TILTING MODULES FOR CYCLOTOMIC SCHUR ALGEBRAS

ANDREW MATHAS

Abstract. This paper investigates the tilting modules of the cyclotomic q–Schur algebras, the Young modules of the Ariki–Koike algebras, and the interconnections between them. The main tools used to understand the tilting modules are contragredient duality, and the Specht filtrations and dual Specht filtrations of certain permutation modules. Surprisingly, Weyl filtrations — which are in general more powerful than Specht filtrations — play only a secondary role. We also develop a theory of Young modules for the Ariki–Koike algebras; as far as we know this is new even for Coxeter groups of type B.

1. Introduction

In [7] Dipper, James and the author introduced the cyclotomic q–Schur algebras as another tool for studying the representations of the Ariki–Koike algebras. These algebras have a rich and beautiful combinatorial representation theory which closely resembles that of the q–Schur algebras. In particular, the cyclotomic q–Schur algebras are quasi– hereditary and so they have a theory of tilting modules by Ringel’s theorem [20]. The purpose of this paper is to the describe these tilting modules.

A special case of the cyclotomic q–Schur algebras are the q–Schur algebras of Dipper and James [5]. The q–Schur algebra 𝒮 (d, n) can be realized as the endomorphism algebra End_{ℋ (𝔖_{n})} (V^{⊗n}), where V is the natural module for the quantum group U_{q}(𝔤𝔩_{d}) and ℋ (𝔖_{n}) is the Iwahori–Hecke algebra of the symmetric group. Donkin [9,10] showed that the tilting modules for 𝒮 (d, n) are the indecomposable direct summands of the exterior powers ∧^{λ}V = ∧^{λ_{1}}V ⊗ ⋯ ⊗ ∧^{λ_{d}}V, where λ is a partition of n.

By definition a cyclotomic q–Schur algebra is the ℋ –module endomorphism algebra of a certain module M(Λ) = ⨁_{λ∈Λ} M(λ) of the Ariki–Koike algebra ℋ . We do not know how to describe M(Λ) as a tensor product; however, in essence all of Donkin’s results generalize to the cyclotomic setting — even though the statements and proofs do not. Instead of exterior powers we consider certain hom–spaces E(α) = Hom_{ℋ} (M(Λ), N(α)), where N(α) is something like an induced “sign representation” for the Ariki–Koike algebra. In order to understand these modules, we work mainly with Specht filtrations and dual Specht filtrations of the modules M(λ) and N(α). Using these filtrations we are able to show that M(λ) and N(α) are both self–dual ℋ –modules; this implies that E(α) is self–dual. Further, we can “lift” these filtrations to show that E(α) has a Weyl filtration. Combined, these two results show that the tilting modules of the cyclotomic q–Schur algebras are the indecomposable direct summands of the E(α).

In more detail the contents of the paper are as follows. Section 2 recalls the notation and basic results from the representation theory of the cyclotomic q–Schur algebras and the Ariki–Koike algebras. Section 3 investigates the Young modules of the Ariki–Koike algebras; these are the indecomposable direct summands of the modules M(λ) mentioned above. The Young module enjoy all of the properties of the Young modules of the symmetric groups introduced by James [14]; they are also closely related to the tilting modules of the cyclotomic q–Schur algebras. The fourth section of the paper deals with a duality operation on the category of ℋ –modules. In the case of the symmetric groups this duality corresponds to tensoring with the sign representation; applying this duality to M(λ) produces the module N(λ). Section 5 studies contragredient duality for the Ariki–Koike algebra; the main result here is that M(λ) and N(λ) are both self–dual ℋ –modules. Finally, building on the previous results, section 6 classifies the tilting modules as the indecomposable direct summands of the E(α) and section 7 describes the Ringel duals of the cyclotomic q–Schur algebras.

1991 Mathematics Subject Classification. 20C08, 20C20, 20G05.

At the referees request I have included an index of the main notation.

Notational index

comp The component of k in 𝔰 ...  16
d(𝔰) Minimal coset representative  4
D(λ) Simple ℋ –module.........  4
E(α) = θ_{α}𝒮 (Λ)................... 20
Ê(α) = Hom_{ℋ} (M(Λ), N(α))..... 20
F_{ω} Schur functor.............  6
ℋ Ariki–Koike algebra...... 2
L_{k} Murphy operator..........  3
L(λ) Simple 𝒮 (Λ)–module ......  6
m_{λ} = u_{λ}^{+}x_{λ}...................... 4
m_{𝔰𝔱} Standard basis element of ℋ  4
m_{S𝔱} Basis element of M(λ) ....  5
M^{⊛} Contragredient dual of M  13
M(λ) The ℋ –module m_{λ}ℋ .....  3
M(Λ) = ⨁_{λ∈Λ} M(λ)............... 4
n_{λ} = u_{λ}^{−}y_{λ}......................10
n_{𝔰𝔱} Twisted standard basis element 10
n_{S𝔱} Basis element of N(λ)..... 11
N(λ) The ℋ –module n_{λ}ℋ ......  11
Ṡ Smallest 𝔰 with μ(𝔰) = S .. 16
S(λ) Specht module.............  4
S′(λ) Dual Specht module ......  10
𝒮 (Λ) Cyclotomic q–Schur algebra  4

𝒮′(Λ) Twisted version of 𝒮 (Λ)..  11
𝔱^{λ} Initial λ–tableau.......... 3
T^{λ} Initial semistandard λ–tableau 6
𝔱′, T′ Conjugate tableaux ......  13
T_{w} Basis element of ℋ (𝔖_{n}) ..  3
T (λ) Tilting module ............ 19
T^{s}(λ) Standard tableaux .......  3
T_{μ}^{rs} (λ) Semistandard tableaux...  4
T_{μ}^{cs}(λ) Column semistandard tableaux 16
Y (λ) Young module .............  8
Y′(λ) Twisted Young module ...  12
Δ(λ) Weyl module ..............  6
Δ′(λ) Twisted Weyl module ....  12
μ(𝔱) Tableau of type μ related to 𝔱  5
φ_{μ} Identity map on M(μ) .....  6
φ_{𝔰𝔱} Basis element of 𝒮 (Λ) ....  5
θ_{α} θ_{α}(h) = n_{α}h for h ∈ ℋ ..... 20
θ_{ST} Basis element of E(α)..... 20
θ′_{AB} Basis element of E(α)..... 24
ω  = ((0), …, (0), (1^{n}))......... 4
∗ Anti–involution on ℋ and 𝒮 5
′  Semi–linear involution on ℋ_{Z} 10
⊵  Dominance order..........  3

2. Cyclotomic Schur algebras

This section is a summary of the definitions and results that we will need from the representation theory of the cyclotomic q–Schur algebras and the Ariki–Koike algebras. The reader is referred to [7] for more details.

Fix positive integers r and n and let 𝔖_{n} be the symmetric group of degree n. Let R be a commutative ring with 1 and let q, Q_{1}, …, Q_{r} be elements of R such that q is invertible. The Ariki–Koike algebra ℋ = ℋ_{r,n} is the associative unital R–algebra with generators T_{0}, T_{1}, …, T_{n−1} and relations

(T_{0} − Q_{1}) … (T_{0} − Q_{r}) = 0,
(T_{i} − q)(T_{i} + q^{−1}) = 0,  for 1 ≤ i ≤ n − 1,
T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0},
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 second relation is often written as (T_{i} − v)(T_{i} + 1) = 0, for 1 ≤ i < n. This presentation may be turned into the one above by renormalizing T_{i} as v^{− \frac{1}{2}} T_{i} and setting q = v^{2}. To do this it is necessary that v have a square root in R; however, every field is a splitting field for ℋ (because ℋ is cellular), so we can adjoin a square root of v without changing the representation theory of ℋ . We use the presentation above because it renormalizes the natural inner product on ℋ and so makes many formulas nicer later on. We convert the formulas that we need from the literature without mention.

For i = 1, …, n − 1 let s_{i} be the transposition (i, i + 1) in 𝔖_{n}; then {s_{1}, …, s_{n−1}} generate 𝔖_{n}. If w ∈ 𝔖_{n} then w = s_{i_{1}} … s_{i_{k}} for some i_{j}; if k is minimal then we say that this expression for w is reduced and that w has length ℓ(w) = k. In this case we set T_{w} = T_{i_{1}} … T_{i_{k}}; then T_{w} is independent of the choice of reduced expression. We also let L_{k} = T_{k−1} … T_{1}T_{0}T_{1} … T_{k−1} for k = 1, 2, …, n. These elements give a basis of ℋ . 2.1 (Ariki–Koike [3, Theorem 3.10]) The Ariki–Koike algebra ℋ is free as an R–module with basis { L_{1}^{a_{1}} … L_{n}^{a_{n}}T_{w} | w ∈ 𝔖_{n} and 0 ≤ a_{i} < r for 1 ≤ i ≤ n }.

Recall that a composition of n is sequence σ = (σ_{1}, σ_{2}, …) of non–negative integers such that |σ| = ∑_{i} σ_{i} = n; σ is a partition if in addition σ_{1} ≥ σ_{2} ≥ · · ·. If σ_{i} = 0 for all i > k then we write σ = (σ_{1}, …, σ_{k}).

A multicomposition of n is an r–tuple λ = (λ^{(1)}, …, λ^{(r)}) of compositions such that |λ^{(1)}| + ⋯ + |λ^{(r)}| = n. A multicomposition λ is a multipartition if each λ^{(i)} is a partition. If λ is a multipartition of n then we write λ ⊢ n. The diagram [λ] of the multicomposition λ is the set [λ] = { (i, j, s) | 1 ≤ λ_{j}^{(s)} ≤ i and 1 ≤ s ≤ r }.

The set of multicompositions of n is partially ordered by dominance; that is, if λ and μ are two multicompositions then λ dominates μ, and we write λ ⊵ μ, if

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

for 1 ≤ s ≤ r and for all i ≥ 1. If λ ⊵ μ and λ ≠ μ then we write λ ▷ μ.

If λ is a multicomposition let 𝔖_{λ} = 𝔖_{λ^{(1)}} × ⋯ × 𝔖_{λ^{(r)}} be the corresponding Young subgroup of 𝔖_{n}. Set

x_{λ} = ∑_{w∈𝔖_{λ}} q^{ℓ(w)}T_{w} and u_{λ}^{+} = ∏_{s=2}^{r} ∏_{k=1}^{a_{s}}(L_{k} − Q_{s}),

where a_{s} = |λ^{(1)}| + ⋯ + |λ^{(s−1)}| for 2 ≤ s ≤ r. Set m_{λ} = x_{λ}u_{λ}^{+} = u_{λ}^{+}x_{λ} and define M(λ) to be the right ideal M(λ) = m_{λ}ℋ of ℋ .

If λ = (λ^{(1)}, …, λ^{(r)}) is a multipartition then a standard λ–tableau is an r–tuple 𝔱 = (𝔱^{(1)}, …, 𝔱^{(r)}) of standard tableau which, collectively, contain the integers 1, 2, …, n and such that 𝔱^{(c)} is a standard λ^{(c)}–tableau, for 1 ≤ c ≤ r. Let T^{s}(λ) be the set of standard λ–tableau.

Let 𝔱^{λ} be the standard λ–tableau with the numbers 1, 2, …, n entered in order from left to right along its rows. If 𝔱 is any standard λ–tableau let d(𝔱) ∈ 𝔖_{n} be the unique permutation such that 𝔱 = 𝔱^{λ}d(𝔱). Finally, let ∗: ℋ ⟶ℋ be the anti–isomorphism given by T_{i}^{∗} = T_{i} , for i = 0, 1, …, n − 1, and set m_{𝔰𝔱} = T_{d(𝔰)}^{∗}m_{λ}T_{d(𝔱)}.

2.2 (Dipper–James–Mathas [7, Theorem 3.26]) The Ariki–Koike algebra ℋ is free as an R–module with (cellular) basis { m_{𝔰𝔱} | 𝔰, 𝔱 ∈ T^{s}(λ) for someλ ⊢ n }.

Here, and below, whenever we write expressions involving a pair of tableaux (such as m_{𝔰𝔱} or φ_{ST}), we implicitly assume that the two tableaux are of the same shape.

The basis {m_{𝔰𝔱}} is the standard basis of ℋ . For each multipartition λ let ℋ (λ) be the R–submodule of ℋ with basis { m_{𝔲𝔳} | 𝔲, 𝔳 ∈ T^{s}(μ) for some μ ▷ λ }; then ℋ (λ) is a two–sided ideal of ℋ .

