ⋯ + |μ^{(r)}| = n. If each μ^{(i)} is a partition then μ a multipartition. Let Λ = Λ(n: r) be the set of multicompositions of n with r–components and let Λ^{+} = Λ^{+}(n: r) ⊂ Λ be the set of multipartitions of n with r–components.

The diagram of a multicomposition μ is the set

[μ] = { (i, j, k) | 1 ≤ k ≤ r and i ≥ 1 and 1 ≤ j ≤ μ_{i}^{(k)} } ,

which we think of as an ordered r–tuple of boxes in the plane. For example, if μ = ((3, 1), (1^{2}), (2, 1)) then

[μ] = (  , ,  ).

In this way, we talk of the rows and columns (of the components) of μ.

Given two multicompositions λ and μ say that λ dominates μ, and write λ ⊵ μ, if for 1 ≤ c ≤ r and for all i ≥ 1

∑_{b=1}^{c−1} |λ^{(b)}| + ∑_{j=1}^{i} λ_{j}^{(c)} ≥ ∑_{b=1}^{c−1} |μ^{(b)}| + ∑_{j=1}^{i} μ_{j}^{(c)}.

If λ ⊵ μ and λ ≠ μ we write λ ▷ μ. This defines a partial order on the sets of multicompositions and multipartitions of n.

If μ is a multicomposition of n then a μ–tableau is a map 𝔱: [μ]⟶{1, 2 …, n}; we write Shape(𝔱) = μ. Generally, we shall think of tableaux as labelled diagrams; for example, when μ = ((3, 1), (1^{2}), (2, 1)) three μ–tableaux are

( 1 2 3 4 , 5 6 , 7 8 9 ), ( 3 5 8 4 , 2 9 , 1 6 7 ) and ( 3 5 8 4 , 7 6 , 1 2 9 ).

Given a tableau 𝔱 we will also write 𝔱 = (𝔱^{(1)}, …, 𝔱^{(r)}) and call 𝔱^{(i)} the ith component of 𝔱. Similarly, given an integer m with 1 ≤ m ≤ n we write comp_{𝔱}(m) = i if m appears in the ith component 𝔱^{(i)} of 𝔱. If 𝔰 is another tableau we write comp_{𝔱} = comp_{𝔰} if comp_{𝔱}(m) = comp_{𝔰}(m) for all m with 1 ≤ m ≤ n.

A μ–tableau 𝔱 is standard if the entries in each row of each component of 𝔱 increase from left to right and the entries in each column of each component of 𝔱 increase from top to bottom (of the tableaux above only the first two are standard). For each multipartition λ let Std(λ) be the set of standard λ– tableaux.

Suppose 𝔱 is a standard λ–tableaux and 𝔰 a standard μ–tableau for λ, μ ∈ Λ. Given an integer m with 1 ≤ m ≤ n let 𝔱↓m be the subtableau of 𝔱 which contains the entries 1, 2, …, m and similarly for 𝔰↓m. Then 𝔰 ⊵ 𝔱 if Shape(𝔰↓m) ⊵ Shape(𝔱↓m), for m = 1, 2, …, n, and we say that 𝔰 dominates 𝔱. Again we write 𝔰 ▷ 𝔱 if 𝔰 ⊵ 𝔱 and 𝔰 ≠ 𝔱.

Let 𝔱^{μ} be the unique μ–tableau such that 𝔱^{μ} ⊵ 𝔱 for all μ–tableau 𝔱. Then 𝔱^{μ} is the tableau which has the numbers 1, 2, …, n entered in order along the rows of [μ]. Note that the symmetric group 𝔖_{n} acts from the right on the set of μ– tableaux. Let 𝔖_{μ} be the row stabilizer of the tableau 𝔱^{μ}; then 𝔖_{μ} is a parabolic subgroup of 𝔖_{n}. In the example above, where μ = ((3, 1), (1^{2}), (2, 1)), the first of the tableaux listed is 𝔱^{μ} and 𝔖_{μ} = 𝔖_{3} × 𝔖_{1} × 𝔖_{1} × 𝔖_{1} × 𝔖_{2} × 𝔖_{1} ↪ 𝔖_{9} (obvious embedding); we will always identify 𝔖_{μ} with a subgroup of 𝔖_{n} in this way.

For each μ–tableau 𝔱 let d(𝔱) be the unique element of 𝔖_{n} such that 𝔱 = 𝔱^{μ}d(𝔱). Then, by [11, Lemma 1.4], d(𝔱) is a distinguished right coset representative of 𝔖_{μ} in 𝔖_{n}; that is, ℓ(wd(𝔱)) = ℓ(w) + ℓ(d(𝔱)) for all w ∈ 𝔖_{μ}.

Given an r–tuple a = (a_{1}, a_{2}, …, a_{r}) of integers, with 0 ≤ a_{i} ≤ n for all i, let u_{a} = u_{a_{1},1}u_{a_{2},2} … u_{a_{r},r} where u_{a,t} = ∏_{k=1}^{a}(L_{k} − Q_{t}) for any a and t. If μ is a multicomposition of n let u_{μ}^{+} = u_{a} where a = (a_{1}, a_{2}, …, a_{r}) is the sequence with a_{t} = |μ^{(1)}| + ⋯ + |μ^{(t−1)}| for t = 1, 2, …, r.

Suppose that λ is a multipartition of n and let x_{λ} = ∑_{w∈𝔖_{λ}} T_{w} and set m_{λ} = u_{λ}^{+}x_{λ}. If 𝔰 and 𝔱 are standard λ–tableaux define m_{𝔰𝔱} = T_{d(𝔰)}^{∗}m_{λ}T_{d(𝔱)}. Then we have the following.

2.3 (Dipper–James–Mathas [16, Theorem 3.26]) Let

M = { m_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ) for some λ ∈ Λ^{+} } .

Then M is a cellular basis of ℋ .

The basis M is called the standard basis of ℋ .

As in [16], let N^{λ} be the R–module with basis the set of all m_{𝔰𝔱} ∈ M where (𝔰, 𝔱) runs over all pairs of standard μ–tableaux with μ ⊵ λ; similarly, let \overline{N}^{λ} be the R–module with basis the set of m_{𝔲𝔳} where 𝔲 and 𝔳 are standard μ–tableau and μ ▷ λ. From the theory of cellular algebras we obtain the following corollary of (2.3).

2.4 ( [16, Corollary 3.22]) The R–modules N^{λ} and \overline{N}^{λ} are two–sided ideals of ℋ .

Given a multipartition λ the Specht module S^{λ} is the submodule of ℋ /\overline{N}^{λ} defined by S^{λ} = z_{λ}ℋ where z_{λ} = m_{λ} +\overline{N}^{λ}. The theory of cellular algebras [18, 24] shows that S^{λ} is free of rank | Std(λ)| (with basis \{ m_{𝔱^{λ}𝔱} + \overline{N}^{λ} | 𝔱 ∈ Std(λ) \}), and that there is an intrinsically defined symmetric ℋ –invariant bilinear form ( , ) on S^{λ}; that is (uh, v) = (u, vh^{∗}) for u, v ∈ S^{λ} and h ∈ ℋ . Let rad S^{λ} be the radical of this form and set D^{λ} = S^{λ}/ rad S^{λ}. Then D^{λ} is an ℋ –module; moreover, the following is true.

2.5 [16, 18] Suppose that R is a field.

(i) If λ is a multipartition of n then D^{λ} is either (0) or absolutely irreducible. (ii) { D^{λ} | λ ∈ Λ^{+} and D^{λ} ≠ (0) } is a complete set of pairwise non–isomorphic

irreducible ℋ –modules.

Given multipartitions λ and μ with D^{μ} ≠ (0) let d_{λμ} = [S^{λ} : D^{μ}] be the decomposition multiplicity of the simple module D^{μ} in the Specht module S^{λ}. The matrix (d_{λμ}) is the decomposition matrix of ℋ . Importantly, the decomposition matrix of ℋ is unitriangular; more precisely, we have the following.

2.6 [16, 18] Suppose that R is a field and that λ and μ are multipartitions such that D^{μ} ≠ (0). Then d_{μμ} = 1 and d_{λμ} ≠ 0 only if λ ⊵ μ.

Let 𝔰 be a tableau and suppose that 1 ≤ k ≤ n appears in row i and column j of the cth component 𝔰^{(c)} of 𝔰. Then the residue of k in 𝔰 is res_{𝔰}(k) = q^{j−i}Q_{c}. The following result underpins much of what follows.

2.7 [22, Prop. 3.7] Let 𝔰 and 𝔱 be standard λ–tableaux, where λ ∈ Λ^{+}, and suppose that 1 ≤ k ≤ n. Then there exist a_{𝔲} ∈ R such that

L_{k}m_{𝔰𝔱} ≡ res_{𝔰}(k)m_{𝔰𝔱} + ∑_{𝔲▷𝔰} a_{𝔲}m_{𝔲𝔱} mod \overline{N}^{λ}.

For each multipartition λ define its content to be the sequence cont(λ) = (c_{r})_{r∈R} where c_{r} is the number of nodes x of the diagram [λ] with res(x) = r. By [18, Theorem 3.7(ii)] all of the irreducible constituents of S^{λ} belong to the same block. Furthermore, by (2.1) every symmetric polynomial f(L), in L_{1}, L_{2}, …, L_{n} belongs to the centre of ℋ ; therefore, by Schur’s Lemma, f(L) acts on S^{λ} as multiplication by a scalar, say α_{f} ∈ R. By (2.7), f(L)m_{λ} ≡ α_{f}m_{λ} mod \overline{N}^{λ}; so it follows that α_{f} depends only upon the content of λ. Hence, we have the following result.

2.8 Corollary (Graham–Lehrer [18, Prop. 5.9(ii)]) Suppose that λ and μ are multipartitions of n. Then S^{λ} and S^{μ} belong to the same block only if cont(λ) = cont(μ).

Grojnowski [19] has recently shown that S^{λ} and S^{μ} are in the same block if and only if cont(λ) = cont(μ). (The definition of residue must be modified slightly in the case q = 1.) When r = 1 this result was already known by [12,21].

For each multicomposition μ of n let M^{μ} = m_{μ}ℋ . To describe how the standard basis of (2.3) can be modified to give a basis of M^{μ} we need to generalize the notion of tableau. Suppose that λ is a multipartition and μ is a multicomposition of n. A λ–tableau of type μ is a map S : [λ] ⟶ {1, …, n} × {1, …, r} such that, for all (i, k), μ_{i}^{(k)} = # { x ∈ [λ] | S(x) = (i, k) }; as before, we will think of S = (S^{(1)}, …, S^{(r)}) as a labelling of [λ] with ordered pairs of integers (i, k). A λ–tableau S of type μ is semistandard if for c = 1 …, r the entries in the cth component S^{(c)} of S are (i) non–decreasing along the rows; (ii) strictly increasing down the columns and, (iii) no entry in S^{(c)} has the form (i, k) with k < c. Let T_{0}(λ, μ) be the set of semistandard λ–tableaux of type μ.

Given a standard λ–tableau 𝔰: [λ] ⟶ {1, …, n} define μ(𝔰) to be the λ– tableau of type μ obtained from 𝔰 by replacing each entry m in 𝔰 by (i, k), if m appears in row i of the kth component of 𝔱^{μ}. Observe that in general the entries in each column of μ(𝔰) will only be non–decreasing so that μ(𝔰) need not be semistandard.

2.9 For example, if we let ω = ((0), …, (0), (1^{n})) then the tableau ω(𝔰) has entries of the form (i, r) where 1 ≤ i ≤ n. Consequently, 𝔰 ⟼ ω(𝔰) gives a bijection from the set of standard λ–tableaux to the set of semistandard λ– tableaux of type ω. Henceforth, we identify λ–tableau of type ω with the λ– tableaux that are maps from [λ] to {1, …, n}; we also use lower case letters 𝔰, 𝔱, … to denote λ–tableaux (of type ω) and upper case letters S, T, … for tableaux of arbitrary type.