Let S(λ) be the Specht module (or cell module) corresponding to the multipartition λ; that is, S(λ) ≅ (m_{λ} + ℋ (λ))ℋ , a submodule of ℋ /ℋ (λ). For each 𝔱 ∈ T^{s}(λ) let m_{𝔱} = m_{𝔱^{λ}𝔱} + ℋ (λ); then S(λ) is free as an R–module with basis { m_{𝔱} | 𝔱 ∈ T^{s}(λ) }. Further, there is an associative symmetric bilinear form on S(λ) which is determined by

⟨m_{𝔰}, m_{𝔱}⟩m_{λ} ≡ m_{𝔱^{λ}𝔰}m_{𝔱𝔱^{λ}} mod ℋ (λ)

for all s, 𝔱 ∈ T^{s}(λ). The radical rad S(λ) of this form is again an ℋ –module, so D(λ) = S(λ)/ rad S(λ) is an ℋ –module. When R is a field, D(λ) is either 0 or absolutely irreducible and all simple ℋ –modules arise uniquely in this way.

We can now give the definition of the cyclotomic q–Schur algebras. A set Λ of multicompositions of n is saturated if Λ is finite and whenever λ is a multipartition such that λ ⊵ μ for some μ ∈ Λ then λ ∈ Λ. If Λ is a saturated set of multicompositions let Λ^{+} be the set of multipartitions in Λ. 2.3 Definition Suppose that Λ is a saturated set of multicompositions of n. The cyclotomic q–Schur algebra with weight poset Λ is the endomorphism algebra

𝒮 (Λ) = End_{ℋ} (M(Λ)), where M(Λ) = ⨁_{λ∈Λ} M(λ).

As we now describe, 𝒮 (Λ) has a basis indexed by pairs of semistandard tableau.

Let λ be a multipartition and μ a multicomposition. A λ–tableau of type μ is a map T : [λ] ⟶ { (i, s) | i ≥ 1 and 1 ≤ s ≤ r } such that μ_{i}^{(s)} = # { x ∈ [λ] | T(x) = (i, s) } for all i ≥ 1 and 1 ≤ s ≤ r. We think of T as being an r–tuple T = (T^{(1)}, …, T^{(r)}), where T^{(s)} is the λ^{(s)}–tableau with T^{(s)}(i, j) = T(i, j, s) for all (i, j, s) ∈ [λ]. In this way we identify the standard tableaux above with the tableaux of type ω = ((0), …, (0), (1^{n})). If T is a tableau of type μ then we write Type(T) = μ.

Given two pairs (i, s) and (j, t) write (i, s) ⪯ (j, t) if either s < t, or s = t and i ≤ j.

2.4 Definition A tableau T is (row) semistandard if, for 1 ≤ t ≤ r, the entries in T^{(t)} are (i) weakly increasing along the rows (with respect to ⪯); (ii) strictly increasing down columns; and, (iii) (i, s) appears in T^{(t)} only if s ≥ t.

Let T_{μ}^{rs}(λ) be the set of semistandard λ–tableau of type μ and let T_{Λ}^{rs}(λ) = ⋃_{μ∈Λ} T_{μ}^{rs}(λ) and T_{μ}^{rs}(Λ^{+}) = ⋃_{λ∈Λ^{+}} T_{μ}^{rs}(λ).

Usually, we will refer to row semistandard tableaux simply as semistandard tableaux. Later we will meet column semistandard tableaux (these are the conjugates of row semistandard tableaux).

Notice that if T_{μ}^{rs}(λ) is non–empty then λ ⊵ μ. This observation will be used many times below.

Suppose that 𝔱 is a standard λ–tableau and let μ be a multicomposition. Let μ(𝔱) be the tableau obtained from 𝔱 by replacing each entry j with (i, k) if j appears in row i of 𝔱^{μ^{(k)}}. The tableau μ(𝔱) is a λ–tableau of type μ; it is not necessarily semistandard.

If S and T are semistandard λ–tableaux of type μ and ν, respectively, and if 𝔱 is a standard λ–tableau let

m_{S𝔱} = ∑_{\begin{matrix}𝔰∈T ^{s}(λ) \\ μ(𝔰)=S \end{matrix}} q^{ℓ(d(𝔰))}m_{𝔰𝔱} and m_{ST} = ∑_{\begin{matrix}𝔰,𝔱∈T ^{s}(λ) \\ μ(𝔰)=S,ν(𝔱)=T \end{matrix}} q^{ℓ(d(𝔰))+ℓ(d(𝔱))}m_{𝔰𝔱}.

Then we have the following two results.

2.5 [7, 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_{μ}^{rs}(λ), 𝔱 ∈ T^{s}(λ) for someλ ⊢ n }.

For S and T as above define φ_{ST} ∈ 𝒮 (Λ) by φ_{ST}(m_{α}h) = δ_{αν}m_{ST}h, for all h ∈ ℋ and all α ∈ Λ. (Here δ_{αν} is the Kronecker delta; so, δ_{αν} = 1 if α = ν and it is zero otherwise.) Then φ_{ST} belongs to 𝒮 (Λ); moreover, these elements give us a basis of 𝒮 (Λ).

2.6 [7, Theorem 6.6] The cyclotomic q–Schur algebra 𝒮 (Λ) is free as an R–module with cellular basis { φ_{ST} | S, T ∈ T_{Λ}^{rs}(λ) for someλ ∈ Λ^{+} } .

The basis {φ_{ST}} is called the semistandard basis of 𝒮 (Λ). Because this basis is cellular the map ∗: 𝒮 (Λ) ⟶ 𝒮 (Λ) which is determined by φ_{ST}^{∗} = φ_{TS} is an anti– isomorphism of 𝒮 (Λ). This involution is closely related to the ∗ involution on ℋ ; explicitly, if φ: M(ν)⟶M(μ) is an ℋ –module homomorphism then φ^{∗}: M(μ)⟶M(ν) is the homomorphism given by φ^{∗}(m_{μ}h) = (φ(m_{ν}))^{∗}h, for all h ∈ ℋ .

In order to understand how 𝒮 (Λ) acts on its representations we need to explain how the multiplication in 𝒮 (Λ) is determined by the multiplication in ℋ . Suppose that S, T, U and V are semistandard tableaux with μ = Type(S), α = Type(U) and ν = Type(V). Then m_{UV} = m_{α}h_{UV}^{α}, for some h_{UV}^{α} ∈ ℋ , and there exist scalars r_{XY} ∈ R such that

(2.7)  m_{ST}h_{UV}^{α} = ∑_{X∈T_{μ}^{rs}(Λ^{+}) ; Y∈T_{ν}^{rs}(Λ^{+})} r_{XY}m_{XY}

by [7, Cor 5.17]. Now, φ_{ST}φ_{UV}(m_{ν}h) = φ_{ST}(m_{UV})h = φ_{ST}(m_{α})h_{UV}^{α}h = m_{ST}h_{UV}^{α}h, for all h ∈ ℋ ; so (2.7) determines the product φ_{ST}φ_{UV} in 𝒮 (Λ). Explicitly, we have

(2.8)  φ_{ST}φ_{UV} = ∑_{X∈T_{μ}^{rs}(Λ^{+}) ; Y∈T_{ν}^{rs}(Λ^{+})} r_{XY}φ_{XY},

where the r_{XY} are given by (2.7). Note that φ_{ST}φ_{UV} = 0 if Type(T) ≠ Type(U). In addition, because {φ_{ST}} is a cellular basis, if r_{XY} ≠ 0 then Shape(X) ⊵ Shape(S), with equality only if X = S. Moreover, if X = S then r_{Y} = r_{SY} depends only on T, U and V; in particular, r_{SY} does not depend on S. These details can be found in [7]; for a complete treatment of the theory of cellular algebras see [13, 17].

For each multipartition λ ∈ Λ^{+} there is a right 𝒮 (Λ)–module Δ(λ), called a Weyl module. The Weyl module Δ(λ) is the submodule of Hom_{ℋ} (M(Λ), S(λ)) with basis the set of maps { φ_{T} | T ∈ T_{μ}^{rs}(λ), μ ∈ Λ }, where φ_{T}(m_{α}h) = δ_{αμ} ∑_{𝔱} m_{𝔱}h and the sum is over those standard λ–tableaux 𝔱 such that μ(𝔱) = T. If T is a semistandard λ–tableau and φ_{UV} is a semistandard basis element then the action of 𝒮 (Λ) on Δ(λ) is determined by

(2.9)  φ_{T}φ_{UV} = ∑_{Y∈T_{ν}^{rs}(Λ^{+})} r_{Y}φ_{Y},

where r_{Y} = r_{SY} is determined by (2.7) and ν = Type(V). (As remarked above, r_{Y} is independent of S.)

For each multicomposition μ let φ_{μ} be the identity map on M(μ). If λ is a multipartition then it is easy to see that φ_{λ} = φ_{T^{λ}}_{T^{λ}}, where T^{λ} = λ(T^{λ}). As with the Specht modules there is an inner product on Δ(λ) which is determined by

⟨φ_{S}, φ_{T}⟩φ_{λ} ≡ φ_{T^{λ}S}φ_{TT^{λ}} mod S^{λ},

where S^{λ} is the R–submodule of 𝒮 (Λ) with basis the set of maps φ_{UV} where U and V are semistandard μ–tableaux with μ ▷ λ. The quotient module L(λ) = Δ(λ)/ rad Δ(λ) is absolutely irreducible and { L(λ) | λ ∈ Λ^{+} } is a complete set of non–isomorphic irreducible 𝒮 (Λ)–modules.

Recall that ω = (ω^{(1)}, …, ω^{(r)}) is the multipartition with ω^{(r)} = (1^{n}) and ω^{(s)} = (0) for 1 ≤ s < r. From the definitions, m_{ω} = 1 and φ_{ω} = φ_{T^{ω}T^{ω}} is the identity map on ℋ ; so, ℋ = M(ω). In particular, φ_{ω} is an idempotent in 𝒮 (Λ) and it is easy to see that ℋ ≅ φ_{ω}𝒮 (Λ)φ_{ω} whenever ω ∈ Λ.

For an algebra A let A–mod be the category of finite dimensional right A–modules. As noted in [15], standard arguments show that there is a functor

F_{ω}: 𝒮 (Λ)–mod ⟶ℋ –mod; M ⟼Mφ_{ω}

which has the following properties. 2.10 (The cyclotomic Schur functor [15]) Suppose that R is a field and that ω ∈ Λ. Let λ ∈ Λ^{+}. Then F_{ω} is an exact functor and as right ℋ –modules

(i) F_{ω}(Δ(λ)) ≅ S(λ);

(ii) F_{ω}(L(λ)) ≅ D(λ).

Furthermore, if D(μ) ≠ 0 then [Δ(λ) : L(μ)] = [S(λ) : D(μ)].

3. Cyclotomic Young modules

This section describes the indecomposable summands of M(μ). We approach this question by considering the right 𝒮 (Λ)–modules M(μ) = Hom_{ℋ} (M(Λ), M(μ)).

All of these results in this section about the modules M(μ) apply without restriction on Λ; however, whenever we apply the Schur functor we implicitly assume that ω ∈ Λ. 3.1 Proposition Suppose that μ ∈ Λ. Then the following hold.

(i) M(μ) is free as an R–module with basis

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

(ii) M(μ) ≅ φ_{μ}𝒮 (Λ) as right 𝒮 (Λ)–modules; in particular, M(μ) is projective. (iii) As ℋ –modules, M(μ) ≅ F_{ω}(M(μ)).

Proof Part (i) is just a restatement of (2.6). For (ii), note that if S ∈ T_{α}^{rs}(λ), for some λ ∈ Λ^{+} and α ∈ Λ, then φ_{μ}φ_{ST} = δ_{αμ}φ_{ST} for all T ∈ T_{Λ}^{rs}(λ). Hence, φ_{μ}𝒮 (Λ) = M(μ) by part (i). As φ_{μ} is an idempotent this also shows that M(μ) is a projective 𝒮 (Λ)– module. Finally, by part (i) again, the ℋ –module F_{ω}(M(μ)) = Hom_{ℋ} (ℋ , M(μ)) is free as an R–module with basis { φ_{S𝔱} | S ∈ T_{μ}^{rs}(λ), 𝔱 ∈ T^{s}(λ) for some λ ∈ Λ^{+} }. Hence, by (2.5), F_{ω}(M(μ)) ≅ M(μ), where the isomorphism is given by the R–linear map determined by φ_{S𝔱} ⟼m_{S𝔱} = φ_{S𝔱}(m_{μ}) for all S ∈ T_{μ}^{rs}(λ) and 𝔱 ∈ T^{s}(λ).

An 𝒮 (Λ)–module X has a Weyl filtration if it has an 𝒮 (Λ)–module filtration

X = X_{1} ⊃ ⋯ ⊃ X_{k} ⊃ X_{k+1} = 0

such that X_{i}/X_{i+1} ≅ Δ(λ_{i}) for some multipartition λ_{i} ∈ Λ^{+}, for 1 ≤ i ≤ k. Since each Weyl module Δ(λ) has simple head L(λ) and [rad Δ(λ) : L(μ)] ≠ 0 only if λ ⊵ μ the equivalence classes of the Weyl modules are a basis of the Grothendieck group of 𝒮 (Λ); consequently, when R is a field the filtration multiplicities

[X : Δ(λ)] = # { 1 ≤ i ≤ k | X_{i}/X_{i+1} ≅ Δ(λ) }

are independent of the choice of filtration. Finally, note that if X has a Weyl filtration as above then F_{ω}(X) = F_{ω}(X_{1}) ⊃ ⋯ ⊃ F_{ω}(X_{k}) ⊃ F_{ω}(X_{k+1}) = 0 is a Specht filtration of F_{ω}(X) by (2.10); that is, F_{ω}(X_{i})/F_{ω}(X_{i+1}) ≅ F_{ω}(X_{i}/X_{i+1}) ≅ S(λ_{i}), for 1 ≤ i ≤ k. 3.2 Lemma Suppose that μ ∈ Λ. Then M(μ) has a Weyl filtration

M(μ) = M_{1} ⊃ M_{2} ⊃ ⋯ ⊃ M_{k} ⊃ M_{k+1} = 0

and there exist multipartitions λ_{1}, …, λ_{k} such that M_{i}/M_{i+1} ≅ Δ(λ_{i}), for i = 1, …, k. Moreover, if λ_{i} ▷ λ_{j} then i > j and # { 1 ≤ i ≤ k | λ_{i} = λ } = #T_{μ}^{rs}(λ) for each multipartition λ. Hence, [M(μ):Δ(λ)] = #T_{μ}^{rs}(λ) when R is a field.

Proof By Proposition 3.1, { φ_{ST} | S ∈ T_{μ}^{rs}(λ), T ∈ T_{ν}^{rs}(λ) for some ν ∈ Λ and λ ∈ Λ^{+} } is a basis of M(μ). Let {S_{1}, …, S_{k}} be the set of semistandard tableaux of type μ ordered so that i > j whenever Shape(S_{i}) ▷ Shape(S_{j}). Let λ_{i} = Shape(S_{i}), for 1 ≤ i ≤ k. Notice that λ_{i} ⊵ μ, for all i, since T_{μ}^{rs}(λ_{i}) ≠ ∅.

Fix an integer i, with 1 ≤ i ≤ k, and let M_{i} be the R–submodule of M(μ) with basis

{ φ_{S_{j}T} | λ_{j} ⊵ λ_{i} and T ∈ T_{Λ}^{rs}(λ_{j}) } .

Then M_{i} is an 𝒮 (Λ)–module by the remarks after (2.8). Further, there is an isomorphism of 𝒮 (Λ)–modules Δ(λ_{i}) ≅ M_{i}/M_{i+1} given by φ_{T} ⟼φ_{S_{i}T} + M_{i+1}, for T ∈ T_{Λ}^{rs}(λ_{i}), because S(λ_{i}) ∩ M_{i} ⊆ M_{i+1}.

The simple 𝒮 (Λ)–modules L(λ) are indexed by the multipartitions λ ∈ Λ^{+}. Fix a set { P (λ) | λ ∈ Λ^{+} } of principal indecomposable 𝒮 (Λ)–modules where P (λ) is the projective cover of L(λ). 3.3 Proposition Suppose that R is a field and let μ be a multipartition of n. Then

M(μ) ≅ P (μ) ⊕ ⨁_{λ▷μ} c_{λμ}P (λ)

for some non–negative integers c_{λμ}. Proof By Proposition 3.1, M(μ) is a projective 𝒮 (Λ)–module; therefore, there exist non– negative integers c_{λμ} such that M(μ) ≅ ⨁_{λ} c_{λμ}P (λ). Now, by [13, Theorem 3.7] (or, more explicitly, [17, Lemma 2.19]), each P (λ) has a Weyl filtration in which Δ(λ) appears with multiplicity 1. On the other hand, Lemma 3.2 M(μ) has a Weyl filtration in which the Weyl module Δ(λ) is a subquotient only if T_{μ}^{rs}(λ) is non–empty; that is, if λ ⊵ μ. Hence, c_{λμ} ≠ 0 only if λ ⊵ μ.

It remains to show that c_{μμ} = 1. First, observe that by Lemma 3.2 Δ(μ) is a top quotient of M(μ); consequently, L(μ) is also top quotient of M(μ). On the other hand, L(μ) is a top quotient of P (λ) if and only if λ = μ: hence, P (μ) is a direct summand of M(μ) and c_{μμ} ≥ 1. Therefore, by Lemma 3.2,

1 = [M(μ) : Δ(μ)] = ∑_{λ} c_{λμ}[P (λ) : Δ(μ)] ≥ c_{μμ} ≥ 1.

We must have equality throughout; so c_{μμ} = 1 and the Proposition follows.

Suppose that S ∈ T_{μ}^{rs}(λ) and T ∈ T_{ν}^{rs}(λ). Then, since M(μ) = Hom_{ℋ} (M(Λ), M(μ)), we can define an 𝒮 (Λ)–module homomorphism Φ_{ST} : M(ν) ⟶ M(μ) by Φ_{ST}(f) = φ_{ST}f for all f ∈ M(ν). In fact, as S and T run over T_{μ}^{rs}(λ) and T_{ν}^{rs}(λ), respectively, these maps give a basis of Hom_{ℋ} (M(ν), M(μ)). 3.4 Lemma Suppose that μ ∈ Λ. Then Hom_{𝒮 (Λ)} (M(ν), M(μ)) is free as an R–module with basis { Φ_{ST} | S ∈ T_{μ}^{rs}(λ), T ∈ T_{ν}^{rs}(λ) for someλ ⊢ n } . Proof By definition each of the maps Φ_{ST} belongs to Hom_{𝒮 (Λ)} (M(ν), M(μ)) and they are certainly linearly independent. It remains to check that these homomorphisms span Hom_{𝒮 (Λ)} (M(ν), M(μ)). Now, if f ∈ Hom_{𝒮 (Λ)} (M(ν), M(μ)) then there exist a_{ST} ∈ R such that f(φ_{ν}) = ∑_{ST} a_{ST}φ_{ST} by Proposition 3.1. Hence, f = ∑ a_{ST}Φ_{ST} and the Lemma is proved.

For each multipartition λ let Y (λ) = F_{ω}(P (λ)). If μ ∈ Λ is a multipartition then M(μ) ≅ F_{ω}(M(μ)) by Proposition 3.1; therefore, by Proposition 3.3,

(3.5)  M(μ) ≅ Y (μ) ⊕ ⨁_{λ▷μ} c_{λμ}Y (λ).

As remarked above, P (λ) has a Weyl filtration. Therefore, Y (λ) has a Specht filtration; in particular, Y (λ) ≠ 0. Following James [14], we call Y (λ) a Young module of ℋ . 3.6 Theorem Suppose that R is a field and let μ be a multipartition of n. Then the following hold.

(i) Each Y (μ) is an indecomposable ℋ –module; (ii) If λ is another multipartition of n then Y (λ) ≅ Y (μ) if and only if λ = μ; and, (iii) The Young module Y (μ) has a Specht filtration

Y (μ) = Y_{1} ⊃ ⋯ ⊃ Y_{k} ⊃ Y_{k+1} = 0

with Y_{i}/Y_{i+1} ≅ S(λ_{i}), for some multipartitions λ_{1}, …, λ_{k}. (iv) The number of λ_{i} equal to λ is the decomposition multiplicity [Δ(λ) : L(μ)].

Proof First note that (2.6) and Lemma 3.4 show that

Hom_{𝒮 (Λ)} (M(μ), M(λ)) ≅ Hom_{ℋ} (M(λ), M(μ))

as R–modules; explicitly, the isomorphism is given by Φ_{ST} ⟼ F_{ω}(Φ_{ST}) = φ_{ST}. Therefore, F_{ω} induces an injective map Hom_{𝒮 (Λ)} (P (μ), P (λ)) ↪ Hom_{ℋ} (Y (λ), Y (μ)). Now Y (λ) is a direct summand of M(μ) so any map from Y (λ) to Y (μ) can be extended to a map in Hom_{ℋ} (M(λ), M(μ)); hence, Hom_{𝒮 (Λ)} (P (μ), P (λ)) and Hom_{ℋ} (Y (λ), Y (μ)) are isomorphic R–modules.

In the special case where λ = ν the last paragraph says that End_{𝒮 (Λ)} (P (μ)) and End_{ℋ} (Y (μ)) are isomorphic rings. This proves (i) as End_{𝒮 (Λ)} (P (μ)) is a local ring because P (μ) is indecomposable. Similarly, part (ii) follows because if Y (μ) ≅ Y (λ) then Hom_{ℋ} (Y (μ), Y (λ)) contains an isomorphism and this lifts to give an isomorphism P (μ) ≅ P (λ), so λ = μ.

We now prove (iii). Recall from the proof of Proposition 3.3 that P (μ) has a Weyl filtration P (μ) = P_{1} ⊃ ⋯ ⊃ P_{k} ⊃ P_{k+1} = 0. Moreover, for each multipartition λ,

# { 1 ≤ i ≤ k | P_{i}/P_{i+1} ≅ Δ(λ) } = [P (μ) : Δ(λ)] = [Δ(λ) : L(μ)],

where the last equality follows from [13, Lemma 2.19] (the cellular algebra analogue of the Brauer–Nesbitt cde–triangle). Setting Y_{i} = F_{ω}(P_{i}), and using (2.10), gives a filtration of Y (μ) with the required properties. Notice that F_{ω}(Δ(λ)) ≅ S(λ) for all λ; therefore, even though F_{ω}(L(λ)) = 0 when D(λ) = 0 the multiplicities in the Specht filtration of Y (μ) are preserved.

In part (iii) we can do slightly better because the arguments of [13, 17] show that P (μ) can be filtered so that each of the quotients is isomorphic to a direct sum of [Δ(λ) : L(μ)] copies of the Weyl module Δ(λ).

We will abuse notation and write [Y (μ) : S(λ)] = [Δ(λ) : L(μ)]. As Y (μ) has a Specht module filtration it is tempting to define [Y (μ) : S(λ)] to be the multiplicity of S(λ) in a Specht filtration of Y (μ); unfortunately this is not well defined because two Specht modules indexed by different multipartitions can be isomorphic when ℋ is not semisimple.

Note that we cannot just define [Y (μ) : S(λ)] to be equal to the number of subquotients which are isomorphic to S(λ) in a Specht filtration of Y (μ) because it can happen that S(λ) ≅ S(ν) even though λ ≠ ν. This is why we have to introduce a modular system.

If μ = (μ^{(1)}, …, μ^{(r)}) is a multicomposition of n let μ⃗ = (μ^{(1)}, …, μ^{(r)}) be the unique multipartition of n such that μ^{(i)} is the partition obtained from μ^{(i)} by reordering its parts. The following result is needed in [8]. 3.7 Corollary Suppose that R is a field and let μ be a multicomposition of n. Then

M(μ) ≅ Y (μ⃗) ⊕ ⨁_{λ▷μ⃗} c_{λμ⃗}Y (λ)

where the integers c_{λμ⃗} are as in Proposition 3.3. Proof If μ is a multipartition then this is just a restatement of (3.5), so suppose that μ is not a multipartition. Then 𝔖_{μ} and 𝔖μ⃗ are conjugate subgroups of 𝔖_{n}; therefore we can find a permutation d ∈ 𝔖_{n} such that 𝔖_{μ} = d^{−1}𝔖μ⃗d and 𝔱^{μ}d^{−1} and 𝔱μ⃗d are both row standard (see, for example, [17, Lemma 3.10]). For this d we have T_{d}m_{μ} = mμ⃗T_{d} (by [7, 2.1(iv)]); consequently, M(μ) ≅ T_{d}^{−1}M(μ⃗) ≅ M(μ⃗) as right ℋ –modules. The general case now follows from (3.5).

4. Twisted cyclotomic Schur algebras

The Ariki–Koike algebra ℋ = ℋ_{r,n} has (at most) 2r one dimensional characters; namely, the R–linear maps χ_{t,α}: ℋ_{r,n} ⟶ R, for 1 ≤ t ≤ r and α ∈ {q, −q^{−1}}, which are determined by χ_{t,α}(T_{0}) = Q_{t} and χ_{t,α}(T_{i}) = α, for 1 ≤ i < n. The character χ_{t,α} is afforded by the Specht module S(λ_{t,α}) where

λ_{t,α}^{(s)} = \{ \begin{matrix}(n), & if s = t and α = q, \\ (1^{n}), & if s = t and α = −q^{−1}, \\ (0), & otherwise. \end{matrix}

Clearly, S(λ_{s,α}) ≅ S(λ_{t,β}) if and only if (Q_{s}, α) = (Q_{t}, β). When ℋ is semisimple all of these representations are pairwise non–isomorphic.