Before we can state the basis theorem for M^{μ} we need one more definition. Let S be a semistandard tableau of type μ and a λ–tableau 𝔱; set

m_{S𝔱} = ∑_{μ(𝔰)=S^{𝔰∈Std(λ)}} m_{𝔰𝔱}.

2.10 [16, Theorem 4.14] Suppose that μ is a multicomposition of n. Then M^{μ} is free as an R–module with basis

{ m_{S𝔱} | S ∈ T_{0}(λ, μ), 𝔱 ∈ Std(λ) for some λ ∈ Λ^{+} } .

The last result that we shall need gives a basis for Hom_{ℋ} (M^{ν}, M^{μ}). Given a semistandard λ–tableau S of type μ and a semistandard tableau T of type ν let

m_{ST} = ∑_{μ(𝔰)=S,ν(𝔱)=T^{𝔰,𝔱∈Std(λ)}} m_{𝔰𝔱}

and define φ_{ST} : M^{ν} ⟶ M^{μ} to be the ℋ –module homomorphism given by φ_{ST}(m_{ν}h) = m_{ST}h for all h ∈ ℋ . It is not completely obvious that φ_{ST} even belongs to Hom_{ℋ} (M^{ν}, M^{ν}); nonetheless, the following is true.

2.11 [16, Theorem 6.6(i)] Suppose that μ and ν are multicompositions of n. Then Hom_{ℋ} (M^{ν}, M^{ν}) is free as an R–module with basis

{ φ_{ST} | S ∈ T_{0}(λ, μ) and T ∈ T_{0}(λ, ν) for some λ ∈ Λ^{+} } .

3 A projective generator for ℋ

For the remainder of the paper we fix an integer s with 1 ≤ s ≤ r.

Following [15], given integers i and j with 1 ≤ i < j < n define

s_{i,j} = s_{i}s_{i+1} … s_{j−1} and s_{j,i} = s_{i,j}^{−1} = s_{j−1}s_{j−2} … s_{i}.

Note that s_{j,i} is the cycle (i, i + 1, …, j). We abbreviate the corresponding elements of ℋ by T_{i,j} = T_{s_{i,j}} and T_{j,i} = T_{s_{j,i}}; so, T_{j,i} = T_{i,j}^{∗}. In passing, we remark that L_{i} = q^{1−i}T_{i,1}T_{0}T_{1,i} for i = 1, …, n.

If a and b are non–negative integers we define w_{a,b} = (s_{a+b,1})^{b}; in particular, w_{a,0} = w_{0,b} = 1. Written as a permutation,

w_{a,b} = ( \begin{matrix}1 & 2 & … & a & a + 1 & a + 2 & … & a + b \\ b + 1 & b + 2 & … & a + b & 1 & 2 & … & b \end{matrix} ) .

For later use we note that w_{a,b}^{−1} = w_{b,a}; consequently, T_{w_{a,b}}^{∗} = T_{w_{b,a}}.

3.1 Lemma Let a and b be non–negative integers with 0 ≤ a + b ≤ n and suppose that i is an integer such that i ≠ a and 1 ≤ i < a + b. Then

T_{i}T_{w_{a,b}} = T_{w_{a,b}}T_{(i)w_{a,b}} = \begin{cases}T_{w_{a,b}}T_{i+b}, & if 1 ≤ i < a, \\ T_{w_{a,b}}T_{i−a}, & if a < i < a + b. \end{cases}

Proof For any w ∈ 𝔖_{n} we have s_{i}w = ww^{−1}s_{i}w = w(iw, (i + 1)w); setting w = w_{a,b} we find

s_{i}w_{a,b} = \begin{cases}w_{a,b}s_{b+i}, & if 1 ≤ i < a, \\ w_{a,b}s_{i−a}, & if a < i < a + b. \end{cases}

The result follows by observing that ℓ(s_{i}w_{a,b}) = ℓ(w_{a,b}) + 1 since w_{a,b} is a distinguished right coset representative of 𝔖_{(a,b)} in 𝔖_{n}; see [15, 2.7].

Equation [15, 2.9] provides a recursive way to find a reduced expression for the permutation w_{a,b}. As a consequence we obtain the following Lemma.

3.2 Lemma Suppose that 1 ≤ b ≤ n − 1. Then there exists w̃ ∈ 𝔖_{(1,n−1)} such that w_{n−b,b} = w̃s_{1,b+1} and ℓ(w_{n−b,b}) = ℓ(w̃) + ℓ(s_{1,b+1}). In particular, T_{w_{n−b,b}} = Tw̃T_{1,b+1}.

Proof By [15, 2.9] if a ≥ 0 and 0 ≤ a + b < n then w_{a,b} = s_{a,a+b}w_{a−1,b} and ℓ(w_{a,b}) = ℓ(s_{a,a+b}) + ℓ(w_{a−1,b}). Therefore,

w_{n−b,b} = s_{n−b,n}w_{n−b−1,b} = s_{n−b,n}s_{n−b−1,n−1}w_{n−b−2,b} = …
= s_{n−b,n}s_{n−b−1,n−1} … s_{2,b+2}w_{1,b} = (s_{n−b,n} … s_{2,b+2})s_{1,b+1},

with the lengths adding throughout. The Lemma follows.

For the next definition recall that we have fixed an integer s with 1 ≤ s ≤ r. For each integer b with 0 ≤ b ≤ n define

u_{n−b}^{−} = ∏_{t=1}^{s}(L_{1} − Q_{t})(L_{2} − Q_{t}) … (L_{n−b} − Q_{t})

and

u_{b}^{+} = ∏_{t=s+1}^{r}(L_{1} − Q_{t})(L_{2} − Q_{t}) … (L_{b} − Q_{t}).

These elements are special instances of the elements u_{a} introduced in the previous section.

We now define the modules which are the cornerstone upon which this paper is built.

3.3 Definition Suppose that 0 ≤ b ≤ n and define v_{b} = u_{n−b}^{−}T_{w_{n−b,b}}u_{b}^{+} and let V^{b} = v_{b}ℋ .

Ultimately we shall show that V^{b} is a projective ℋ –module and that its endomorphism ring is isomorphic to a tensor product of smaller Ariki–Koike algebras; this will imply our main result. At this point it is not even clear that v_{b} is non–zero; we will deduce this important fact latter. We begin by establishing some key properties of v_{b}.

3.4 Proposition Suppose that 0 ≤ b ≤ n.

(i) If 1 ≤ i < n − b then T_{i}v_{b} = v_{b}T_{i+b}. (ii) If n − b < i ≤ n then T_{i}v_{b} = v_{b}T_{i−n+b}. (iii) If 1 ≤ k ≤ n − b then L_{k}v_{b} = v_{b}L_{k+b}. (iv) If n − b + 1 ≤ k ≤ n then L_{k}v_{b} = v_{b}L_{k−n+b}.

Proof First, observe that parts (i) and (ii) follow from parts (i) and (ii) of Lemma 3.1, respectively, together with (2.1)(iv).

Next, consider part (iii). If b = 0 or b = n then v_{b} is central in ℋ by parts (ii) and (iii) of (2.1); so our claims follow in these two cases and we may assume that 1 ≤ b ≤ n − 1.

We first consider the case k = 1; that is, L_{1}v_{b}. We write w_{n−b,b} = w̃s_{1,b+1}, as in Lemma 3.2; then w̃ ∈ S_{(1,n−1)} so that L_{1} and Tw̃ commute by (2.1). Now each L_{i} commutes with u_{n−b}^{−} since the L_{i} generate an abelian subalgebra of ℋ ; therefore,

L_{1}v_{b} = L_{1}u_{n−b}^{−}T_{w_{n−b,b}}u_{b}^{+}
= u_{n−b}^{−}L_{1}Tw̃T_{1,b+1}u_{b}^{+}
= u_{n−b}^{−}Tw̃L_{1}T_{1,b+1}u_{b}^{+}
= u_{n−b}^{−}Tw̃T_{b+1,1}^{−1}T_{b+1,1}L_{1}T_{1,b+1}u_{b}^{+}
= q^{b}u_{n−b}^{−}Tw̃T_{b+1,1}^{−1}L_{b+1}u_{b}^{+}
= q^{b}u_{n−b}^{−}Tw̃(T_{1}^{−1} … T_{b}^{−1})u_{b}^{+}L_{b+1}.

Now q^{b}(T_{1}^{−1} … T_{b}^{−1}) = (T_{1} − q + 1) … (T_{b} − q + 1) = T_{1} … T_{b} + h, where h ∈ ℋ is an R–linear combination of terms of the form T_{x}T_{y} such that (x, y) is an element of 𝔖_{i} × 𝔖_{b−i} = 𝔖_{(i,b−i)} for some i with 0 < i < b. Write u_{n−b}^{−} = u_{1}ũ, where u_{1} = ∏_{t=1}^{s}(L_{1} − Q_{t}), and suppose (x, y) ∈ 𝔖_{(i,b−i)} for some i > 0. Then T_{x}T_{y} = T_{xy} = T_{yx} = T_{y}T_{x} and, by (2.1), T_{x} commutes with u_{b}^{+}, T_{y} commutes with ũ and Tw̃ commutes with u_{1}. Therefore,

u_{n−b}^{−}Tw̃T_{xy}u_{b}^{+} = u_{1}ũTw̃T_{y}T_{x}u_{b}^{+} = ũTw̃u_{1}T_{y}T_{x}u_{b}^{+} = ũTw̃T_{y}u_{1}u_{b}^{+}T_{x} = 0;

the last equality following because ∏_{t=1}^{r}(L_{1} − Q_{t}) = 0 is a factor of u_{1}u_{b}^{+}. Consequently, q^{b}u_{n−b}^{−}Tw̃hu_{b}^{+}L_{b+1} = 0. Hence,

L_{1}v_{b} = u_{n−b}^{−}Tw̃(T_{1} … T_{b})u_{b}^{+}L_{b+1} = u_{n−b}^{−}T_{w_{n−b,b}}u_{b}^{+}L_{b+1} = v_{b}L_{b+1}

as claimed.

Next consider L_{k}v_{b} for some k with 1 < k ≤ n−b. By induction and part (i),

L_{k}v_{b} = q^{−1}T_{k−1}L_{k−1}T_{k−1}v_{b} = q^{−1}T_{k−1}v_{b}L_{b+k−1}T_{b+k−1} = v_{b}L_{b+k},

proving (iii).

Part (iv) can be proved similarly; however, here is a better argument. As there is no essential difference between u_{b}^{+} and u_{n−b}^{−} — and hence between v_{b} and v_{b}^{∗} — it follows from part (iii) that if n − b < k ≤ n then L_{k}v_{b} = (v_{b}^{∗}L_{k})^{∗} = (L_{k−n+b}v_{b}^{∗})^{∗} = v_{b}L_{k−n+b}, giving (iv).

Define L_{A}(m) = { x ∈ A | xm = 0 } to be the left annihilator of m ∈ M in A; L_{A}(h) is a left ideal of A. We also let Ann_{A}(M) = ⋂_{m∈M} L_{A}(m) be the annihilator of M; this is an ideal of A. We will apply these definitions in the case where M = V^{b} and A = H_{b} is the subalgebra of ℋ generated by { T_{i}, T_{j}, L_{k} | 1 ≤ i < n − b < j < n and 1 ≤ k ≤ n }. Observe that by the Proposition V^{b} is invariant under left multiplication by H_{b} and hence a left H_{b}-module.

3.5 Corollary Suppose that 0 ≤ b ≤ n and let the subalgebra H_{b} of ℋ be defined as above. Then H_{b} acts on V^{b} by left multiplication and L_{ℋ} (v_{b}) ∩ H_{b} = Ann_{H_{b}}(V^{b}); consequently, the algebra Ĥ_{b} = H_{b}/ Ann_{H_{b}}(V^{b}) is a subalgebra of End_{ℋ} (V^{b}).

In fact, in Theorem 4.7 below we will prove that

Ĥ_{b} ≅ End_{ℋ} (V^{b}) ≅ ℋ_{q,(Q_{1},...,Q_{s})}(b) ⊗ ℋ_{q,(Q_{s+1},...,Q_{r})}(n − b).