Given any ℋ –module M we can use the character χ_{t,α} to twist the ℋ –action to give a new ℋ –module M_{t,α} on which h ∈ ℋ acts as χ_{t,α}(h)h. By considering characters, in the semisimple case the effect of this operation on the Specht modules amounts to a cyclic permutation of the components of the corresponding multipartitions, and taking conjugates when α = −q^{−1}. In contrast, when ℋ is not semisimple the twisted Specht module S(λ_{t,α}) is not necessarily isomorphic to another Specht module.

This section investigates what happens when we twist modules by the ‘sign representation’ χ_{r,−q^{−1}} of ℋ . These twisted modules play a key role in understanding the tilting modules of the cyclotomic Schur algebras.

Let Z = ℤ[q̇, q̇^{−1}, Q̇_{1}, …, Q̇_{r}], where q̇, Q̇_{1}, …, Q̇_{r} are indeterminates over ℤ, and let ℋ_{Z} be the Ariki–Koike algebra over Z with parameters q̇, Q̇_{1}, …, Q̇_{r}. The relations of ℋ imply that ℋ has a ℤ–algebra involution′ which is determined by

T′_{i} = T_{i}, q̇′ = −q̇^{−1}, and Q̇′_{s} = Q̇_{r−s+1},

for 0 ≤ i < n and 1 ≤ s ≤ r. Then L′_{k} = L_{k} and T′_{w} = T_{w}, for all 1 ≤ k ≤ n and w ∈ 𝔖_{n}. We emphasize that the involution ′ is only defined generically (i.e. over Z) and that ℋ does not have a corresponding involution when R is not a free Z–module under specialization. Nevertheless, specialization arguments will allow us to transport the effects of′ into ℋ_{R}.

Suppose that λ is a multicomposition and define

y_{λ} = ∑_{w∈𝔖_{λ}}(−q̇)^{−ℓ(w)}T_{w} and u_{λ}^{−} = ∏_{s=1}^{r−1} ∏_{k=1}^{b_{s}}(L_{k} − Q̇_{s}),

where b_{s} = |λ^{(s+1)}| + ⋯ + |λ^{(r)}| for 2 ≤ k ≤ r. Then y_{λ} = (x_{λ})′ and u_{λ}^{−} = (u_{λ}^{+})′; in particular, it follows that y_{λ}u_{λ}^{−} = u_{λ}^{−}y_{λ}. Set n_{λ} = y_{λ}u_{λ}^{−} and, if 𝔰 and 𝔱 are standard λ–tableaux, define n_{𝔰𝔱} = T_{d(𝔰)}^{∗}n_{λ}T_{d(𝔱)}; then n_{𝔰𝔱} = (m_{𝔰𝔱})′. Therefore, because ′ is a ℤ–algebra involution, {n_{𝔰𝔱}} is a cellular basis of ℋ_{Z} by (2.2).

Returning to the general case, any ring R with a choice of parameters q, Q_{1}, …, Q_{r} is naturally a Z–module under specialization: that is, q̇ acts on R as multiplication by q, and Q̇_{s} acts as multiplication by Q_{s}, for 1 ≤ s ≤ r. Moreover, because ℋ is R–free this induces a isomorphism of R–algebras ℋ_{R} ≅ ℋ_{Z} ⊗_{Z} R via T_{i} ⟼T_{i} ⊗1_{R}, for 0 ≤ i < n. We say that ℋ_{R} is a specialization of ℋ_{Z}.

Hereafter, we drop the distinction between q and q̇, and Q̇_{s} and Q_{s}, and we identify the algebras ℋ = ℋ_{R} and ℋ_{Z} ⊗_{Z} R via the isomorphism T_{i} ⟼ T_{i} ⊗ 1_{R} above. Thus, we have elements y_{λ}, u_{λ}^{−} and n_{𝔰𝔱} in ℋ and by (2.2), and the specialization argument above, we have the following.

4.1 (Du–Rui [11, 2.7]) The Ariki–Koike algebra ℋ is free as an R–module with cellular basis { n_{𝔰𝔱} | 𝔰, 𝔱 ∈ T^{s}(λ) for someλ ⊢ n }.

Since {n_{𝔰𝔱}} is a cellular basis it gives us a second collection of cell modules for ℋ ; namely, for each multipartition λ define the dual Specht module S′(λ) to be the right ℋ –module (n_{λ} + ℋ ′(λ))ℋ , where ℋ ′(λ) = (ℋ (λ))′ is the two–sided ideal of ℋ with basis n_{𝔲𝔳} with Shape(𝔲) = Shape(𝔳) ▷ λ. Then S′(λ) is R–free with basis { n_{𝔱} | 𝔱 ∈ T^{s}(λ) }, where n_{𝔱} = n_{𝔱^{λ}𝔱}+ℋ ′(λ). This terminology is justified in Corollary 5.7 below which shows that S′(λ) is isomorphic to the contragredient dual of S(λ′), where λ′ is the multipartition conjugate to λ. We remark when ℋ is semisimple a straightforward calculation using characters shows that S′(λ) ≅ S(λ′)_{r,−q^{−1}}.

Let D′(λ) = S′(λ)/ rad S′(λ), where rad S′(λ) is the radical of the bilinear form on S′(λ); the form is defined in terms of the structure constants of the cellular basis {n_{𝔰𝔱}}. Once again, the theory of cellular algebras says that the non–zero D′(λ) are a complete set of pairwise non–isomorphic irreducible ℋ –modules.

For any multicomposition μ let N(μ) = n_{μ}ℋ . If S is a semistandard λ–tableau of type μ and 𝔱 is a standard λ–tableau define

n_{S𝔱} = ∑_{\begin{matrix}𝔰∈T ^{s}(λ) \\ μ(𝔰)=S \end{matrix}}(−q)^{−ℓ(d(𝔰))}n_{𝔰𝔱}.

From the definitions, n_{S𝔱} = m′_{S𝔱} in ℋ_{Z}; therefore, (2.5) and the usual specialization argument show that the following holds. 4.2 Corollary Suppose that μ is a multicomposition of n. Then N(μ) is free as an R– module with basis { n_{S𝔱} | S ∈ T_{μ}^{rs}(λ), 𝔱 ∈ T^{s}(λ) for someλ ⊢ n }.

Just as in [7, Cor. 4.15], this implies that N(μ) has a dual Specht filtration in which the number of subquotients equal to S′(λ) is #T_{μ}^{rs}(λ). This filtration can also be obtained by specializing the corresponding Specht filtration of the ℋ_{Z}–module M(μ).

Mirroring Definition 2.3, if Λ is a saturated set of multicompositions Λ define the twisted cyclotomic q–Schur algebra to be the endomorphism algebra

𝒮′(Λ) = End_{ℋ} (N(Λ)), where N(Λ) = ⨁_{μ∈Λ} N(μ).

If S ∈ T_{μ}^{rs}(λ) and T ∈ T_{ν}^{rs}(λ) are semistandard tableaux let

n_{ST} = ∑_{\begin{matrix}𝔰,𝔱∈T ^{s}(λ) \\ μ(𝔰)=S, ν(𝔱)=T \end{matrix}}(−q)^{−ℓ(d(𝔰))−ℓ(d(𝔱))}n_{𝔰𝔱}.

Now define the homomorphism φ′_{ST} ∈ 𝒮′(Λ) by φ′_{ST}(n_{α}h) = δ_{αν}n_{ST}h, for all h ∈ ℋ and all α ∈ Λ. Then φ′_{ST} belongs to 𝒮′(Λ).

Write 𝒮′(Λ)_{Z} for the twisted cyclotomic Schur algebra over Z. Similarly, we write M(μ)_{Z}, N(μ)_{Z}, … whenever we have a free R–module whose rank is independent of R, q and Q_{1}, …, Q_{r}. 4.3 Proposition Suppose that Λ is a saturated set of multicompositions.

(i) The twisted cyclotomic q–Schur algebra 𝒮′(Λ) is free as an R–module with cellular basis

{ φ′_{ST} | S ∈ T_{μ}^{rs}(λ), T ∈ T_{ν}^{rs}(λ) for someμ, ν ∈ Λ and some λ ∈ Λ^{+} } .

Consequently, 𝒮′(Λ) ≅ 𝒮′(Λ)_{Z} ⊗_{Z} R. (ii) The twisted cyclotomic Schur algebra 𝒮′(Λ) is quasi–hereditary. (iii) The R–algebras 𝒮′(Λ) and 𝒮 (Λ) are canonically isomorphic.

Proof Using Corollary 4.2, an easy modification of the argument of [7, Prop. 6.3] shows that { n_{ST} | S ∈ T_{μ}^{rs}(λ), T ∈ T_{ν}^{rs}(λ) for some λ ⊢ n } is a basis of N(ν)^{∗} ∩ N(μ). Part (i) now follows exactly as in the proof (2.6); see [7, Theorem 6.6]. In particular, notice that because 𝒮′(Λ) ≅ 𝒮′(Λ)_{Z} ⊗_{Z} R we can now use specialization arguments.

Part (ii) follows from (i) using the argument of [7, Cor. 6.18]; alternatively, it may be deduced by the specialization of a hereditary chain of the algebra 𝒮 (Λ)_{Z}.

Finally, (iii) follows because when R = Z,

Hom_{ℋ} (M(ν), M(μ)) ≅ Hom_{ℋ} (M(ν)′, M(μ)′) = Hom_{ℋ} (N(ν), N(μ)); explicitly, the isomorphism is given by φ_{ST} ⟼ φ′_{ST}, for semistandard tableaux S and T. As 𝒮 (Λ) ≅ 𝒮 (Λ)_{Z} ⊗ R and 𝒮′(Λ) ≅ 𝒮′(Λ)_{Z} ⊗ R this implies the general case.

Let Δ′(λ) and L′(λ), respectively, be the Weyl modules and simple modules of 𝒮′(Λ); these are defined in exactly the same way as the corresponding modules for 𝒮 (Λ). As in (2.10), if ω ∈ Λ then there is a functor

𝒮′(Λ)–mod ⟶ ℋ –mod; N ⟼Nφ′_{ω},

where φ′_{ω} is the identity map on ℋ . Because φ′_{ω} = φ_{ω} we abuse notation and again denote this functor by F_{ω}. As in (2.10), we have F_{ω}(Δ′(λ)) ≅ S′(λ), F_{ω}(L′(λ)) ≅ D′(λ) and [Δ′(λ) : L′(μ)] = [S′(λ) : D′(μ)] whenever D′(μ) ≠ 0.

In latter sections we will be particularly interested in the analogues of the Young modules in this setup. For each λ ∈ Λ^{+} let P′(λ) be the projective cover of L′(λ). Suppose that μ ∈ Λ^{+} and let N (μ) = φ′_{μ}𝒮′(Λ), where φ′_{μ} is the identity map on N(μ). Then φ′_{μ} is an idempotent so N (μ) is a projective 𝒮′(Λ)–module. Therefore, there exist non–negative integers c_{λμ} ≥ 0 such that

N (μ) ≅ P′(μ) ⊕ ⨁_{λ▷μ} c_{λμ}P′(λ).

In fact, because N (μ)_{Z} = M(μ)′_{Z} it follows by a specialization argument that the integers c_{λμ} are the same as those appearing in Proposition 3.3. We call Y′(μ) = F_{ω}(P′(λ)) a

twisted Young module.

Define the filtration multiplicities [Y′(λ) : S′(μ)] exactly as in section 3.

4.4 Proposition Suppose that R is a field and let λ and μ be multipartitions of n. Then

(i) N(μ) ≅ Y′(μ) ⊕ ⨁_{λ▷μ} Y′(ν)^{c_{λ}μ} where the integers c_{λμ} are the same as those appearing in Proposition 3.3;

(ii) Y′(μ) is indecomposable; (iii) Y′(λ) ≅ Y′(μ) if and only if λ = μ; (iv) the Young module Y′(μ) has a dual Specht filtration in which the number of

subquotients equal to S′(λ) is [Δ′(λ) : L′(μ)]; and,

Proof This can be proved in exactly the same way as in Theorem 3.6; alternatively, one can use a specialization argument.

As before, we set [Y′(μ) : S′(λ)] = [Δ′(λ) : L′(μ)].

We remark that the set of Young modules { Y (λ) | λ ⊢ n } and the set of twisted Young modules { Y′(λ) | λ ⊢ n } do not usually coincide; however, we always have that

{ Y (λ) | D(λ) ≠ 0 } = { Y′(λ) | D′(λ) ≠ 0 } ,

because these modules are the indecomposable direct summands of ℋ . To see this use Corollary 3.7 to show that if λ is a multipartition and D(λ) ≠ 0 then Y (λ) is the projective cover of D(λ) and, similarly, that Y′(μ) is the projective cover of D′(μ). It follows that D(λ) ≅ D′(μ) if and only if Y (λ) ≅ Y′(μ). By the results of [1, 2, 8, 12], the correspondence between these two different labelings of the simple ℋ –modules is given by a generalization of Kleshchev’s version of the Mullineux map (that is, in terms of paths in the associated crystal graphs).

5. Contragredient duality

We now investigate contragredient duality for the category of ℋ –modules; this will give us the connection between the Specht modules with the dual Specht modules constructed in the previous section. The aim of the section is really to construct a dual Specht filtration of M(λ); in essence, this is the main tool that we need in order to understand the tilting modules of the cyclotomic Schur algebras.

Recall that ∗ is the unique anti–isomorphism of ℋ such that T_{i}^{∗} = T_{i} for 0 ≤ i < n. Given a right ℋ –module M define its contragredient dual M^{⊛} to be the dual module Hom_{R}(M, R) equipped with the right ℋ –action (φh)(m) = φ(mh^{∗}) for all φ ∈ M^{⊛}, h ∈ ℋ and m ∈ M. A module M is self–dual if M ≅ M^{⊛}. By standard arguments, M is self–dual if and only if M possesses a non–degenerate associative bilinear form (the form ⟨ , ⟩ is associative if ⟨xh, y⟩ = ⟨x, yh^{∗}⟩ for all x, y ∈ M and h ∈ ℋ ).

If M is a submodule of ℋ the reader should be careful not to confuse the dual module M^{⊛} with M^{∗} = { m^{∗} | m ∈ M }.

Constructing dual bases inside ℋ is, in general, quite hard. We are going to do it by comparing the two bases {m_{𝔰𝔱}} and {n_{𝔰𝔱}} of ℋ . First we need to introduce some notation for conjugate multipartitions and tableaux.

Recall that the conjugate of a composition σ is the partition σ′ = (σ′_{1}, σ′_{2}, …) where σ′_{i} is the number of nodes in column i of the diagram of σ. If λ = (λ^{(1)}, …, λ^{(r)}) is a multicomposition then the conjugate of λ is the multipartition λ′ = ((λ^{(r)})′, …, (λ^{(1)})′). Observe that if λ ⊵ μ then μ′ ⊵ λ′ (and conversely, if λ and μ are multipartitions).

If T = (T^{(1)}, …, T^{(r)}) is a λ–tableau of type μ then the conjugate of T is the λ′– tableau T′ = (T′^{(1)}, …, T′^{(r)}) where T′(i, j, s) = T(j, i, r − s + 1) for all (i, j, s) ∈ [λ′]; that is, T′^{(s)} is the tableau obtained by interchanging the rows and columns of T^{(r−s+1)}. Notice that Type(T′) = Type(T). Further, if 𝔱 is a tableau of type ω then 𝔱 is standard if and only if 𝔱′ is standard.

For each multipartition λ let 𝔱_{λ} = (𝔱^{λ′})′; thus, 𝔱_{λ} is the standard λ–tableau with the numbers 1, 2, …, n entered in order first down the columns of 𝔱_{λ}^{(r)} and then the columns of 𝔱_{λ}^{(r−1)} and so on. Observe that if 𝔱 is a standard λ–tableau then 𝔱^{λ} ⊵ 𝔱 ⊵ 𝔱_{λ}. We set w_{λ} = d(𝔱_{λ}). The following Lemma is well–known; it can be proved by induction on 𝔱.

5.1 Lemma Suppose that λ is a multipartition of n and that 𝔱 is a standard λ–tableau. Then d(𝔱)d(𝔱′)^{−1} = w_{λ} and ℓ(w_{λ}) = ℓ(d(𝔱)) + ℓ(d(𝔱′)).

We need to extend the dominance ordering to standard tableaux. For any standard tableau 𝔰 and positive integer k let 𝔰_{k} be the subtableau of 𝔰 which contains the integers 1, …, k. If 𝔰 and 𝔱 are standard tableaux then we say that 𝔰 dominates 𝔱, and write 𝔰 ⊵ 𝔱, if either Shape(𝔰) ▷ Shape(𝔱), or Shape(𝔰) = Shape(𝔱) and Shape(𝔰_{k}) ⊵ Shape(𝔱_{k}) for k = 1, …, n. If 𝔰 ⊵ 𝔱 and 𝔰 ≠ 𝔱 we also write 𝔰 ▷ 𝔱. Note that 𝔰 ⊵ 𝔱 if and only if 𝔱′ ⊵ 𝔰′.

We also extend the dominance order to pairs of tableaux in the usual way.

If 𝔱 is a standard tableau and k an integer with 1 ≤ k ≤ n then the residue of k in 𝔱 is defined to be res_{𝔱}(k) = q^{2(j−i)}Q_{s} if k appears in row i and column j of component s of 𝔱. Residues are important because of the following result.

5.2 (James–Mathas [15, Prop. 3.7]) Suppose that 𝔰 and 𝔱 are standard λ–tableau and that 1 ≤ k ≤ n. Then there exist r_{𝔲𝔳} ∈ R such that

m_{𝔰𝔱}L_{k} = res_{𝔱}(k)m_{𝔰𝔱} + ∑_{(𝔲,𝔳)▷(𝔰,𝔱)} r_{𝔲𝔳}m_{𝔲𝔳}.

Let K = ℚ(q, Q_{1}, …, Q_{r}) and, following [18], define F_{𝔱} ∈ ℋ_{K} by

F_{𝔱} = ∏_{k=1}^{n} ∏_{\begin{matrix}c∈R(k) \\ c≠res_{𝔱}(k) \end{matrix}} \frac{L_{k} − c}{res_{𝔱}(k) − c},

where R(k) = { q^{2d}Q_{s} | 1 ≤ s ≤ r and |d| < k and d ≠ 0 if r = 1 and k = 2, 3 }. Finally, given two standard λ–tableau 𝔰 and 𝔱 set f_{𝔰𝔱} = F_{𝔰}m_{𝔰𝔱}F_{𝔱} and g_{𝔰𝔱} = F_{𝔰′}n_{𝔰𝔱}F_{𝔱′}. From the definitions, ( res_{𝔱}(k))′ = res_{𝔱′}(k) in Z, for all tableau 𝔱 and all k. This implies that F′_{𝔱} = F_{𝔱′} and hence that g_{𝔰𝔱} = f′_{𝔰𝔱} in ℋ_{K}; see [18].

Using (5.2) we obtain the following.

5.3 (Mathas [18]) Suppose that ℋ = ℋ_{K}.

(i) m_{𝔰𝔱} = f_{𝔰𝔱} + ∑_{𝔞,𝔟} r_{𝔞𝔟}f_{𝔞𝔟} for some r_{𝔞𝔟} ∈ R with r_{𝔞𝔟} ≠ 0 only if (𝔞, 𝔟) ▷ (𝔰, 𝔱). (ii) n_{𝔰𝔱} = g_{𝔰𝔱} + ∑_{𝔞,𝔟} r_{𝔞𝔟}g_{𝔞𝔟} for some r_{𝔞𝔟} ∈ R with r_{𝔞𝔟} ≠ 0 only if (𝔞, 𝔟) ▷ (𝔰, 𝔱). (iii) Suppose that 𝔰, 𝔱, 𝔲 and 𝔳 are standard tableaux. Then f_{𝔰𝔱}g_{𝔲𝔳} = 0 unless 𝔱 = 𝔲′.

By (i) and (ii), the sets {f_{𝔰𝔱}} and {g_{𝔰𝔱}} are both bases of ℋ_{K}. In fact, by [18], both bases are self–orthogonal with respect to the bilinear form ⟨ , ⟩ which we introduce below.

5.4 Lemma Suppose that 𝔰 and 𝔱 are standard λ–tableaux and that 𝔳 and 𝔲 are standard μ–tableaux such that m_{𝔰𝔱}n_{𝔳𝔲} ≠ 0. Then 𝔳′ ⊵ 𝔱.

Proof Now, m_{𝔰𝔱}n_{𝔳𝔲} ≠ 0 in ℋ only if m_{𝔰𝔱}n_{𝔳𝔲} ≠ 0 in ℋ_{Z} since ℋ ≅ ℋ_{Z} ⊗_{ℤ} R and specialization maps the standard basis of ℋ_{Z} to the standard basis of ℋ and, similarly, for the n_{𝔳𝔲} basis elements. Hence, by embedding ℋ_{Z} into ℋ_{K} in the natural way, we may assume that ℋ = ℋ_{K}. By parts (i) and (ii) of (5.3) there exists scalars r_{𝔞𝔟}, r_{𝔠𝔡} ∈ R such that

0 ≠ m_{𝔰𝔱}n_{𝔳𝔲} = (f_{𝔰𝔱} + ∑_{(𝔞,𝔟)▷(𝔰,𝔱)} r_{𝔞𝔟}f_{𝔞𝔟})(g_{𝔳𝔲} + ∑_{(𝔠,𝔡)▷(𝔳,𝔲)} r_{𝔠𝔟}f_{𝔠𝔡}).

Therefore, there exist (𝔞, 𝔟) ⊵ (𝔰, 𝔱) and (𝔠, 𝔡) ⊵ (𝔳, 𝔲) such that f_{𝔞𝔟}g_{𝔠𝔡} ≠ 0; so, 𝔠′ = 𝔟 by (5.3)(iii). Consequently, 𝔳′ ⊵ 𝔠′ = 𝔟 ⊵ 𝔱 as required.

It is possible to give a direct proof of Lemma 5.4 without using the two orthogonal bases {f_{𝔰𝔱}} and {g_{𝔰𝔱}} of ℋ (cf. [19, Lemma 4.11]); however, the proof above is simpler because it avoids long calculations with the relations in ℋ . (The proof of (5.3) is straightforward and does not use such calculations.)

Recall from (2.1) that { L_{1}^{a_{1}} … L_{n}^{a_{n}}T_{w} | 0 ≤ a_{i} < r and w ∈ 𝔖_{n} } is a basis of ℋ . Define τ : ℋ ⟶R to be the R–linear map determined by

τ(L_{1}^{a_{1}} … L_{n}^{a_{n}}T_{w}) = \{ \begin{matrix}1, & if a_{1} = ⋯ = a_{n} = 0 and w = 1, \\ 0, & otherwise. \end{matrix}

This map was introduced by Bremke and Malle [4] who showed that τ is a trace form; that is, τ(ab) = τ(ba) for all a, b ∈ ℋ . (The definition above is slightly different from Bremke and Malle’s; it is shown in [16] that the two definitions coincide.) Combining the definition with the fact that τ is a trace form shows that τ(h^{∗}) = τ(h), for all h ∈ ℋ .

Define a bilinear form ⟨ , ⟩: ℋ × ℋ ⟶ ℋ on ℋ by ⟨h_{1}, h_{2}⟩ = τ(h_{1}h_{2}^{∗}) for all h_{1}, h_{2} ∈ ℋ . Then ⟨ , ⟩ is an associative bilinear form on ℋ ; further, ⟨ , ⟩ is symmetric because τ is a trace form.

For each multipartition λ set

Q_{λ} = (−1)^{n(r−1)} ∏_{s=1}^{r} Q_{s}^{n−|λ^{(s)}|}.

Then Q_{λ} ∈ R and Q_{λ} is a unit if and only if Q_{s} is a unit whenever |λ^{(s)}| < n. Many of the results which follow rely upon the next result.

5.5 Theorem Suppose that (𝔰, 𝔱) is a pair of λ–tableau and that (𝔲, 𝔳) are μ–tableaux. Then

⟨m_{𝔰𝔱}, n_{𝔲𝔳}⟩ = \{ \begin{matrix}Q_{λ}, & if(𝔲′, 𝔳′) = (𝔰, 𝔱), \\ 0, & if(𝔲′, 𝔳′) ⋭ (𝔰, 𝔱). \end{matrix}

Proof Suppose first that ⟨m_{𝔰𝔱}, n_{𝔲𝔳}⟩ ≠ 0. Now ⟨m_{𝔰𝔱}, n_{𝔲𝔳}⟩ = τ(m_{𝔰𝔱}n_{𝔳𝔲}), so m_{𝔰𝔱}n_{𝔳𝔲} ≠ 0; hence, 𝔳′ ⊵ 𝔱 by Lemma 5.4. Now τ is a trace form and τ(h) = τ(h^{∗}), for all h ∈ ℋ ; so, applying these two facts, we have τ(m_{𝔰𝔱}n_{𝔳𝔲}) = τ(n_{𝔳𝔲}m_{𝔰𝔱}) = τ(m_{𝔱𝔰}n_{𝔲𝔳}); hence, m_{𝔱𝔰}n_{𝔲𝔳} ≠ 0 and 𝔲′ ⊵ 𝔰 by Lemma 5.4. Therefore, if ⟨m_{𝔰𝔱}, n_{𝔲𝔳}⟩ ≠ 0 then (𝔲′, 𝔳′) ⊵ (𝔰, 𝔱).