The next result shows that when L_{n−b+1} and L_{1}act on V^{b} by left multiplication they each satisfy one of the relations of the generators 1 ⊗ T_{0} and T_{0} ⊗ 1, respectively, in the tensor product above.

3.6 Corollary Suppose that 0 ≤ b ≤ n. Then (i) (L_{1} − Q_{s+1}) … (L_{1} − Q_{r})v_{b} = 0; (ii) (L_{n−b+1} − Q_{1}) … (L_{n−b+1} − Q_{s})v_{b} = 0; (iii) v_{b}(L_{1} − Q_{1}) … (L_{1} − Q_{s}) = 0; and, (iv) v_{b}(L_{b+1} − Q_{s+1}) … (L_{b+1} − Q_{r}) = 0.

Proof Parts (i) and (iii) follow from the relation ∏_{t=1}^{r}(L_{1} − Q_{t}) = 0 and the definition of v_{b}; for parts (ii) and (iv) apply the Proposition 3.4 to parts (iii) and (i) respectively.

3.7 Corollary Let 0 ≤ b < c ≤ n. Then u_{n−b}^{−}T_{w_{n−b,b}}u_{c}^{+} = 0.

Proof Let h = ∏_{t=s+1}^{r}(L_{b+2} − Q_{t}) ⋯ (L_{c} − Q_{t}). Then

u_{n−b}^{−}T_{w_{n−b,b}}u_{c}^{+} = u_{n−b}^{−}T_{w_{n−b,b}} ∏_{t=s+1}^{r}(L_{1} − Q_{t})(L_{2} − Q_{t}) ⋯ (L_{c} − Q_{t})
= u_{n−b}^{−}T_{w_{n−b,b}}u_{b}^{+}(L_{b+1} − Q_{s+1}) ⋯ (L_{b+1} − Q_{r})h
= v_{b}(L_{b+1} − Q_{s+1}) ⋯ (L_{b+1} − Q_{r})h = 0,

where the last equality comes from Corollary 3.6(iv).

We will study the ideals V^{b} by thinking of them as quotients of one of the modules M^{μ}. Let ω_{b} = (ω_{b}^{(1)}, …, ω_{b}^{(r)}) be the multipartition of n with

ω_{b}^{(t)} = \begin{cases}(1^{b}), & if t = s, \\ (1^{n−b}), & if t = r, \\ (0), & otherwise. \end{cases}

Then u_{b}^{+} = u_{ω_{b}}^{+} = m_{ω_{b}}; consequently, v_{b} = u_{n−b}^{−}T_{w_{n−b,b}}m_{ω_{b}} and V^{b} is a quotient of M^{ω_{b}}. This motivates the following definition.

3.8 Definition Suppose that 0 ≤ b ≤ n. Let θ_{b}: M^{ω_{b}} ⟶ V^{b} be the map given by θ_{b}(h) = u_{n−b}^{−}T_{w_{n−b,b}}h for all h ∈ M^{ω_{b}}.

Thus, θ_{b} is a surjective ℋ –module homomorphism from M^{ω_{b}} onto V^{b}. The map θ_{b} is the main tool we need to understand the modules V^{b}; first we set up some notation.

Let λ be a multipartition of n and set

Std_{b}(λ) = { 𝔱 ∈ Std(λ) | comp_{𝔱}(k) ≤ s whenever 1 ≤ k ≤ b };

that is, 𝔱 ∈ Std_{b}(λ) if and only if the numbers 1, 2, …, b all appear in one of the first s components of 𝔱. Similarly, let

Std_{b,n−b}(λ) = { 𝔱 ∈ Std_{b}(λ) | comp_{𝔱}(k) > s whenever b < k ≤ n } .

Let Λ_{b}^{+} = { λ | λ ∈ Λ^{+} and |λ^{(1)}| + ⋯ + |λ^{(s)}| = b }; then Std_{b,n−b}(λ) is non– empty if and only if λ ∈ Λ_{b}^{+}. On the other hand, Std_{b}(λ) is non–empty if and only if |λ^{(1)}| + ⋯ + |λ^{(s)}| ≥ b; we set

\overline{Λ}_{b}^{+} = \{ λ | λ ∈ Λ^{+} and |λ^{(1)}| + ⋯ + |λ^{(s)}| > b \} .

Then \overline{Λ}_{b}^{+} is a coideal in Λ^{+}; that is, if μ ∈ Λ^{+} and μ ⊵ λ for some λ ∈ Λ_{b}^{+} then μ ∈ Λ_{b}^{+}. In contrast, Λ_{b}^{+} is not a coideal; however, Λ_{b}^{+} ∪ \overline{Λ}_{b}^{+} is a coideal and Λ_{b}^{+} and (Λ_{b}^{+} ∪ \overline{Λ}_{b}^{+})/\overline{Λ}_{b}^{+} are isomorphic posets.

Observe that if λ ∈ Λ_{b}^{+} then Std_{b,n−b}(λ) = Std_{b}(λ); however, we will continue to write Std_{b,n−b}(λ) in order to emphasize the restrictions on the components of these tableaux.

3.9 Lemma Suppose that 0 ≤ b ≤ n. Then M^{ω_{b}} is free as an R–module with basis \{ m_{𝔰𝔱} | 𝔰 ∈ Std_{b}(λ) and 𝔱 ∈ Std(λ) for some λ ∈ Λ_{b}^{+} ∪ \overline{Λ}_{b}^{+} \} .

Proof As noted in (2.10), M^{ω_{b}} is free as an R–module with basis m_{S𝔱}, where S ∈ T_{0}(λ, ω_{b}) and 𝔱 ∈ Std(λ) for some multipartition λ of n. It follows from the definitions that T_{0}(λ, ω_{b}) is non–empty if and only if λ ∈ Λ_{b}^{+} ∪ \overline{Λ}_{b}^{+}; further, if λ ∈ Λ_{b}^{+} ∪ \overline{Λ}_{b}^{+} then there is a bijection between Std_{b}(λ) and T_{0}(λ, ω_{b}) given by 𝔰 ⟼ ω_{b}(𝔰) (cf. (2.9)). Consequently, if S ∈ T_{0}(λ, ω_{b}) then S = ω_{b}(𝔰) for a uniquely determined 𝔰 ∈ Std_{b}(λ) and m_{S𝔱} = m_{𝔰𝔱}. Combining these observations gives the result.

Next we identify some elements in the kernel of θ_{b}; shortly we will see that these elements are actually a basis of ker θ_{b}.

3.10 Lemma Suppose that 0 ≤ b ≤ n and λ ∈ \overline{Λ}_{b}^{+}. Let 𝔰 and 𝔱 be standard λ–tableaux with 𝔰 ∈ Std_{b}(λ). Then θ_{b}(m_{𝔰𝔱}) = 0.

Proof Suppose h = θ_{b}(m_{𝔰𝔱}) ≠ 0. Let c = |λ^{(1)}| + ⋯ + |λ^{(s)}|; then λ ∈ Λ_{c}^{+} so that b < c. Now, because λ ∈ Λ_{c}^{+} and 𝔰 ∈ Std_{b}(λ), we can find a permutation w ∈ 𝔖({b+1, …, n}) such that 𝔰w ⊵ 𝔰 and 𝔰w is a standard λ–tableau with the numbers 1, …, c all appearing in the first s components of 𝔰_{w}. (If a_{1} < ⋯ < a_{k} are the numbers between b+1 and c which appear in the last r−s components of 𝔰 and b_{1} < ⋯ < b_{k} are the k smallest numbers larger than b which appear in the first s components of 𝔰 then we can set w = (a_{1}, b_{1}) … (a_{k}, b_{k}).) Let 𝔰_{w} = 𝔰w; then d(𝔰) = d(𝔰_{w})w^{−1} and m_{𝔰𝔱} = T_{w}m_{𝔰_{w}𝔱}. Also let w̃ = w_{n−b,b}^{−1}ww_{n−b,b}; then w̃ ∈ 𝔖_{n−b} and T_{w_{n−b,b}}T_{w} = Tw̃T_{w_{n−b,b}} by Lemma 3.2. Therefore,

θ_{b}(m_{𝔰𝔱}) = u_{n−b}^{−}T_{w_{n−b,b}}m_{𝔰𝔱} = u_{n−b}^{−}T_{w_{n−b,b}}T_{w}m_{𝔰_{w}𝔱} = u_{n−b}^{−}Tw̃T_{w_{n−b,b}}m_{𝔰_{w}𝔱}.

Now, w̃ ∈ 𝔖_{n−b} so u_{n−b}^{−}Tw̃ = Tw̃u_{n−b}^{−} by (2.1)(iv); consequently

θ_{b}(m_{𝔰𝔱}) = Tw̃u_{n−b}^{−}T_{w_{n−b,b}}m_{𝔰_{w}𝔱} = Tw̃θ_{b}(m_{𝔰_{w}𝔱}).

Since Tw̃ is invertible, θ_{b}(m_{𝔰𝔱}) ≠ 0 if and only if Tw̃θ_{b}(m_{𝔰_{w}𝔱}) ≠ 0; therefore, we may assume that 𝔰 = 𝔰_{w}. Thus, it is enough to show that θ(m_{𝔰𝔱}) = 0 whenever 𝔰 ∈ Std_{c,n−c}(λ) and 𝔱 ∈ Std(λ). However, if 𝔰 ∈ Std_{c,n−c}(λ) then m_{𝔰𝔱} ∈ M^{ω_{c}} by Lemma 3.9; so m_{𝔰𝔱} = u_{ω_{c}}^{+}h for some h ∈ ℋ . Now u_{ω_{c}}^{+} = u_{c}^{+}; so θ_{b}(m_{𝔰𝔱}) = u_{n−b}^{−}T_{w_{n−b,b}}m_{𝔰𝔱} = u_{n−b}^{−}T_{w_{n−b,b}}u_{c}^{+}h = 0, by Corollary 3.7, as desired.

Recall from Theorem 1.6 that given an integer s, with 1 ≤ s ≤ r, we let

f_{s}(q, Q) = ∏_{1≤i≤s<j≤r} ∏_{−n<a<n}(q^{a}Q_{i} − Q_{j}a).

For the remainder of this paper we assume that f_{s}(q, Q) is an invertible element of R.

Suppose that 𝔰 ∈ Std_{b,n−b}(λ) for some multipartition λ. Then 𝔰′ = 𝔰w_{b,n−b} is a standard λ–tableau which has the numbers n − b + 1, …, n appearing in its first s components and the remaining numbers 1, …, n − b appearing in its last r − s components.

3.11 Lemma Suppose that λ ∈ Λ_{b}^{+} and let 𝔰 and 𝔱 be standard λ–tableaux with 𝔰 ∈ Std_{b,n−b}(λ). Let 𝔰′ = 𝔰w_{b,n−b}. Then

(i) T_{w_{n−b,b}}m_{𝔰𝔱} = m_{𝔰′𝔱}; and, (ii) there exists an invertible element α ∈ R such that

θ_{b}(m_{𝔰𝔱}) ≡ αm_{𝔰′𝔱} + ∑_{𝔲′▷𝔰′} a_{𝔲′}m_{𝔲′𝔱} mod \overline{N}^{λ},

for some a_{𝔲′} ∈ R.

Proof (i) First note that d(𝔰) ∈ 𝔖_{(b,n−b)} and recall that w_{b,n−b} is a distinguished right coset representative of 𝔖_{(b,n−b)}. Therefore, ℓ(d(𝔰′)) = ℓ(d(𝔰)) + ℓ(w_{b,n−b}); consequently, T_{w_{n−b,b}}m_{𝔰𝔱} = m_{𝔰′𝔱} since w_{n−b,b}^{−1} = w_{b,n−b}.

(ii) Part (i) together with (2.7) implies that, modulo \overline{N}^{λ},

θ_{b}(m_{𝔰𝔱}) = u_{n−b}^{−}T_{w_{n−b,b}}m_{𝔰𝔱} = ( ∏_{t=1}^{s}(L_{1} − Q_{t}) … (L_{n−b} − Q_{t}))m_{𝔰′𝔱}

≡ ( ∏_{t=1}^{s}(res_{𝔰′}(1) − Q_{t}) … (res_{𝔰′}(n − b) − Q_{t}))m_{𝔰′𝔱} + ∑_{𝔲′▷𝔰′} a_{𝔲′}m_{𝔲′𝔱}

for some a_{𝔲′} ∈ R. Let α = ∏_{t=1}^{s}(res_{𝔰′}(1) − Q_{t}) … (res_{𝔰′}(n − b) − Q_{t}); then α is the coefficient of m_{𝔰′𝔱} in θ_{b}(m_{𝔰𝔱}). Now, 1, 2, …, n − b all belong to one of the last r − s components of 𝔰′; so, for 1 ≤ k ≤ n − b, res_{𝔰′}(k) = q^{j}Q_{c} for some c and j with s < c ≤ r and −n < j < n. Therefore, α is a product of terms of the form (q^{j}Q_{c} − Q_{t}), with 1 ≤ t ≤ s. As each of these factors divides f_{s}(q, Q), it follows that α is invertible; so the Lemma is proved.

Observe that the invertible element α in part (ii) of the Lemma depends only on λ = Shape(𝔰), rather than 𝔰 itself. Notice also that

v_{b} = u_{n−b}^{−}T_{w_{n−b,b}}m_{ω_{b}} = u_{n−b}^{−}T_{w_{n−b,b}}m_{𝔱^{ω_{b}} 𝔱^{ω_{b}}} ≡ αm_{𝔱𝔱^{ω_{b}}} mod \overline{N}^{λ},

where 𝔱 = 𝔱^{ω_{b}}w_{b,n−b}; in particular, we have finally proved that v_{b} is non–zero.

3.12 Definition Suppose that 0 ≤ b ≤ n and let 𝔰 ∈ Std_{b,n−b}(λ) and 𝔱 ∈ Std(λ) for some multipartition λ ∈ Λ_{b}^{+}. Let v_{𝔰𝔱} = θ_{b}(m_{𝔰𝔱}) = u_{n−b}^{−}T_{w_{n−b,b}}m_{𝔰𝔱}.

It follows from part (ii) of the Lemma that the v_{𝔰𝔱} are linearly independent elements in V^{b}. In fact, they are a basis of V^{b}.

3.13 Theorem Suppose that f_{s}(q, Q) is invertible in R and let b be an integer with 0 ≤ b ≤ n. Then V^{b} is free as an R–module with basis

{ v_{𝔰𝔱} | 𝔰 ∈ Std_{b,n−b}(λ) and 𝔱 ∈ Std(λ) for some λ ∈ Λ_{b}^{+} }

and ker θ_{b} is free as an R–module with basis

\{ m_{𝔲𝔳} | 𝔲 ∈ Std_{b}(μ) and 𝔳 ∈ Std(μ) for some μ ∈ \overline{Λ}_{b}^{+} \} .

Proof The homomorphism θ_{b} is surjective, so by Lemma 3.9 V^{b} is spanned by the elements θ_{b}(m_{𝔲𝔳}), where 𝔲 ∈ Std_{b}(μ), 𝔳 ∈ Std(μ) and μ ∈ Λ_{b}^{+} ∪ \overline{Λ}_{b}^{+}. Furthermore, θ_{b}(m_{𝔲𝔳}) = 0 whenever μ ∈ \overline{Λ}_{b}^{+}, by Lemma 3.10. Finally, the elements { v_{𝔰𝔱} = θ_{b}(m_{𝔰𝔱}) | 𝔰 ∈ Std_{b,n−b}(λ), 𝔱 ∈ Std(λ) and λ ∈ Λ_{b}^{+} } are linearly independent (and non–zero) by Lemma 3.11(ii). Combing these three statements proves the Theorem.

Let \overline{N}^{b} = ⋂_{λ∈Λ_{b}^{+}} \overline{N}^{λ} = ∑_{μ∈\overline{Λ}_{b}^{+}} N^{μ}. Then \overline{N}^{b} is a two–sided ideal in ℋ and it is free as an R–module with basis \{ m_{𝔲𝔳} | 𝔲, 𝔳 ∈ Std(μ) for some μ ∈ \overline{Λ}_{b}^{+} \}.

3.14 Corollary Suppose that 0 ≤ b ≤ n. Then ker θ_{b} = M^{ω_{b}} ∩ \overline{N}^{b}.

3.15 Remark The proof of Theorem 3.13 relies on the assumption that f_{s}(q, Q) is an invertible element of R. Without assuming that f_{s}(q, Q) is invertible it is possible to prove via a brute force calculation that V^{b} is free as an R–module with basis

{ v_{b}L_{1}^{d_{1}} … L_{n}^{d_{n}}T_{w} | w ∈ 𝔖_{n}, 0 ≤ d_{i} < s for 1 ≤ i ≤ b and 0 ≤ d_{i} < r − s for b < i ≤ n }.

(A straightforward argument using the Robinson–Schensted correspondence verifies that the rank of V^{b} is the same in both cases.) That this set is a basis of V^{b} can also be deduced from Proposition 4.8 below; however, this argument requires that f_{s}(q, Q) be invertible in R.

In [16, Corollary 4.15] Specht filtrations of the right ideals M^{μ} of ℋ were constructed as a consequence of (2.10); we now refine the filtration of M^{ω_{b}} to give Specht filtrations of the modules V^{b} and ker θ_{b}.

3.16 Theorem Suppose that f_{s}(q, Q) is invertible in R and let b be an integer with 0 ≤ b ≤ n.

(i) There is a filtration V^{b} = V_{1} ⊃ V_{2} ⊃ ⋯ ⊃ V_{k} ⊃ V_{k+1} = 0 of V^{b} such that for each 1 ≤ i ≤ k there exists a multipartition λ_{i} ∈ Λ_{b}^{+} with V_{i}/V_{i+1} ≅ S^{λ_{i}}. Moreover, for each λ ∈ Λ_{b}^{+} the number of i with λ_{i} = λ is | Std_{b,n−b}(λ)|.

(ii) There is a filtration ker θ_{b} = K_{1} ⊃ K_{2} ⊃ ⋯ ⊃ K_{l} ⊃ K_{l+1} = 0 of ker θ_{b} such that for each 1 ≤ i ≤ l there exists a multipartition μ_{i} ∈ \overline{Λ}_{b}^{+} with K_{i}/K_{i+1} ≅ S^{μ_{i}}. Moreover, for each μ ∈ \overline{Λ}_{b}^{+} the number of i with μ_{i} = μ is | Std_{b}(μ)|.

Proof We recall the construction from [16, Cor. 4.15], using Lemma 3.9 to adapt the notation. Let 𝔰_{1}, 𝔰_{2}, …, 𝔰_{N} be the tableaux in ⋃_{λ∈Λ_{b}^{+}∪\overline{Λ}_{b}^{+}} Std_{b}(λ), ordered so that j > i whenever λ_{i} ▷ λ_{j}; here we set λ_{i} = Shape(𝔰_{i}) for all i. For i = 1, 2, …, N let M_{i} be the R–submodule of M^{ω_{b}} with basis { m_{𝔰_{j}𝔱} | j ≥ i and 𝔱 ∈ Std(λ_{j}) } . Then the proof of [16, Cor. 4.15] shows that each M_{i} is an ℋ –submodule of M^{ω_{b}} and that M_{i}/M_{i+1} ≅ S^{λ_{i}} for all i.

Choose k to be maximal such that λ_{k} ∈ Λ_{b}^{+}; then M_{k+1} = ker θ_{b} and λ_{i} ∈ \overline{Λ}_{b}^{+} if and only if i > k. By Corollary 3.14 we can set V_{i} = M_{i}/ ker θ_{b}, for i = 1, 2, …, k+1, to obtain a filtration of V^{b} with the required properties. Similarly, setting K_{j} = M_{k+j}, for j = 1, 2, …, N −k, gives the promised filtration of ker θ_{b}.

Recall from before Corollary 2.8 that the content cont(λ) of λ is the sequence (c_{r})_{r∈R}, where c_{r} = # { x ∈ [λ] | res(x) = r } for all r ∈ R. Importantly, two Specht modules belong to the same block only if the corresponding multipartitions have the same content by Corollary 2.8.

3.17 Corollary Suppose that f_{s}(q, Q) is invertible and let b and c be distinct integers with 0 ≤ b, c ≤ n. Then Hom_{ℋ} (V^{b}, V^{c}) = 0.

Proof Let R_{s} = { q^{d}Q_{i} | −n < d < n and 1 ≤ i ≤ s }; then R_{s} is the complete set of possible residues res(x), where x runs through the nodes in the first s–components of the diagram of any multipartition of n. In addition, since f_{s}(q, Q) is invertible, if y is a node appearing in one of the last r −s components of some multipartition then res(y) ∉ R_{s}; consequently, if λ is a multipartition and cont(λ) = (c_{r})_{r∈R} then λ ∈ Λ_{b}^{+} if and only if b = ∑_{r∈R_{s}} c_{r}. Therefore, if λ ∈ Λ_{b}^{+} and μ ∈ Λ_{c}^{+} then cont(λ) ≠ cont(μ) — note that by assumption b ≠ c.

Now consider Hom_{ℋ} (V^{b}, V^{c}). By Theorem 3.16, V^{b} has a Specht filtration indexed by multipartitions in Λ_{b}^{+} and V^{c} has a Specht filtration indexed by the multipartitions in Λ_{c}^{+}. Therefore, the simple composition factors of V^{b} and V^{c} belong to different blocks by the last paragraph and Corollary 2.8; hence, Hom_{ℋ} (V^{b}, V^{c}) = 0 by Schur’s lemma.

Notice that \overline{Λ}_{b}^{+} = ⋃_{c=b+1}^{n} Λ_{c}^{+}; therefore, by a similar argument, again using Theorem 3.16 and Corollary 2.8, the composition factors of V^{b} and ker θ_{b} belong to different blocks. Hence, we also have the following.

3.18 Corollary Suppose f_{s}(q, Q) is invertible in R and that 0 ≤ b ≤ n. Then the composition factors of V^{b} and ker θ_{b} belong to different blocks of ℋ ; consequently, M^{ω_{b}} ≅ V^{b} ⊕ ker θ_{b} and End_{ℋ} (M^{ω_{b}}) ≅ End_{ℋ} (V^{b}) ⊕ End_{ℋ} (ker θ_{b}).

In fact, this allows us to determine a basis of End_{ℋ} (V^{b}). Given two standard λ–tableaux 𝔰 and 𝔱 in Std_{b,n−b}(λ) let θ_{𝔰𝔱}: V^{b} ⟶V^{b} be the R–linear map given by θ_{𝔰𝔱}(v_{b}h) = v_{𝔰𝔱}h for all h ∈ ℋ ; then θ_{𝔰𝔱} is an R–module homomorphism. A priori there is no reason to expect that θ_{𝔰𝔱} is even well defined; nevertheless, it is and these elements give a basis of End_{ℋ} (V^{b}).

3.19 Theorem Suppose f_{s}(q, Q) is invertible in R and let 0 ≤ b ≤ n. Then End_{ℋ} (V^{b}) is free as an R–module with basis

{ θ_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std_{b,n−b}(λ) for some λ ∈ Λ_{b}^{+} } .

Proof A basis of End_{ℋ} (M^{ω_{b}}) is given by (2.11); in light of Lemma 3.9, we see that End_{ℋ} (M^{ω_{b}}) has as basis the maps

\{ φ_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std_{b}(λ) for some λ ∈ Λ_{b}^{+} ∪ \overline{Λ}_{b}^{+} \} ,

where φ_{𝔰𝔱} is given by φ_{𝔰𝔱}(m_{ω_{b}}h) = m_{𝔰𝔱}h for all h ∈ ℋ .