Now assume that (𝔲′, 𝔳′) = (𝔰, 𝔱). Then T_{w_{λ}} = T_{d(𝔱)}T_{d(𝔱′)}^{∗} = T_{d(𝔰′)}T_{d(𝔰)}^{∗} by Lemma 5.1. Therefore, once again using the fact that τ is a trace form,

⟨m_{𝔰𝔱}, n_{𝔰′𝔱′}⟩ = τ(m_{𝔰𝔱}n_{𝔱′𝔰′}) = τ(T_{d(𝔰)}^{∗}m_{λ}T_{d(𝔱)}T_{d(𝔱′)}^{∗}n_{λ′}T_{d(𝔰′)})
= τ(T_{d(𝔰′)}T_{d(𝔰)}^{∗}m_{λ}T_{w_{λ}}n_{λ′}) = τ(T_{w_{λ}}^{∗}m_{λ}T_{w_{λ}}n_{λ′}).

Finally, τ(T_{w_{λ}}^{∗}m_{λ}T_{w_{λ}}n_{λ′}) = Q_{λ} by [18, Prop. 5.12], so we’re done.

As a first consequence we obtain a new proof that ℋ is a symmetric algebra.

5.6 Corollary (Malle–Mathas [16]) Suppose that q, Q_{1}, …, Q_{r} are invertible elements of R. Then ⟨ , ⟩ is a non–degenerate associative symmetric bilinear form on ℋ . Therefore, ℋ is a symmetric algebra; in particular, it is self–dual.

At first sight, this proof of Corollary 5.6 is considerably easier than the original proof in [16]; however, all of the work is hidden in the calculation of τ(T_{w_{λ}}^{∗}m_{λ}T_{w_{λ}}n_{λ′}) from [18] and this is quite involved. The payoff for this extra effort is Theorem 5.5 which shows that the two bases {m_{𝔰𝔱}} and {n_{𝔰𝔱}} are almost orthogonal; this fact will be used many times in what follows.

Let λ be a multipartition. We next show that S′(λ) ≅ S(λ′)^{⊛}, and so justify the term dual Specht module. Recall that N(λ) = n_{λ}ℋ and that S′(λ) is a quotient of N(λ).

5.7 Corollary Suppose that q, Q_{1}, …, Q_{r} are invertible elements of R and let λ be a multipartition of n. Then S′(λ) ≅ S(λ′)^{⊛}.

Proof By definition S(λ′) is a submodule of ℋ /ℋ (λ′) and S′(λ) is a submodule of ℋ /ℋ ′(λ). By Theorem 5.5(i) the modules ℋ (λ′) and ℋ ′(λ) are orthogonal with respect to the form ⟨ , ⟩, as are M(λ′) and ℋ ′(λ), and N(λ) and ℋ (λ′). Therefore, ⟨ , ⟩ induces an associative bilinear form ⟨ , ⟩_{S(λ)} : S(λ′) × S′(λ)⟶R given by

⟨a + ℋ (λ), b + ℋ ′(λ)⟩_{S(λ)} = ⟨a, b⟩ = τ(ab^{∗}).

In particular, if 𝔰 ∈ T^{s}(λ′) and 𝔱 ∈ T^{s}(λ) then

⟨m_{𝔰}, n_{𝔱}⟩_{S(λ)} = \{ \begin{matrix}Q_{λ}, & if 𝔱′ = 𝔰, \\ 0, & unless 𝔱′ ⊵ 𝔰, \end{matrix}

by Theorem 5.5. Hence, ⟨ , ⟩_{S(λ)} is non–degenerate and S′(λ) ≅ S(λ′)^{⊛} as required.

Recall from (2.5) that M(λ) is free as an R–module with basis

{ m_{S𝔱} | S ∈ T_{μ}^{rs}(λ) and 𝔱 ∈ T^{s}(λ) for λ ⊢ n } .

It was shown in [7, Cor. 4.15] that this basis gives rise to a Specht filtration of M(μ). Similarly, the basis of Corollary 4.2 produces a dual Specht filtration of N(λ). We next produce another basis of M(μ) which exhibits a dual Specht filtration of M(μ) and, similarly, a basis of N(μ) which exhibits a Specht filtration of N(μ). As a byproduct we will also obtain a non–degenerate associative bilinear form on each of these modules and hence see that they are both self–dual.

A λ–tableau T is column semistandard if T′ is semistandard. If μ ∈ Λ and λ ∈ Λ^{+} let

T_{μ}^{cs}(λ) = { T | T′ ∈ T_{μ}^{rs}(λ′) }

be the set of column semistandard λ–tableaux of type μ. Observe that if T_{μ}^{cs}(λ) ≠ ∅ then T_{μ}^{rs}(λ′) ≠ ∅, so λ′ ⊵ μ; equivalently, μ′ ⊵ λ. We also set T_{μ}^{cs}(Λ^{+}) = ⋃_{λ∈Λ^{+}} T_{μ}^{cs}(λ).

As a final piece of notation, if 𝔳 is any tableau and 1 ≤ k ≤ n then write comp_{𝔳}(k) = s if k appears in component s of 𝔳. 5.8 Lemma Suppose that μ is a multicomposition and that m_{μ}n_{𝔲𝔳} ≠ 0 or n_{μ}m_{𝔲𝔳} ≠ 0 for some standard tableaux 𝔲 and 𝔳. Then μ(𝔲) is column semistandard. Proof As in the proof of Lemma 5.4 we may assume that ℋ = ℋ_{Z}. We consider only the case where m_{μ}n_{𝔲𝔳} ≠ 0; the other case can be proved by applying the involution′.

Before we begin the proof proper we remark that it is well–known, and easy enough to check, that if s_{i} ∈ 𝔖_{μ} then m_{μ}T_{i} = qm_{μ}; similarly, if s_{j} ∈ 𝔖_{λ} then T_{j}n_{λ} = −q^{−1}n_{λ}.

First, because 𝔲′ is standard the entries of μ(𝔲′) are weakly increasing along rows. Suppose that the entries of μ(𝔲′) are not strictly increasing down columns. Then we can find integers i < j such that i and j are in the same row of 𝔱^{μ} and the same column of 𝔲′. The entries in 𝔱^{μ} are consecutive so this means that there exists an integer i such that i and i + 1 are in the same row of 𝔱^{μ} and the same row of 𝔲. Therefore,

qm_{μ}n_{𝔲𝔳} = (m_{μ}T_{i})n_{𝔲𝔳} = m_{μ}(T_{i}n_{𝔲𝔳}) = −q^{−1}m_{μ}n_{𝔲𝔳}.

Consequently, m_{μ}n_{𝔲𝔳} = 0 since ℋ_{Z} is Z-free.

It remains to show that μ(𝔲′) satisfies condition (iii) of Definition 2.4. If μ is a multipartition then m_{μ} = m_{𝔱^{μ}𝔱^{μ}}; so m_{𝔱^{μ}𝔱^{μ}}n_{𝔲𝔳} ≠ 0 and 𝔲′ ⊵ 𝔱^{μ} by Lemma 5.4. Looking at the definitions, we see that μ(𝔲′) satisfies condition Definition 2.4(iii) because 𝔲′ ⊵ 𝔱^{μ}. Hence, 𝔲 is column semistandard as claimed.

If μ is a multicomposition (and not a multipartition) let μ⃗ = (μ^{(1)}, …, μ^{(r)}) be the multipartition obtained by ordering the parts in each component μ^{(s)} of μ. Then we can find a permutation d of minimal length in 𝔖_{|μ|} = 𝔖_{|μ^{(1)}|} × ⋯ × 𝔖_{|μ^{(r)}|} such that d𝔖_{μ} = 𝔖μ⃗d. Then T_{d}m_{μ} = mμ⃗T_{d}. Now, m_{μ}n_{𝔲𝔳} is non–zero so T_{d}m_{μ}n_{𝔲𝔳} = mμ⃗T_{d}n_{𝔲𝔳} is also non–zero. Therefore, there exists tableaux 𝔞 and 𝔟 such that n_{𝔞𝔟} is a non–zero summand of T_{d}n_{𝔲𝔳} and mμ⃗n_{𝔞𝔟} ≠ 0. By the last paragraph μ⃗(𝔞) satisfies Definition 2.4(iii). This implies that μ(𝔲) also satisfies Definition 2.4(iii) because comp_{𝔲}(k) = comp_{𝔞}(k), for 1 ≤ k ≤ n, by [7, Prop. 3.18] since d ∈ 𝔖_{|μ|}. Hence, μ(𝔲) is column semistandard.

If S is a λ–tableau of type μ let Ṡ be the unique standard tableau such that μ(Ṡ) = S and Ṡ ⊵ 𝔰 whenever 𝔰 is a standard λ–tableau with μ(𝔰) = S. The tableau Ṡ is denoted first(S) in [15]. The permutation d(Ṡ) is a distinguished (𝔖_{λ}, 𝔖_{μ})–double coset representative; that is, it is the unique element of minimal length in 𝔖_{λ}d(Ṡ)𝔖_{μ}. We emphasize that Ṡ is a standard tableau. 5.9 Proposition Suppose that μ is a multicomposition of n. Then M(μ) is free as an R– module with basis { m_{μ}n_{Ṡ𝔱} | S ∈ T_{μ}^{cs}(λ) and 𝔱 ∈ T^{s}(λ) for someλ ⊢ n } and N(μ) is free as an R–module with basis { n_{μ}m_{Ṡ𝔱} | S ∈ T_{μ}^{cs}(λ) and 𝔱 ∈ T^{s}(λ) for someλ ⊢ n }. Proof We only prove for the claim for M(μ); the second statement can be proved by a similar argument, or by specialization.

By (5.3)(iii) {n_{𝔰𝔱}} is a basis of ℋ , so M(μ) is spanned by the elements m_{μ}n_{𝔰𝔱}, where 𝔰 and 𝔱 range over all pairs of standard tableaux of the same shape. Furthermore, if m_{μ}n_{𝔰𝔱} ≠ 0 then μ(𝔰) is column semistandard by Lemma 5.8. Hence, M(μ) is spanned by the elements m_{μ}n_{𝔰𝔱} with μ(𝔰) column semistandard. Now, if d(𝔰) and d(𝔲) are in the same (𝔖_{λ}, 𝔖_{μ})–double coset then m_{μ}T_{d(𝔰)}n_{λ} = ±q^{a}m_{μ}T_{d(𝔲)}n_{λ} for some integer a; see the remarks at the start of the proof of Lemma 5.8. By definition d(Ṡ) is the unique element of minimal length in its double coset; therefore, the elements in the statement of the Lemma span M(μ). However, now we are done because M(μ) is R–free and the number of elements in our spanning set is exactly the rank of M(μ) by (2.5).

Combining Lemma 5.8 and the Proposition we have. 5.10 Corollary Suppose that μ is a multicomposition and that 𝔰 and 𝔱 are standard tableaux. Then m_{μ}n_{𝔰𝔱} ≠ 0 if and only if μ(𝔰) is column semistandard. Similarly, n_{μ}m_{𝔰𝔱} ≠ 0 if and only if μ(𝔰) is column semistandard.

Using Proposition 5.9, the argument of Lemma 3.2 produces the following result. 5.11 Corollary Suppose that μ is a multicomposition of n. Then there exist filtrations

M(μ) = M_{1} ⊃ ⋯ ⊃ M_{k} ⊃ M_{k+1} = 0 and N(μ) = N_{1} ⊃ ⋯ ⊃ N_{k} ⊃ N_{k+1} = 0

of M(μ) and N(μ), respectively, and multipartitions λ_{1}, …, λ_{k}, such that μ′ ⊵ λ_{i}, M_{i}/M_{i+1} ≅ S′(λ_{i}) and N_{i}/N_{i+1} ≅ S(λ_{i}), for 1 ≤ i ≤ k. Moreover, if λ is any multipartition of n then # { 1 ≤ i ≤ k | λ_{i} = λ } = #T_{μ}^{cs}(λ). 5.12 Remark As an R–module, N_{i} is the submodule of N(μ) with basis the set of elements n_{μ}m_{Ṡ𝔱} with Shape(S) ⊵ λ_{i}, for 1 ≤ i ≤ k. In particular, S(μ′) ≅ N_{k} is spanned by { n_{μ}m_{𝔱_{μ′} 𝔱} | 𝔱 ∈ T^{s}(μ′) }; note that μ(𝔱_{μ′}) is the unique column standard μ′–tableau of type μ. Therefore, S(μ′) ≅ n_{μ}T_{w_{μ′}}^{∗} m_{μ′}ℋ ; this is a result of Du and Rui [11]. Similarly, S′(μ′) ≅ m_{μ}T_{w_{μ′}}^{∗} n_{μ′}ℋ .