By Corollary 3.18 the homomorphism θ_{b}: M^{ω_{b}} ⟶ V^{b} splits, so θ_{b} has a right inverse; we abuse notation and write θ_{b}^{−1} for this one sided inverse. Let φ be any map in End_{ℋ} (M^{ω_{b}}); then θ_{b}φθ_{b}^{−1} belongs to End_{ℋ} (V^{b}) and every homomorphism in End_{ℋ} (V^{b}) is of this form. Now, θ_{b}(m_{ω_{b}}) = v_{b}; so there exists an h_{b} ∈ ker θ_{b} such that θ_{b}^{−1}(v_{b}h) = (m_{ω_{b}} + h_{b})h for all h ∈ ℋ . Observe that φ_{𝔰𝔱}(h_{b}) ∈ ker θ_{b} since ker θ_{b} are V^{b} are in different blocks by Corollary 3.18. Therefore, by Theorem 3.13,

θ_{b}φ_{𝔰𝔱}θ_{b}^{−1}(v_{b}h) = θ_{b}φ_{𝔰𝔱}(m_{ω_{b}} + h_{b})h = θ_{b}(m_{𝔰𝔱})h = \begin{cases}v_{𝔰𝔱}h, & if λ ∈ Λ_{b}^{+}, \\ 0, & if λ ∈ \overline{Λ}_{b}^{+}. \end{cases}

Consequently, θ_{b}φ_{𝔰𝔱}θ_{b}^{−1} = θ_{𝔰𝔱} if λ ∈ Λ_{b}^{+} and θ_{b}φ_{𝔰𝔱}θ_{b}^{−1} = 0 if λ ∈ \overline{Λ}_{b}^{+}; in particular, θ_{𝔰𝔱} ∈ End_{ℋ} (V^{b}) whenever 𝔰 and 𝔱 belong to Std_{b,n−b}(λ). As the elements { v_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std_{b,n−b}(λ) for some λ ∈ Λ_{b}^{+} } are linearly independent, so are the corresponding homomorphisms {θ_{𝔰𝔱}}. The Theorem follows.

The argument also shows that a basis of End_{ℋ} (ker θ_{b}) is given by restricting the maps \{ φ_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std_{b}(μ) for some μ ∈ \overline{Λ}_{b}^{+} \} to ker θ_{b}.

The final property of the modules V^{b} that we need is that they are projective ℋ –modules. To prove this let 𝒟_{(b,n−b)} be the set of distinguished right coset representatives of 𝔖_{(b,n−b)} in 𝔖_{n}; see, for example, [24, Prop. 3.3].

3.20 Theorem Suppose that f_{s}(q, Q) is invertible. Then ℋ ≅ ⨁_{b=0}^{n} \binom{n}{b}V^{b}.

Proof Suppose that 0 ≤ b ≤ n. As in the proof of Theorem 3.19 let θ_{b}^{−1} be a right inverse to θ_{b} and set V′_{b} = θ_{b}^{−1}(V^{b}); then V′_{b} ≅ V^{b} is a submodule of M^{ω_{b}}. For each multipartition λ ∈ Λ_{b}^{+} and tableaux 𝔰 ∈ Std_{b,n−b}(λ) and 𝔱 ∈ Std(λ) let v′_{𝔰𝔱} = θ_{b}^{−1}(v_{𝔰𝔱}); then v′_{𝔰𝔱} ∈ V′_{b} and v′_{𝔰𝔱} = m_{𝔰𝔱} + h_{𝔰𝔱} for some h_{𝔰𝔱} ∈ ker θ_{b}. Moreover, the set of these elements is a basis of V′_{b}.

We claim that ℋ = ⨁_{b=0}^{n} ⨁_{w∈𝒟_{(n−b,b)}} T_{w}^{∗}V′_{b}; since [𝔖_{n} : 𝔖_{(b,n−b)}] = \binom{n}{b} this will establish the Theorem. Note that T_{w}^{∗}V′_{b} ≅ V^{b} as right ℋ –modules.

If w ∈ 𝒟_{(b,n−b)} and 𝔰 ∈ Std_{b,n−b}(λ) then 𝔰_{w} = 𝔰w is again a standard λ– tableau, d(𝔰_{w}) = d(𝔰)w and ℓ(d(𝔰_{w})) = ℓ(d(𝔰)) + ℓ(w). Therefore, T_{w}^{∗}V′_{b} has as basis the elements

T_{w}^{∗}v′_{𝔰𝔱} = T_{w}^{∗}(m_{𝔰𝔱} + h_{𝔰𝔱}) = m_{𝔰_{w}𝔱} + T_{w}^{∗}h_{𝔰𝔱},

where 𝔰, 𝔱 ∈ Std_{b,n−b}(λ) for some λ ∈ Λ_{b}^{+}. Notice that ker θ_{b} ⊆ \overline{N}^{b} by Corollary 3.14 and that \overline{N}^{b} is a two–sided ideal of ℋ ; so T_{w}^{∗}h_{𝔰𝔱} ∈ \overline{N}^{b}.

Now, if λ ∈ Λ_{b}^{+} then Std(λ) = ∐_{w∈𝒟_{(b,n−b)}} Std_{b,n−b}(λ)w. Therefore, the elements

⋃_{b=0}^{n} { T_{w}^{∗}v′_{𝔰𝔱} | w ∈ 𝒟_{(b,n−b)}, 𝔰 ∈ Std_{b,n−b}(λ) and 𝔱 ∈ Std(λ) for some λ ∈ Λ_{b}^{+} }

are linearly independent; moreover, by (2.3) and the above remarks, this set is also a basis of ℋ . Consequently, ℋ = ⨁_{b=0}^{n} ⨁_{w∈𝒟_{(b,n−b)}} T_{w}^{∗}V′_{b}, proving our claim and hence the Theorem.

A similar argument shows that M^{ω_{a}} ≅ ⨁_{b=a}^{n} \binom{n−a}{b}V^{b} whenever 0 ≤ a ≤ n; the Theorem is the case a = 0 since m_{ω_{0}} = 1.

Recall that a progenerator, or projective generator, for ℋ is a projective ℋ –module which contains every principal indecomposable ℋ –module as a direct summand. Every direct summand of ℋ is automatically projective and every principal indecomposable ℋ –module is a direct summand of ℋ ; hence, Theorem 3.20 yields the following.

3.21 Corollary Suppose that f_{s}(q, Q) is invertible. Then (i) V^{b} is a projective ℋ –module for b = 0, 1, …, n; and,

(ii) V = ⨁_{b=0}^{n} V^{b} is a progenerator for ℋ .

4 The Morita equivalence

We are almost ready to prove our main results; in fact, we have already done most of the hard work. In Corollary 3.21(ii) we showed that V = ⨁_{b=0}^{n} V^{b} is a progenerator of ℋ ; hence, it follows that ℋ is Morita equivalent to End_{ℋ} (V). To prove Theorem 1.1 we show that End_{ℋ} (V) is isomorphic to a direct sum of tensor products of smaller Ariki–Koike algebras.

4.1 Proposition Suppose that f_{s}(q, Q) is invertible. Then ℋ is Morita equivalent to ⨁_{b=0}^{n} End_{ℋ} (V^{b}).

Proof Let V = ⨁_{b=0}^{n} V^{b}. As remarked above, it follows from Corollary 3.21(ii) that ℋ is Morita equivalent to End_{ℋ} (V); see [7, Lemma 2.2.3]. However, by Corollary 3.17,

End_{ℋ} (V) = ⨁_{0≤b,c≤n} Hom_{ℋ} (V^{b}, V^{c}) ≅ ⨁_{b=0}^{n} End_{ℋ} (V^{b}),

giving the result.

For b = 0, 1 …, n let ℋ_{b} ⊗ ℋ_{n−b} = ℋ_{q,Q_{1}}(b) ⊗ ℋ_{q,Q_{2}}(n − b), where Q_{1} = (Q_{1}, …, Q_{s}) and Q_{2} = (Q_{s+1}, …, Q_{r}). We have already computed End_{ℋ} (V^{b}) in Theorem 3.19; in order to prove Theorem 1.6, and hence our main result, we will use this result to show that End_{ℋ} (V^{b}) ≅ ℋ_{b} ⊗ ℋ_{n−b}.

The subalgebra ℋ (𝔖_{b}) ⊗ ℋ (𝔖_{n−b}) of ℋ_{b} ⊗ ℋ_{n−b} is isomorphic to the subalgebra ℋ (𝔖_{(n−b,b)}) of ℋ spanned by { T_{w} | w ∈ 𝔖_{(n−b,b)} } — note that ℋ (𝔖_{(b,n−b)}) and ℋ (𝔖_{(n−b,b)}) are isomorphic algebras, the reason for introducing the twist is that ℋ (𝔖_{(n−b,b)})v_{b} = v_{b}ℋ (𝔖_{(b,n−b)}) by Proposition 3.4. In general ℋ has no subalgebra isomorphic to ℋ_{b} ⊗ ℋ_{n−b}; however, ℋ does have an R–submodule isomorphic to ℋ_{b} ⊗ ℋ_{n−b} and this we can exploit. Let ℋ_{n−b,b} be the R–submodule of ℋ spanned by the elements

{ L_{1}^{d_{1}} … L_{n}^{d_{n}}T_{w} | w ∈ 𝔖_{(n−b,b)}, 0 ≤ d_{i} < r − s for 1 ≤ i ≤ n − b and 0 ≤ d_{i} < s for n − b < i ≤ n }.

We emphasize that typically ℋ_{n−b,b} is not a subalgebra of ℋ ; however, ℋ_{n−b,b} does generate the subalgebra H_{b} of Corollary 3.5.

4.2 Suppose that 0 ≤ b ≤ n. Then ℋ_{b} ⊗ ℋ_{n−b} and ℋ_{n−b,b} are isomorphic as R–modules via the R–linear map Θ_{b}: ℋ_{b} ⊗ ℋ_{n−b} ⟶ℋ_{n−b,b} determined by

L_{1}^{d_{1}} … L_{b}^{d_{b}}T_{x} ⊗ L_{1}^{e_{1}} … L_{n−b}^{e_{n−b}}T_{y} ⟼(L_{1}^{e_{1}} … L_{n−b}^{e_{n−b}}T_{y})(L_{n−b+1}^{d_{1}} … L_{n}^{d_{b}}T_{x′}),

where x′ = w_{n−b,b}xw_{b,n−b} = w_{b,n−b}^{−1}xw_{b,n−b}; here x ∈ 𝔖_{b}, y ∈ 𝔖_{n−b}, 0 ≤ d_{i} < s for i = 1, …, b and 0 ≤ e_{j} < r − s for j = 1, …, n − b.

Maintaining the notation of (4.2) notice that

(L_{1}^{e_{1}} … L_{n−b}^{e_{n−b}}T_{y})(L_{n−b+1}^{d_{1}} … L_{n}^{d_{b}}T_{x′}) = L_{1}^{e_{1}} … L_{n−b}^{e_{n−b}}L_{n−b+1}^{d_{1}} … L_{n}^{d_{b}}T_{x′y}

by (2.1); in particular, the map 𝔖_{b} × 𝔖_{n−b} ⟶ 𝔖_{(n−b,b)} : (x, y) ⟼ x′y = yx′ is an isomorphism of groups. Hereafter, we identify elements of the R–modules ℋ_{b} ⊗ ℋ_{n−b} and ℋ_{n−b,b} via the map Θ_{b}.

4.3 Lemma Suppose that 0 ≤ b ≤ n. Then V^{b} becomes a left ℋ_{b} ⊗ ℋ_{n−b}– module via hv = Θ_{b}(h)v for all h ∈ ℋ_{b} ⊗ ℋ_{n−b} and v ∈ V^{b}. Consequently, Θ_{b}(h_{1}h_{2})v = Θ_{b}(h_{1})Θ_{b}(h_{2})v for all h_{1}, h_{2} ∈ ℋ_{b} ⊗ ℋ_{n−b} and all v ∈ V^{b}.

Proof It is enough to check that the relations in ℋ_{b} ⊗ ℋ_{n−b} are preserved. As an algebra ℋ_{b} ⊗ ℋ_{n−b} is generated by the two sets of commuting elements T_{i} ⊗ 1 and 1 ⊗ T_{j}, where 0 ≤ i < b and 0 ≤ j < n − b. Now

Θ_{b}(T_{i} ⊗ 1) = \begin{cases}L_{n−b+1}, & if i = 0, \\ T_{n−b+i}, & otherwise, \end{cases} and Θ_{b}(1 ⊗ T_{j}) = \begin{cases}L_{1}, & if j = 0, \\ T_{j}, & otherwise. \end{cases}

Therefore, the relations in ℋ and (2.1)(ii) ensure that all of the relations in ℋ_{b} ⊗ ℋ_{n−b} are satisfied except possibly that (T_{0} ⊗ 1 − Q_{1}) … (T_{0} ⊗ 1 − Q_{s}) and (1 ⊗ T_{0} − Q_{s+1}) … (1 ⊗ T_{0} − Q_{r}) must both act as zero on V^{b}. However, this is precisely the content of parts (ii) and (i), respectively, of Corollary 3.6. (Notice that when Θ_{b} is applied to the braid relation T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0} in ℋ_{b} it becomes qL_{n−b+1}L_{n−b+2} = qL_{n−b+2}L_{n−b+1} in ℋ ; this holds by virtue of (2.1)(ii).)

Therefore, V^{b} is an (ℋ_{b} ⊗ ℋ_{n−b}, ℋ )–bimodule. Evidently, the left action of ℋ_{b} ⊗ ℋ_{n−b} on V^{b} commutes with the right action of ℋ so we have a homomorphism from ℋ_{b} ⊗ ℋ_{n−b} into End_{ℋ} (V^{b}). Furthermore, Lemma 4.3 implies that for all h_{1}, h_{2} ∈ ℋ_{b} ⊗ ℋ_{n−b} there exists some h_{3} ∈ ℋ such that Θ_{b}(h_{1}h_{2}) = Θ_{b}(h_{1})Θ_{b}(h_{2}) + h_{3} and h_{3} ∈ L_{H_{b}}(V^{b}); cf. Corollary 3.5.

Shortly we will see that ℋ_{b} ⊗ ℋ_{n−b} acts faithfully on V^{b}; before we can show this we need some more notation. Recall that Λ^{+} = Λ^{+}(n: r) and, more generally, that Λ^{+}(b: s) is the set of multipartitions of b with s–components. There is an evident bijection Λ^{+}(b: s)×Λ^{+}(n−b: r −s) ⟶ Λ_{b}^{+} which maps the pair σ = (σ^{(1)}, …, σ^{(s)}) ∈ Λ^{+}(b: s) and τ = (τ^{(1)}, …, τ^{(r−s)}) ∈ Λ^{+}(n − b: r − s) to the multipartition (σ, τ) = (σ^{(1)}, …, σ^{(s)}, τ^{(1)}, …, τ^{(r−s)}) ∈ Λ_{b}^{+}.

4.4 Lemma Suppose that 0 ≤ b ≤ n and that λ ∈ Λ_{b}^{+}. Then λ = (σ, τ) for unique multipartitions σ ∈ Λ^{+}(b: s) and τ ∈ Λ^{+}(n − b: r − s); moreover, we have θ_{b}(u_{λ}^{+}) = Θ_{b}(u_{σ}^{+} ⊗ u_{τ}^{+})v_{b}.

Proof The uniqueness of σ and τ follows from the remarks above. For the remaining statement first recall that u_{α,t} = ∏_{k=1}^{α}(L_{k} − Q_{t}) for any α and t. Let α_{t} = |λ^{(1)}| + ⋯ + |λ^{(t−1)}|, for 1 ≤ t ≤ r, and let β_{t} = α_{t+r−s} − b for t = 1, …, r − s. Then u_{λ}^{+} = u_{α_{1},1} … u_{α_{r},r} and, abusing notation, u_{σ}^{+} ⊗ u_{τ}^{+} = u_{α_{1},1} … u_{α_{s},s} ⊗ u_{β_{1},s+1} …, u_{β_{r−s},r} ∈ ℋ_{b} ⊗ ℋ_{n−b}.

Now, by Proposition 3.4(iv), if 1 ≤ t ≤ s then

Θ_{b}(u_{α_{t},t} ⊗ 1)v_{b} = ( ∏_{k=n−b+1}^{n−b+α_{t}}(L_{k} − Q_{t}))v_{b} = v_{b}u_{α_{t},t};

so, Θ_{b}(u_{σ}^{+} ⊗ 1)v_{b} = v_{b}u_{α_{1},1} … u_{α_{s},s}. On the other hand, by Proposition 3.4(iii),

Θ_{b}(1 ⊗ u_{β_{t},t})v_{b} = ( ∏_{k=1}^{β_{t}}(L_{k} − Q_{t}))v_{b} = v_{b} ∏_{k=b+1}^{b+β_{t}}(L_{k} − Q_{t}),

for t = s + 1, …, r. Now, u_{α_{t},t} = ∏_{k=1}^{b+β_{t}}(L_{k} − Q_{t}) for t = s + 1, …, r; therefore,

Θ_{b}(1 ⊗ u_{τ}^{+})v_{b} = v_{b} ∏_{t=s+1}^{r} ∏_{k=b+1}^{b+β_{t}}(L_{k} − Q_{t})

= u_{n−b}^{−}T_{w_{n−b,b}}u_{b}^{+} ∏_{t=s+1}^{r} ∏_{k=b+1}^{b+β_{t}}(L_{k} − Q_{t})

= u_{n−b}^{−}T_{w_{n−b,b}}u_{α_{s+1},s+1} … u_{α_{r},r}.

Hence, Θ_{b}(u_{σ}^{+} ⊗ u_{τ}^{+})v_{b} = u_{n−b}^{−}T_{w_{n−b,b}}u_{α_{1},1} … u_{α_{r},r} = θ_{b}(u_{λ}^{+}) as claimed.

We want to extend this result to the standard basis of ℋ_{b} ⊗ ℋ_{n−b}. Suppose that λ ∈ Λ_{b}^{+} and, as above, write λ = (σ, τ) for multipartitions σ ∈ Λ^{+}(b: s) and τ ∈ Λ^{+}(n−b: r −s). Given standard tableaux 𝔰_{1} = (𝔰_{1}^{(1)}, …, 𝔰_{1}^{(s)}) ∈ Std(σ) and 𝔰_{2} = (𝔰_{2}^{(1)}, …, 𝔰_{2}^{(r−s)}) ∈ Std(τ) let (𝔰_{1}, 𝔰′_{2}) be the λ–tableau

(𝔰_{1}, 𝔰′_{2}) = (𝔰_{1}^{(1)}, …, 𝔰_{1}^{(s)}, 𝔰_{2}^{(1)}w_{n−b,b}, …, 𝔰_{2}^{(r−s)}w_{n−b,b});

then (𝔰_{1}, 𝔰′_{2}) ∈ Std_{b,n−b}(λ) and a straightforward calculation reveals the following.

4.5 Lemma Suppose that λ = (σ, τ) ∈ Λ_{b}^{+} as above. Then the map

Std(σ) × Std(τ) ⟶ Std_{b,n−b}(λ) : (𝔰_{1}, 𝔰_{2})⟼𝔰 = (𝔰_{1}, 𝔰′_{2})

is a bijection. Furthermore, d(𝔰) = d(𝔰_{1})w_{n−b,b}^{−1}d(𝔰_{2})w_{n−b,b} ∈ 𝔖_{(b,n−b)} and ℓ(d(𝔰)) = ℓ(d(𝔰_{1})) + ℓ(d(𝔰_{2})).

We can now give the connection between θ_{b} and Θ_{b}.

4.6 Lemma Suppose λ = (σ, τ) ∈ Λ_{b}^{+} as above. Let 𝔰, 𝔱 ∈ Std_{b,n−b}(λ) and write 𝔰 = (𝔰_{1}, 𝔰′_{2}) and 𝔱 = (𝔱_{1}, 𝔱′_{2}) as in Lemma 4.5. Then

θ_{b}(m_{𝔰𝔱}) = Θ_{b}(m_{𝔰_{1}𝔱_{1}} ⊗ m_{𝔰_{2}𝔱_{2}})v_{b}.

Proof By Proposition 3.4, ℋ (𝔖_{(n−b,b)})v_{b} = v_{b}ℋ (𝔖_{(b,n−b)}); therefore,

Θ_{b}(x_{σ} ⊗ x_{τ})v_{b} = x_{(τ,σ)}v_{b} = v_{b}x_{(σ,τ)} = v_{b}x_{λ}.

Furthermore, Θ_{b}(u_{σ}^{+}⊗u_{τ}^{+})v_{b} = θ_{b}(u_{λ}^{+}) by Lemma 4.4; therefore, using Lemma 4.3 to combine these two equations shows that

Θ_{b}(m_{σ} ⊗ m_{τ})v_{b} = Θ_{b}(x_{σ}u_{σ}^{+} ⊗ x_{τ} u_{τ}^{+})v_{b} = Θ_{b}(x_{σ} ⊗ x_{τ})Θ_{b}(u_{σ}^{+} ⊗ u_{τ}^{+})v_{b}
= Θ_{b}(x_{σ} ⊗ x_{τ})v_{b}u_{λ}^{+} = v_{b}x_{λ}u_{λ}^{+} = v_{b}m_{λ}.

Finally, note that if x ∈ 𝔖_{b} and y ∈ 𝔖_{n−b} then, by (4.2) and Proposition 3.4,

Θ_{b}(T_{x} ⊗ T_{y})v_{b} = T_{w_{b,n−b}^{−1}xw_{b,n−b}}T_{y}v_{b} = v_{b}T_{x}T_{w_{n−b,b}^{−1}yw_{n−b,b}}.

Since m_{𝔰_{1}𝔱_{1}} ⊗ m_{𝔰_{2}𝔱_{2}} = (T_{d(𝔰_{1})}^{∗} ⊗ T_{d(𝔱_{1})})(m_{σ} ⊗ m_{τ})(T_{d(𝔰_{2})}^{∗} ⊗ T_{d(𝔱_{2})}), another application of Lemma 4.3, together with Lemma 4.5, now completes the proof.

We have finally reached the summit.

4.7 Theorem Suppose that f_{s}(q, Q) is invertible in R and let b be an integer with 0 ≤ b ≤ n. Then

(i) V^{b} is a faithful left ℋ_{b} ⊗ ℋ_{n−b}–module; (ii) End_{ℋ} (V^{b}) ≅ ℋ_{b} ⊗ ℋ_{n−b}; and,

(iii) ℋ is Morita equivalent to ℋ_{s} = ⨁_{b=0}^{n} ℋ_{b} ⊗ ℋ_{n−b}.

Proof By Lemma 4.3 ℋ_{b}⊗ℋ_{n−b} acts on V^{b} by left multiplication; this action is faithful because, by Lemma 4.6 and Theorem 3.13, the elements of the standard basis of ℋ_{b} ⊗ ℋ_{n−b} map v_{b} to linearly independent elements in V^{b}. Further, because the action is faithful, End_{ℋ} (V^{b}) ≅ ℋ_{b} ⊗ ℋ_{n−b} by Theorem 3.19 and Lemma 4.6. Finally, part (iii) follows from part (ii) and Proposition 4.1.

This proves Theorem 1.6 and hence the main result of this paper. The remainder of this section examines this Morita equivalence more closely.

For an algebra A let Mod_{A} be the category of (finite dimensional) right A–modules. Then we have shown that there exist functors (in fact, category equivalences),

H_{s} : Mod_{ℋ} ⟶Mod_{ℋ_{s}} and Ĥ_{s} : Mod_{ℋ_{s}} ⟶Mod_{ℋ} .

These functors are described in terms of the (ℋ_{s}, ℋ )–bimodule V = ⨁_{b=0}^{n} V^{b}; explicitly, H_{s}(M) = Hom_{ℋ} (V, M) and Ĥ_{s}(X) = X ⊗_{ℋ_{s}} V. (Here we consider V as a left ℋ_{s}–module by specifying that ℋ_{b} ⊗ ℋ_{n−b} annihilates V^{c} when b ≠ c.) For this, and other standard facts about Morita equivalences, see [7, Section 2.2].