Because S(λ′) ≅ S(λ′)^{⊛}, the Specht filtrations of M(μ) given by [7, Cor. 4.15] and the last result suggest that M(λ) is self–dual. A similar remark applies to N(λ). When r = 1 it is clear that both of these modules are self–dual because they are induced representations from parabolic subalgebras.

We need a non–degenerate associative bilinear form. Let ⟨ , ⟩_{μ} be the bilinear map on M(μ) determined by

⟨m_{S𝔱}, m_{μ}n_{U̇𝔳}⟩_{μ} = ⟨m_{S𝔱}, n_{U̇𝔳}⟩,

where m_{S𝔱} and m_{μ}n_{U̇𝔳} run over the bases of (2.5) and Proposition 5.9, respectively.

If S is a semistandard tableau let S_{(i,s)} be the subtableau of S consisting of those entries (j, t) with (j, t) ⪯ (i, s) (see Definition 2.4). We extend the dominance order to the set of semistandard tableaux by defining S ⊵ T if Shape(S_{(i,s)}) ⊵ Shape(T_{(i,s)}) for all (i, s). This definition coincides with our previous definition of dominance when S is a standard tableau (recall that we are identifying standard tableaux and semistandard tableaux of type ω).

5.13 Proposition Suppose that Q_{1}, …, Q_{r} are invertible elements of R and that μ is a multicomposition. Then ⟨ , ⟩_{μ} is a non–degenerate associative bilinear form on M(λ). In particular, M(μ) is self–dual. Similarly, N(μ) is self–dual.

Proof We prove the Proposition only for M(μ); the result for N(μ) can be obtained using a similar argument or by specialization.

Suppose that S ∈ T_{μ}^{rs}(λ), 𝔱 ∈ T^{s}(λ), U ∈ T_{μ}^{cs}(ρ) and 𝔳 ∈ T^{s}(ρ) for some multipartitions λ and ρ. Applying the definitions we find that

⟨m_{S𝔱}, m_{μ}n_{U̇𝔳}⟩_{μ} = ⟨m_{S𝔱}, n_{U̇𝔳}⟩ = ∑_{\begin{matrix}𝔰∈T ^{s}(λ) \\ μ(𝔰)=S \end{matrix}}⟨m_{𝔰𝔱}, n_{U̇𝔳}⟩.

Therefore, by Theorem 5.5, ⟨m_{S𝔱}, m_{μ}n_{U̇𝔳}⟩_{μ} = 0 unless there is a standard tableau 𝔰 such that (U̇′, 𝔳′) ⊵ (𝔰, 𝔱) and μ(𝔰) = S; hence, (U′, 𝔳′) ⊵ (S, 𝔱). (Here U̇′ = (U̇)′ and not (U′)·; in general these tableaux are different.)

Next suppose that (U′, 𝔳′) = (S, 𝔱). Then U̇ = (S′)^{·} ⊵ 𝔰′ whenever 𝔰′ is a standard tableau with μ(𝔰′) = S′; therefore, 𝔰 ⊵ U̇′ whenever 𝔰 is a standard tableau such that μ(𝔰) = S. Therefore, if (U, 𝔳) = (S′, 𝔱′) then

⟨m_{S𝔱}, n_{U̇𝔳}⟩_{μ} = ∑_{\begin{matrix}𝔰∈T ^{s}(λ) \\ μ(𝔰)=S \end{matrix}}⟨m_{𝔰𝔱}, n_{U̇𝔳}⟩ = ⟨m_{U̇′𝔳′}, n_{U̇𝔳}⟩ = Q_{λ}

by Theorem 5.5.

Combining the last two paragraphs shows that the matrix (⟨m_{S𝔱}, m_{μ}n_{U̇𝔳}⟩_{μ}) is invertible and, hence, that the form ⟨ , ⟩_{μ} on M(μ) is non–degenerate.

The harder part is to prove that ⟨ , ⟩_{μ} is associative. Now, the form ⟨ , ⟩ on ℋ is associative, so if h ∈ ℋ then

⟨m_{S𝔱}h, m_{μ}n_{U̇𝔳}⟩_{μ} = ⟨m_{S𝔱}h, n_{U̇𝔳}⟩ = ⟨m_{S𝔱}, n_{𝔲𝔳}h^{∗}⟩ = ∑_{𝔞,𝔟} r_{𝔞𝔟}⟨m_{S𝔱}, n_{𝔞𝔟}⟩,

where n_{𝔞𝔟}h^{∗} = ∑_{𝔞,𝔟} r_{𝔞𝔟}n_{𝔞𝔟} for some r_{𝔞𝔟} ∈ R. Write m_{S𝔱} = m_{μ}h_{S𝔱}^{μ}, for some h_{S𝔱}^{μ} ∈ ℋ . Then

⟨m_{S𝔱}, n_{𝔞𝔟}⟩ = τ(m_{S𝔱}n_{𝔟𝔞}) = τ(n_{𝔟𝔞}m_{S𝔱}) = τ(n_{𝔟𝔞}m_{μ}h_{S𝔱}^{μ}) = τ(h_{𝔱S}^{μ}m_{μ}n_{𝔞𝔟}),

where the second equality uses the fact that τ is a trace form and the last equality follows because τ(a) = τ(a^{∗}) for all a ∈ ℋ . Therefore, if ⟨m_{S𝔱}, n_{𝔞𝔟}⟩ ≠ 0 then μ(𝔞) is column semistandard by Lemma 5.8. Let A = μ(𝔞) and recall that m_{μ}n_{𝔞𝔟} = ±q^{i}m_{μ}n_{Ȧ𝔟} for some integer i (which depends on 𝔞); write m_{μ}n_{𝔞𝔟} = λ_{𝔞}m_{μ}n_{Ȧ𝔟} and set r_{A𝔟} = ∑_{𝔞} λ_{a}r_{𝔞𝔟}, where the sum runs over those standard tableaux with μ(𝔞) = A. Then we have shown that

⟨m_{S𝔱}h, m_{μ}n_{U̇𝔳}⟩_{μ} = ∑_{\begin{matrix}𝔟∈T ^{s}(λ) \\ A∈T_{μ}^{cs}(λ) \end{matrix}} r_{A𝔟}⟨m_{S𝔱}, n_{Ȧ𝔟}⟩.

On the other hand, by Lemma 5.8 again,

⟨m_{S𝔱}, m_{μ}n_{U̇𝔳}h^{∗}⟩_{μ} = ∑_{𝔞,𝔟} r_{𝔞𝔟}⟨m_{S𝔱}, m_{μ}n_{𝔞𝔟}⟩_{μ} = ∑_{\begin{matrix}𝔞,𝔟∈T ^{s}(λ) \\ μ(𝔞)∈T_{μ}^{cs}(λ) \end{matrix}} r_{𝔞𝔟}⟨m_{S𝔱}, m_{μ}n_{𝔞𝔟}⟩_{μ}

= ∑_{\begin{matrix}𝔟∈T ^{s}(λ) \\ A∈T_{μ}^{cs}(λ) \end{matrix}} r_{A𝔟}⟨m_{S𝔱}, m_{μ}n_{Ȧ𝔟}⟩_{μ} = ∑_{\begin{matrix}𝔟∈T ^{s}(λ) \\ A∈T_{μ}^{cs}(λ) \end{matrix}} r_{A𝔟}⟨m_{S𝔱}, n_{Ȧ𝔟}⟩.

Hence, ⟨m_{S𝔱}h, m_{μ}n_{U̇𝔳}⟩_{μ} = ⟨m_{S𝔱}, m_{μ}n_{U̇𝔳}h^{∗}⟩_{μ}, so the form is associative as claimed.

5.14 Corollary Suppose that R is a field and let λ be a multipartition of n. Then Young module Y (λ) is self–dual. Similarly. twisted Young module Y′(λ) is self–dual. Proof By the Proposition, M(λ) and N(λ) are both self–dual. Hence, the result follows by induction on λ using Theorem 3.6 and Proposition 4.4, respectively.

6. The cyclotomic tilting modules

Let (A, X^{+}) be a quasi–hereditary algebra, where X^{+} is the poset of weights for A; see, for example, [10, Appendix]. For each λ ∈ X^{+} there is a standard module Δ(λ) with simple head L(λ) and a costandard module with simple socle L(λ). An A–module M has a Δ–filtration if it has a filtration in which every subquotient isomorphic to a standard module; similarly, M has a ∇–filtration if every subquotient is isomorphic to a costandard module. An A–module T is a tilting module if it has both a Δ–filtration and a ∇–filtration. 6.1 (Ringel [20]) Suppose that R is a field and that (A, X^{+}) is a quasi–hereditary algebra. Then, for each λ ∈ X^{+}, there is a unique indecomposable tilting module T (λ) such that [T (λ) : Δ(λ)] = 1 and [T (λ) : Δ(μ)] ≠ 0 only ifλ ≥ μ Moreover, if T is any tilting module then

T ≅ ⨁_{λ∈X^{+}} T (λ)^{a_{λ}}

for some non–negative integers a_{λ}.

The T (λ) are the partial tilting modules of A. A full tilting module for A is any tilting

module which contains every T (λ), for λ ∈ X^{+}, as a direct summand.

By [7, Cor. 6.18] the cyclotomic Schur algebras are quasi–hereditary algebras with weight poset Λ^{+}. The standard modules of 𝒮 (Λ) are the Weyl modules and the costandard modules are their contragredient duals. In this section we will describe the partial tilting modules of 𝒮 (Λ).

First consider the case r = 1. Suppose that d ≥ 1 and let Λ_{d,n} be the set of compositions of n into at most d parts and let V be a free R–module of rank d. Then ℋ (𝔖_{n}) acts on V^{⊗n} (by q–analogues of place permutations) and V^{⊗n} ≅ M(Λ); see [6]. The Dipper– James [5] q–Schur algebra 𝒮 (d, n) is the cyclotomic q–Schur algebra 𝒮 (Λ_{d,n}); by the above remarks 𝒮 (d, n) ≅ End_{ℋ (𝔖_{n})} (V^{⊗n}). Donkin [9] has shown that when d ≥ n the tilting modules for 𝒮 (d, n) are the indecomposable direct summands of the exterior powers ∧^{λ}V = ∧^{λ_{1}}V ⊗ ⋯ ⊗ ∧^{λ_{d}}V. In the cyclotomic case we do not have a description of M(Λ) = ⨁_{μ} M(μ) as a tensor product; nevertheless, we do have an analogue of the exterior powers. 6.2 Definition Suppose that α is a multicomposition of n.

(i) Let Ê(α) = Hom_{ℋ} (M(Λ), N(α)).

(ii) Let θ_{α} ∈ Hom_{ℋ} (ℋ , N(α)) be the map θ_{α}(h) = n_{α}h, for all h ∈ ℋ .

(iii) If ω ∈ Λ let E(α) = θ_{α}𝒮 (Λ)

By definition, Ê(α) is a right 𝒮 (Λ)–module and E(α) is a submodule of Ê(α) — when ω ∈ Λ. In fact, we will see that most of the time E(α) = Ê(α); however, for some choices of the parameters (for example, if q^{2} = −1) this need not be the case.

Recall that α⃗ is the multipartition obtained by reordering the parts of α^{(s)} for each s. By the argument of Corollary 3.7, N(α) ≅ N(α⃗); therefore, Ê(α) ≅ E(α⃗). Hence, there is no loss in assuming that α is a multipartition. The Ê(α) are very similar to the modules M(μ) = Hom_{ℋ} (M(Λ), M(μ)) = φ_{μ}𝒮 (Λ) of Proposition 3.1. The E(α) play the role of exterior powers and the M(μ) the symmetric powers.

We will show that E(α) has a Weyl filtration and that it is self–dual; hence, it also has a dual Weyl filtration. This will enable us to show that the tilting modules of 𝒮 (Λ) are the indecomposable summands of the E(λ) as λ runs over the multipartitions in Λ^{+}.

The first step is to produce a basis for E(α).

6.3 Definition Suppose that λ is a multipartition of n and that α and μ are two multicompositions of n. For tableaux S ∈ T_{α}^{cs}(λ) and T ∈ T_{μ}^{rs}(λ) let θ_{ST} be the homomorphism in E(α) determined by θ_{ST}(m_{ν}h) = δ_{νμ}n_{α}m_{ṠT}h, for all h ∈ ℋ and all ν ∈ Λ.

6.4 Proposition Suppose that ω ∈ Λ and that α is a multicomposition of n. Then E(α) is free as an R–module with basis { θ_{ST} | S ∈ T_{α}^{cs}(λ) and T ∈ T_{Λ}^{rs}(λ) for someλ ∈ Λ^{+} } .