By Theorem 3.20 we can write ℋ = ⨁_{b=0}^{n} ℋ (b) where ℋ (b) is the smallest two–sided ideal of ℋ which contains V^{b} as a direct summand (for 0 ≤ b ≤ n). Therefore, Mod_{ℋ} = ⨁_{b=0}^{n} Mod_{ℋ(b)} . The modules in Mod_{ℋ(b)} are mapped into Mod_{ℋ_{b}⊗ℋ_{n−b}} by H_{s}, so we have subfunctors

H_{s,b} : Mod_{ℋ(b)} ⟶Mod_{ℋ_{b}⊗ℋ_{n−b}} and Ĥ_{s,b} : Mod_{ℋ_{b}⊗ℋ_{n−b}} ⟶Mod_{ℋ(b)}

given by H_{s,b}(M) = Hom_{ℋ} (V^{b}, M) and Ĥ_{s,b}(X) = X ⊗_{ℋ_{b}⊗ℋ_{n−b}} V^{b}. Each of these functors induces a Morita equivalence.

In general, V is not free as a left ℋ_{s}–module; however, as the next result shows, V^{b} is free as a left ℋ_{b} ⊗ ℋ_{n−b}–module. Recall that 𝒟_{(b,n−b)} is the set of distinguished right coset representatives of 𝔖_{(b,n−b)} ≅ 𝔖_{b} × 𝔖_{n−b} in 𝔖_{n}. For each w ∈ 𝒟_{(b,n−b)} let V_{w}^{b} be the R–submodule Θ_{b}(ℋ_{b} ⊗ ℋ_{n−b})v_{b}T_{w} of V^{b}.

4.8 Proposition Suppose that 0 ≤ b ≤ n and let w ∈ 𝒟_{(b,n−b)}. Then V_{d}^{b} is a left ℋ_{b} ⊗ ℋ_{n−b}–submodule of V^{b} which is isomorphic to the left regular representation of ℋ_{b} ⊗ ℋ_{n−b}. A basis of V_{d}^{b} is given by

{ v_{𝔰𝔱} | 𝔰 ∈ Std_{b,n−b}(λ) and 𝔱 ∈ Std(λ)w for some λ ∈ Λ_{b}^{+} } .

Moreover, as a left ℋ_{b} ⊗ ℋ_{n−b}–module,

V^{b} = ⨁_{w∈𝒟_{(b,n−b)}} V_{w}^{b}.

Consequently, V^{b} is a free ℋ_{b} ⊗ ℋ_{n−b}–module of rank \binom{n}{b}.

Proof Applying Lemma 4.6 and Theorem 3.13 we see that V_{w}^{b} has basis { v_{𝔰𝔱}T_{w} | 𝔰, 𝔱 ∈ Std_{b,n−b}(λ) for some λ ∈ Λ_{b}^{+} }. Now, if 𝔱 ∈ Std_{b,n−b}(λ) then d(𝔱) ∈ 𝔖_{b,n−b} so that ℓ(d(𝔱)w) = ℓ(d(𝔱)) = ℓ(w) since w is a distinguished right coset representative of 𝔖_{(b,n−b)} in 𝔖_{n}. Therefore, if 𝔰 and 𝔱 in Std_{b,n−b}(λ) then m_{𝔰𝔱}T_{w} = m_{𝔰𝔲} where 𝔲 = 𝔱w ∈ Std(λ); so, v_{𝔰𝔱}T_{w} = v_{𝔰𝔲}. Therefore, V_{w}^{b} has the required basis and V^{b} = ⨁_{w∈𝒟_{(b,n−b)}} V_{w}^{b}. Finally, V_{w}^{b} affords the left regular representation of ℋ_{b} ⊗ ℋ_{n−b} by Theorem 4.7(i).

Suppose that { h_{i} | i ∈ I } is a basis of ℋ_{b} ⊗ℋ_{n−b}; then, by the Proposition,

{ Θ_{b}(h_{i})v_{b}T_{w} | i ∈ I and w ∈ 𝒟_{(b,n−b)} }

is a basis of V^{b}. In particular, taking { h_{i} | i ∈ I } to be the Ariki–Koike basis of ℋ_{b} ⊗ ℋ_{n−b} we obtain the basis of V^{b} mentioned in Remark 3.15.

In particular, since V^{b} is free as a left ℋ_{b}⊗ℋ_{n−b}–module by Proposition 4.8, tensoring with V^{b} is an exact functor. Hence, we have the following.

4.9 Corollary Suppose that X is a right ideal of ℋ_{b} ⊗ ℋ_{n−b}. Then

Ĥ_{s,b}(X) = Θ_{b}(X)V^{b} = ⨁_{w∈𝒟_{(b,n−b)}} Θ_{b}(X)v_{b}T_{w},

where Θ_{b}(X)v_{b}T_{w} = Θ_{b}(X)V_{w}^{b} ≅ X as an R–module. In particular, if X is a free R–module of rank ℓ then Ĥ_{s,b}(X) is a free R–module of rank ℓ\binom{n}{b}.

More generally, if 0 ⟶ X ⟶ Y ⟶ Y/X ⟶ 0 is an exact sequence of ℋ_{b}⊗ℋ_{n−b}–modules then Ĥ_{s,b}(Y/X) ≅ Ĥ_{s,b}(Y)/Ĥ_{s,b}(X) ≅ Θ_{b}(Y)V^{b}/Θ_{b}(X)V^{b}.

We are now ready to describe how the Specht modules S^{λ} and simple ℋ – modules D^{μ} of ℋ_{b} ⊗ ℋ_{n−b} “multiply up” to give ℋ –modules.

4.10 Lemma Suppose that R is a field, 0 ≤ b ≤ n and that λ ∈ Λ_{b}^{+}. Then S^{λ} ∈ Mod_{ℋ(b)} ; moreover, if D^{λ} ≠ (0) then D^{λ} ∈ Mod_{ℋ(b)} .

Proof First note that by [18, Theorem 3.7(ii)] all of the composition factors of S^{λ} belong to the same block; therefore, by Theorem 3.16(i) if λ ∈ Λ_{b}^{+} then S^{λ} ∈ Mod_{ℋ(b)} . Finally, if D^{λ} ≠ (0) then S^{λ} and D^{λ} belong to the same block, so D^{λ} also belongs to Mod_{ℋ(b)} .

Consequently, if λ ∈ Λ_{b}^{+} then Ĥ_{s}(S^{λ}) = Ĥ_{s,b}(S^{λ}) and Ĥ_{s}(D^{λ}) = Ĥ_{s,b}(D^{λ}). Recall that d_{λμ} = [S^{λ} : D^{μ}] is the decomposition multiplicity of the simple

module D^{μ} in the Specht module S^{λ}.

4.11 Proposition Suppose that R is a field and that f_{s}(q, Q) ≠ 0. Let λ ∈ Λ_{b}^{+} and μ ∈ Λ_{c}^{+} be multipartitions of n and, as in Lemma 4.4, write λ = (σ, τ) and μ = (α, β), where σ ∈ Λ^{+}(b: s), τ ∈ Λ^{+}(n − b: r − s), α ∈ Λ^{+}(c: s) and β ∈ Λ^{+}(n − c: r − s) (and 0 ≤ b, c ≤ n). Then the following hold.

(i) S^{λ} ≅ Ĥ_{s}(S^{σ} ⊗ S^{τ}) = Ĥ_{s,b}(S^{σ} ⊗ S^{τ}).

(ii) D^{μ} ≅ Ĥ_{s}(D^{α}⊗D^{β}) = Ĥ_{s,c}(D^{α}⊗D^{β}). Consequently, D^{μ} ≠ (0) if and only

if D^{α} ≠ (0) and D^{β} ≠ (0); in addition, dim D^{μ} = \binom{n}{c}(dim D^{α})(dim D^{β}). (iii) Suppose that D^{μ} ≠ (0). Then

d_{λμ} = \begin{cases}d_{σα}d_{τβ}, & if b = c, \\ 0, & if b ≠ c. \end{cases}

Proof For part (i) we don’t actually need to assume that R is a field. By definition, S^{σ} ⊗ S^{τ} = (z_{σ} ⊗ z_{τ})ℋ_{b} ⊗ ℋ_{n−b} where z_{σ} ⊗ z_{τ} = m_{σ} ⊗ m_{τ} + \overline{N}^{σ}⊗ \overline{N}^{τ}. By Corollary 4.9 and Lemma 4.6, Ĥ_{s,b}(\overline{N}^{σ}⊗ \overline{N}^{τ}) ≅ Θ_{b}(\overline{N}^{σ}⊗ \overline{N}^{τ})V^{b} ≅ V^{b}\overline{N}^{λ}; therefore, Ĥ_{s,b}(S^{σ} ⊗ S^{τ}) ≅ S^{λ} by the remarks following Corollary 4.9. (Notice that in this case the formula dim S^{λ} = \binom{n}{b} dim(S^{σ}⊗S^{τ}) is just the combinatorial identity | Std(λ)| = \binom{n}{b}| Std(σ)| · | Std(τ)|.)

For (ii) note that because Ĥ_{s,c} is an equivalence of categories it takes simple ℋ_{b} ⊗ ℋ_{n−b}–modules to simple ℋ –modules. Hence, D^{μ} ≅ Ĥ_{s,c}(D^{α} ⊗ D^{β}) by induction on the dominance ordering using part (i) and (2.6); the dimension formula now follows from Corollary 4.9.

Finally, as Ĥ_{s} takes composition series to composition series, part (iii) follows from (i) and (ii) and the decomposition Mod_{ℋ} = ⨁_{a=0}^{n} Mod_{ℋ(a)} .

Note that part (iii) of the Proposition provides a recipe for calculating the decomposition matrix of ℋ from the decomposition matrices of the “smaller” Ariki–Koike algebras ℋ_{q,Q_{1}}(b) and ℋ_{q,Q_{2}}(n − b) for 0 ≤ b ≤ n.

Proposition 4.11 and all of the other consequences of Theorem 1.6 can be extended to the general case of Theorem 1.1, where the parameter set Q is partitioned into an arbitrary number of pieces. The notation needed to describe this is rather cumbersome so we leave the details to the reader.

5 The cyclotomic q–Schur algebra

We now extend the Morita equivalence of the previous section to cyclotomic q–Schur algebras. Let Γ ⊆ Λ be a finite set of multicompositions of n with the property that whenever μ ∈ Λ^{+} and μ ⊵ λ for some λ ∈ Γ then μ ∈ Γ. The cyclotomic q–Schur algebra associated with Γ is the algebra

𝒮_{q,Q}(Γ) = End_{ℋ} ( ⨁_{λ∈Γ} M^{λ}).

For the statement of Theorem 1.5 we set 𝒮_{q,Q}(n) = 𝒮_{q,Q}(Λ^{+}).

Let Γ^{+} = Γ ∩ Λ^{+} be the set of multipartitions in Γ. Then, by [16, Theorem 6.6] (see also (2.11)), the algebra 𝒮_{q,Q}(Γ) has (cellular) basis

{ φ_{ST} | S ∈ T_{0}(λ, μ) and T ∈ T_{0}(λ, ν) for some μ, ν ∈ Γand someλ ∈ Γ^{+} } ,

where φ_{ST} is the ℋ –module homomorphism given by φ_{ST}(m_{α}h) = δ_{αν}m_{ST}h. The basis {φ_{ST}} is the semistandard basis of 𝒮_{q,Q}(Γ).

As before, we fix an integer s, with 1 ≤ s ≤ r, such that f_{s}(q, Q) is invertible in R and let Γ_{b} = { (λ^{(1)}, …, λ^{(r)}) ∈ Γ | b = |λ^{(1)}| + ⋯ + |λ^{(s)}| } . We need to define analogues of the sets Λ^{+}(b: s); however, we must be a little careful. Let

Γ_{l}(b: s) = { σ ∈ Λ(b: s) | (σ, τ) ∈ Γ for some τ ∈ Λ(n − b: r − s) }

and

Γ_{r}(n − b: r − s) = { τ ∈ Λ(n − b: r − s) | (σ, τ) ∈ Γ for some σ ∈ Λ(b: s) } .

Also let Γ_{l}^{+}(b: s) = Γ_{l}(b: s) ∩ Λ^{+}(b: s) and Γ_{r}^{+}(n − b: r − s) = Γ_{r}(n − b: r − s) ∩ Λ^{+}(n − b: r − s). Consider Γ_{l}(b: s) × Γ_{r}(n − b: r − s) as a poset in the obvious way.

5.1 Lemma Suppose that 0 ≤ b ≤ n. Then Γ_{l}(b: s)×Γ_{r}(n−b: r−s) and Γ_{b} are naturally isomorphic posets. In particular, if (α, β) ∈ Λ^{+}(b: s)×Λ^{+}(n−b: r−s) and (α, β) ⊵ (σ, τ) for some (σ, τ) ∈ Γ_{l}(b: s) × Γ_{r}(n − b: r − s) then (α, β) ∈ Γ_{l}(b: s) × Γ_{r}(n − b: r − s).

Proof The isomorphism is given by

((σ^{(1)}, …, σ^{(s)}), (τ^{(1)}, …, τ^{(r−s)}))⟼(σ^{(1)}, …, σ^{(s)}, τ^{(1)}, …, τ^{(r−s)}).

This is a poset isomorphism because |σ^{(1)}| + ⋯ + |σ^{(s)}| = b whenever (σ, τ) is an element of Γ_{l}(b: s) × Γ_{r}(n − b: r − s).

Let 𝒮_{b}(Γ) ⊗ 𝒮_{n−b}(Γ) = 𝒮_{q,Q_{1}}(Γ_{l}(b: s)) ⊗ 𝒮_{q,Q_{2}}(Γ_{r}(n − b: r − s)), where Q_{1} = (Q_{1}, …, Q_{s}) and Q_{2} = (Q_{s+1}, …, Q_{r}). The point of the Lemma is that it allows us to index the representations of 𝒮_{b}(Γ) ⊗ 𝒮_{n−b}(Γ) with the elements of Γ_{b}. For convenience we identity Γ_{b} and Γ_{l}(b: s) × Γ_{r}(n − b: r − s) in the sequel.

We can now give the analogue of Theorem 1.6 for a cyclotomic q–Schur algebra. Rather than introduce the notation necessary to state the general result, we consider only the special case where Q is partitioned into two pieces. See Theorem 1.5 for a special case of the more general result.

5.2 Theorem Let 1 ≤ s ≤ r and suppose that f_{s}(q, Q) is an invertible element of R. Then the cyclotomic q–Schur algebra 𝒮_{q,Q}(Γ) is Morita equivalent to

𝒮_{s}(Γ) = ⨁_{b=0}^{n} 𝒮_{b}(Γ) ⊗ 𝒮_{n−b}(Γ).

Before we can give the proof we require some preparation. The basic idea is to investigate how the functor H_{s} acts on ⨁_{λ∈Γ} M^{λ}; in fact, it is easier to work with Ĥ_{s}.

5.3 Lemma Suppose that 0 ≤ b ≤ n and let λ = (σ, τ) ∈ Γ_{b}. Then

Ĥ_{s,b}(M^{σ} ⊗ M^{τ}) = θ_{b}(M^{λ}).

Proof We first note that u_{b}^{+} = m_{ω_{b}} is a left factor of u_{λ}^{+} because λ ∈ Λ_{b}^{+}; consequently, M^{λ} is a submodule of M^{ω_{b}} and so θ_{b}(M^{λ}) makes sense. Now Θ_{b}(m_{σ} ⊗ m_{τ})v_{b} = θ_{b}(m_{λ}) by Lemma 4.6; therefore, Ĥ_{s,b}(M^{σ} ⊗ M^{τ}) = θ_{b}(m_{λ})ℋ = θ_{b}(M^{λ}) by Corollary 4.9.

Combining Theorem 3.16 and Corollary 3.18 we see that θ_{b} projects M^{ω_{b}} onto V^{b} and consequently that V^{b} is the unique direct summand of M^{ω_{b}} which belongs to Mod_{ℋ(b)} . Now, ℋ = ⨁_{c=0}^{n} ℋ (c) so we can write M^{λ} = ⨁_{c=0}^{n} M^{λ}(c), where M^{λ}(c) is the largest direct summand of M^{λ} which is contained in ℋ (c). Furthermore, since M^{λ} is a quotient of M^{ω_{b}}, M^{λ}(c) = (0) whenever c < b by Corollary 3.18. To proceed we need to understand the direct summands M^{λ}(c) of M^{λ}.

Given a multicomposition μ let \overline{μ} = (\overline{μ}^{(1)}, …, \overline{μ}^{(r)}) be the (unique) multipartition where \overline{μ}^{(c)} is obtained by ordering the parts of μ^{(c)}, for 1 ≤ c ≤ r.

5.4 [25, Corollary 3.5] There exists a family { Y^{λ} | λ ∈ Λ^{+} } of indecomposable ℋ –modules which are uniquely determined by the property that, for each μ ∈ Λ,

M^{μ} ≅ Y \overline{μ} ⊕ ⨁_{λ▷\overline{μ}^{λ∈Λ^{+}}} c_{λ\overline{μ}}Y^{λ}

for some non–negative integers c_{λ\overline{μ}} (which depend only on λ and \overline{μ}). Moreover, S^{λ} is a quotient of Y^{λ}.

The Y^{λ} are generalizations of the Young modules of the symmetric groups. For the case of the q–Schur algebra (that is, when r = 1) this result is proved in [13].

Set c_{λλ} = 1 and c_{λ\overline{μ}} = 0 if λ ⋭ \overline{μ}. We can now identify the summand of M^{μ} which is contained in ℋ (b).

5.5 Corollary Suppose that μ ∈ Γ. Then M^{μ}(b) ≅ ⨁_{λ∈Γ_{b}^{+}} c_{λ\overline{μ}}Y^{λ} for 0 ≤ b ≤ n.

Proof If λ ∈ Λ_{b}^{+} then S^{λ} ∈ Mod_{ℋ(b)} by Lemma 4.10; therefore, Y^{λ} also belongs to Mod_{ℋ(b)} , since Y^{λ} is indecomposable and S^{λ} is a quotient of Y^{λ}. (Observe that \overline{μ} ⊵ μ; consequently, if λ ∈ Λ^{+} and λ ⊵ \overline{μ} then λ ⊵ μ and so λ ∈ Γ^{+}.) As the decomposition of M^{μ} into a direct sum of indecomposables is unique up to isomorphism, the result follows.

Let M_{Γ} = ⨁_{μ∈Γ} M^{μ} and M_{Γ_{b}} = ⨁_{(σ,τ)∈Γ_{b}} M^{σ} ⊗ M^{τ}. Then, by definition, 𝒮_{q,Q}(Γ) = End_{ℋ} (M_{Γ}) and 𝒮_{s}(Γ) = ⨁_{b=0}^{n} End_{ℋ_{b}⊗ℋ_{n−b}} (M_{Γ_{b}}). Finally, we set M_{Γ,s} = ⨁_{b=0}^{n} Ĥ_{s,b}(M_{Γ_{b}}). These modules are the analogues of q–tensor space for the various cyclotomic q–Schur algebras; compare with [14].

5.6 Lemma Suppose that f_{s}(q, Q) is invertible. Then M_{Γ,s} and M_{Γ} have the same set of indecomposable direct summands.

Proof Applying the definitions,

M_{Γ,s} = ⨁_{b=0}^{n} Ĥ_{𝔰,b}(M_{Γ_{b}}) ≅ ⨁_{b=0}^{n} ⨁_{(σ,τ)∈Γ_{b}} Ĥ_{𝔰,b}(M^{σ} ⊗ M^{τ}).

Now, if μ = (σ, τ) ∈ Γ_{b} then Ĥ_{𝔰,b}(M^{σ} ⊗ M^{τ}) = M^{μ}(b) by Lemma 5.3; on the other hand, M^{μ}(b) ≅ ⨁_{λ∈Γ_{b}^{+}} c_{λ\overline{μ}}Y^{λ} by Corollary 5.5. Therefore,

M_{Γ,s} ≅ ⨁_{b=0}^{n} ⨁_{μ∈Γ_{b}} ⨁_{λ∈Γ_{b}^{+}} c_{λ\overline{μ}}Y^{λ}.

Consequently, { Y^{λ} | λ ∈ Γ^{+} } is a complete set of isomorphism classes of indecomposable direct summands of M_{Γ,s}. By (5.4) this is also the set of indecomposable direct summands of M_{Γ}, so this proves the Lemma.

Proof of Theorem 5.2 By Lemma 5.6, the sets of indecomposable direct summands of the ℋ –modules M_{Γ} and M_{Γ,s} coincide, except that the multiplicities of each Y^{λ} in the two modules will typically differ. Therefore, by a general argument, the ℋ –module endomorphism rings of M_{Γ} and M_{Γ,s} are Morita equivalent. On the other hand, since Ĥ_{s} is an equivalence of categories, it induces an isomorphism of endomorphism rings; so,

𝒮_{s}(Γ) = ⨁_{b=0}^{n} End_{ℋ_{b}⊗ℋ_{n−b}} (M_{Γ_{b}}) ≅ ⨁_{b=0}^{n} End_{ℋ} (Ĥ_{s,b}(M_{Γ_{b}})) ≅ End_{ℋ} (M_{Γ,s}),

where the last isomorphism is a consequence of the decomposition Mod_{ℋ} = ⨁_{b=0}^{n} Mod_{ℋ(b)} . As we have already observed that End_{ℋ} (M_{Γ,s}) is Morita equivalent to 𝒮_{q,Q}(Γ) = End_{ℋ} (M_{Γ}), the proof of Theorem 5.2 is complete.

To conclude, we note that Proposition 4.11 also generalizes to the cyclotomic q–Schur algebra case. Formally, the proof is similar to the Ariki–Koike case, so we omit the details.

Recall that the cell modules of 𝒮_{q,Q}(Γ) are called Weyl modules and are denoted by W^{λ}, for λ ∈ Γ^{+}. As with the Specht modules, there is a symmetric associative bilinear form on W^{λ}; let F^{λ} = W^{λ}/ rad W^{λ}, where rad W^{λ} is the radical of this form. Then { F^{λ} | λ ∈ Γ^{+} } is a complete set of pairwise non– isomorphic irreducible 𝒮_{q,Q}(Γ)–modules by [16, Theorem 6.16]. Furthermore, by [25, Theorem 2.3] if D^{μ} ≠ (0) then [W^{λ} : F^{μ}] = [S^{λ} : D^{μ}]; because of this we abuse notation and write d_{λμ} = [W^{λ} : F^{μ}] for all λ, μ ∈ Γ^{+}.

Again, we write Ĥ_{s} and Ĥ_{s,b} for the induced functors between the categories Mod_{𝒮_{s}(Γ)} and Mod_{𝒮_{q,Q}(Γ)} and their subcategories.

5.7 Corollary Suppose that R is a field and that f_{s}(q, Q) ≠ 0. Suppose that λ = (σ, τ) ∈ Γ_{b}^{+} and μ = (α, β) ∈ Γ_{c}^{+} where 0 ≤ b, c ≤ n. Then we have the following.

(i) W^{λ} ≅ Ĥ_{s}(W^{σ} ⊗ W^{τ}) = Ĥ_{s,b}(W^{σ} ⊗ W^{τ}). (ii) F^{μ} ≅ Ĥ_{s}(F^{α} ⊗ F^{β}) = Ĥ_{s,c}(F^{α} ⊗ F^{β}). Consequently, F^{μ} ≠ (0) if and only

if F^{α} ≠ (0) and F^{β} ≠ (0). (iii) Finally, if b = c then d_{λμ} = d_{σα}d_{τβ}; otherwise, d_{λμ} = 0.

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R.D. Mathematisches Institut B, Universität Stuttgart,

D-70550 Stuttgart. Deutschland.

        rdipper@mathematik.uni-stuttgart.de

A.M. School of Mathematics F07, University of Sydney,

Sydney N.S.W. 2006. Australia.

        mathas@maths.usyd.edu.au