Proof By definition, and (2.6), E(α) is spanned by the maps θ_{α}φ_{ST} where S, T ∈ T_{Λ}^{rs}(λ) for some λ ∈ Λ^{+}. Now θ_{α}φ_{ST} = 0 unless S is a semistandard tableau of type ω so E(α) is spanned by the elements θ_{α}φ_{𝔰T}, where 𝔰 ∈ T^{s}(λ) and T ∈ T_{Λ}^{rs}(λ) for some λ ∈ Λ^{+}. Furthermore, θ_{α}φ_{𝔰T}(m_{μ}h) = n_{α}m_{𝔰T}h, for all h ∈ ℋ ; so, θ_{α}φ_{𝔰T} ≠ 0 if and only if α(𝔰) is column semistandard by Corollary 5.10 and in this case θ_{α}φ_{𝔰T} = θ_{ST} where S = α(𝔰). Finally, by Proposition 5.9 the maps {θ_{ST}} are linearly independent, so the Proposition follows.

6.5 Theorem Suppose that ω ∈ Λ and that Q_{1}, …, Q_{r} are all distinct. Then there exist multipartitions λ_{1}, …, λ_{k} in Λ and an 𝒮 (Λ)–module filtration

E(α) = E_{1} ⊃ ⋯ ⊃ E_{k} ⊃ E_{k+1} = 0

such thatE_{i}/E_{i+1} ≅ Δ(λ_{i}) and α′ ⊵ λ_{i}, for i = 1, …, k. Moreover, if λ is any multipartition in Λ^{+} then # { 1 ≤ i ≤ k | λ_{i} = λ } = #T_{α}^{cs}(λ).

Proof Let S_{1}, …, S_{k} be the complete set of column semistandard tableaux of type α ordered so that i > j whenever Shape(S_{i}) ⊵ Shape(S_{j}). For each i let let λ_{i} = Shape(S_{i}); then α′ ⊵ λ_{i} since T_{α}^{cs}(λ_{i}) ≠ ∅.

For i = 1, …, k let E_{i} be the R–submodule of E(α) spanned by the elements θ_{S_{j}T} with j ≥ i and T ∈ T_{Λ}^{rs}(λ_{j}). To prove the Theorem it is enough to show that E_{i} is a submodule of E(α) and that E_{i}/E_{i+1} ≅ Δ(λ_{i}) for each i.

Suppose that i ≥ 1 and let T ∈ T_{μ}^{rs}(λ_{i}), U ∈ T_{σ}^{rs}(ρ) and V ∈ T_{ν}^{rs}(ρ), for some multicompositions μ, ν, σ ∈ Λ and ρ ∈ Λ^{+}. Consider the product θ_{S_{i}T}φ_{UV}. By definition, θ_{S_{i}T}φ_{UV} = 0 unless μ = Type(T) = Type(U) = σ; so suppose that σ = μ. In order to write θ_{S_{i}T}φ_{UV} as a linear combination of the basis elements of E(α) it suffices to consider (θ_{S_{i}T}φ_{UV})(m_{ν}). Write m_{UV} = m_{μ}h_{UV}^{μ}, for some h_{UV}^{μ} ∈ ℋ . Then we have

(θ_{S_{i}T}φ_{UV})(m_{ν}) = θ_{S_{i}T}(m_{UV}) = θ_{S_{i}T}(m_{μ})h_{UV}^{μ} = n_{α}m_{Ṡ_{i}T}h_{UV}^{μ}

≡ n_{α}( ∑_{Y∈T_{ν}^{rs}(λ_{i})} r_{S_{i}Y}m_{Ṡ_{i}Y}) mod H^{λ_{i}}

= ∑_{Y∈T_{ν}^{rs}(λ_{i})} r_{S_{i}Y}n_{α}m_{Ṡ_{i}Y},

where the second line follows from (2.9). Hence, Theorem 6.9 implies that

θ_{S_{i}T}φ_{UV} ≡ ∑_{Y∈T_{ν}^{rs}(λ_{i})} r_{S_{i}Y}θ_{S_{i}Y} mod E_{i+1}.

All of our claims now follow.

6.6 Remark By Theorem 6.5 the module Δ(λ) is a subquotient of E(α) whenever T_{α}^{cs}(λ) ≠ ∅; thus, α′ ⊵ λ. Moreover, since T_{α}^{cs}(ω) is always non–empty this means that Δ(ω) is always a subquotient of E(α); consequently, the assumption that ω ∈ Λ is necessary in Theorem 6.5. If ω ∉ Λ then Ê(α) is still an 𝒮 (Λ)–module (for all multicompositions α); however, we are not able to give a Weyl filtration of Ê(α) even with the restrictions on the parameters from Theorem 6.9. 6.7 Corollary Suppose that R is a field, ω ∈ Λ and that λ and μ are multipartitions of n. Then [E(λ):Δ(λ′)] = 1 and [E(λ):Δ(μ)] ≠ 0 only if λ′ ⊵ μ. Proof By the Theorem, [E(λ):Δ(μ)] = #T_{λ}^{cs}(μ). Therefore, if [(λ):Δ(μ)] ≠ 0 then T_{λ}^{cs}(μ) ≠ ∅ and μ′ ⊵ λ. Finally, [E(λ):Δ(λ′)] = 1 because λ(𝔱_{λ′}) = (T^{λ})′ is the unique column semistandard λ′–tableau of type λ (just as T^{λ} = λ(𝔱^{λ}) is the unique semistandard λ–tableau of type λ).

In order to show that E(α) is self–dual (and hence has a dual Weyl filtration) we have to work much harder. This is where the module E(α) enters the picture.

As is well–known, if f ∈ N(α)∩M(μ)^{∗} then left multiplication by f is an ℋ –module homomorphism from M(μ) into N(α); in fact, every element of Ê(α) arises in this way. 6.8 Lemma Suppose that α and μ are multicompositions of n. Then there is an isomorphism of R–modules Hom_{ℋ} (M(μ), N(α)) ≅ N(α) ∩ M(μ)^{∗} given by θ ⟼ θ(m_{μ}).

Proof This follows from Theorem 5.16 and Lemma 5.2 of [7]. (Theorem 5.16 says that the double annihilator, { h ∈ ℋ | hs = 0 whenever m_{μ}s = 0 }, of m_{μ} is ℋ m_{μ}; Lemma 5.2 observes that this property of the double annihilator implies the Lemma.)

Therefore, to give a basis of Ê(α) it is enough to find a basis of N(α)∩M(μ)^{∗}. To give a basis for this R–module we need to impose some constraints on the algebras parameters. 6.9 Theorem Suppose that Q_{1}, …, Q_{r} are all distinct and that q^{2} ≠ −1. Let α and μ be multicompositions of n. Then N(α) ∩ M(μ)^{∗} is free as an R–module with basis

{ n_{α}m_{ṠT} | S ∈ T_{α}^{cs}(λ), T ∈ T_{μ}^{rs}(λ) for someλ ⊢ n } .

Proof By (2.5), if S ∈ T_{α}^{cs}(λ) and T ∈ T_{μ}^{rs}(λ), for some multipartition λ, then n_{α}m_{ṠT} ∈ N(α) ∩ M(μ)^{∗} since N(α) is a right ideal and M(μ)^{∗} is a left ideal of ℋ . Moreover, as S and T run over the possible choices above, these elements remain linearly independent because the elements n_{α}m_{Ṡ𝔱} are a basis of N(α) by Proposition 5.9.

It remains to show that these elements span N(α) ∩ M(μ)^{∗}. By Proposition 5.9, if x ∈ N(α) ∩ M(μ)^{∗} then x = ∑ r_{U𝔳}n_{α}m_{U̇𝔳}, for some r_{U𝔳} ∈ R, where the sum is over all pairs (U, 𝔳) of tableaux with U column semistandard of type α and 𝔳 standard. Now, if (i, i + 1) ∈ 𝔖_{μ} then m_{μ}T_{i} = q; so, xT_{i} = qx. Using this and the argument of [17, Prop. 3.18], it is easy to see that if i and i+1 are in different columns of 𝔳 then r_{U𝔳} = r_{U𝔳′}, where 𝔳′ = 𝔳s_{i}, and

n_{α}(m_{U̇𝔳} + m_{U̇𝔳′})T_{i} = qn_{α}(m_{U̇𝔳} + m_{U̇𝔳′}).

Let x′ = ∑ r_{U𝔳}n_{α}m_{U̇𝔳} where the sum is over those pairs (U, 𝔳) for which i and i+1 are in different columns of 𝔳. By the above remarks, x′T_{i} = qx′. Therefore, if we set x′′ = x−x′ then x′′T_{i} = qx′′ as well. Suppose that x′′ ≠ 0. Then we can choose (U, 𝔳) so that n_{α}m_{U̇𝔳} appears in x′′ and n_{α}m_{Ṡ𝔱} does not appear in x′′ if (U, 𝔳) ⋭ (S, 𝔱). By the definition, i and i + 1 are in the same column of 𝔳; so, by [17, Cor. 3.21],

n_{α}m_{U̇𝔳}T_{i} = −q^{−1}n_{α}m_{U̇𝔳}T_{i} + n_{α}h,

where h is a linear combination of terms m_{𝔰𝔱} with 𝔱 ▷ 𝔳. Using the minimality of (U, 𝔳) to compare the coefficient of n_{α}m_{U̇𝔳} on both sides of the equation qx′′ = x′′T_{i}, we have a contradiction since q ≠ −q^{−1} (that is, q^{2} ≠ −1). Hence, x′′ = 0; so x = x′ and r_{U𝔳} ≠ 0 only if V = μ(𝔳) satisfies conditions (i) and (ii) of Definition 2.4.

It will be convenient to let T_{μ}^{as}(Λ^{+}) be the set of tableaux of type μ which satisfy conditions (i) and (ii) of Definition 2.4 but not condition (iii) — these are “almost” semistandard tableaux. We have now shown that if x ∈ N(α) ∩ M(μ)^{∗} then x = ∑ r_{UV}n_{α}m_{U̇V} where the sum is over pairs (U, V) with U ∈ T_{α}^{cs}(Λ^{+}) and V ∈ T_{μ}^{rs}(Λ^{+}) ∪ T_{μ}^{as}(Λ^{+}). By the first paragraph of the proof we can assume that r_{UV} = 0 unless V ∈ T_{μ}^{as}(Λ^{+}). Thus, we are reduced to showing that if we have an element x ∈ N(α) ∩ M(μ)^{∗} which can be written in the form

x = ∑_{\begin{matrix}U∈T_{α}^{cs}(Λ^{+}) \\ V∈T_{μ}^{as}(λ) \end{matrix}} r_{UV}n_{α}m_{U̇V},

for some r_{UV} ∈ R, then x = 0. By way of contradiction, suppose that x ≠ 0.

Fix (S, T) with r_{ST} ≠ 0 such that r_{UV} = 0 whenever (S, T) ▷ (U, V) for U ∈ T_{α}^{cs}(Λ^{+}) and V ∈ T_{μ}^{as}(Λ^{+}). Let T̈ be the unique standard tableau such that μ(T̈) = T and 𝔱 ⊵ T̈ whenever μ(𝔱) = T; the tableau T̈ is denoted last(T) in [15]. Define i to be the smallest positive integer such that c = compT̈(i) > comp_{𝔱^{μ}}(i); such an i exists because T does not satisfy Definition 2.4(iii). If j < i then compT̈(j) < compT̈(i) by the minimality of i and the fact that comp_{𝔱^{μ}}(j) ≤ comp_{𝔱^{μ}}(i); in particular, this implies that i must appear in the first row and first column of T̈^{(c)}. Following [7] define

y_{i} = T_{i−1} … T_{1} ∏_{s=1}^{comp_{𝔱^{μ}} (i)}(L_{1} − Q_{s}).

Then y_{i} ≠ 0 since comp_{𝔱^{μ}}(i) < c ≤ r. Moreover, m_{μ}y_{i} = 0 by [7, Lemma 5.8] (when translating into the notation of [7] note that γ_{i} = comp_{𝔱^{μ}}(i)). Let 𝔱 = T̈s_{i−1} … s_{1}; then res_{𝔱}(1) = resT̈(i) = Q_{c} and ℓ(d(𝔱)) = ℓ(d(T̈)) + i − 1; so, m_{ṠT̈}T_{i−1} … T_{1} = m_{Ṡ𝔱}. Furthermore, by the cancellation property of the Bruhat–Chevalley order, 𝔳 ⊵ T̈ if and only if 𝔳s_{i−1} … s_{1} ⊵ 𝔱 since ℓ(d(𝔱)) = ℓ(d(T̈)) + i − 1; see, for example, [17, Cor. 3.9]. Therefore, using (5.2) for the third equality, we have

xy_{i} = (r_{ST}n_{α}m_{ṠT̈} + ∑_{\begin{matrix}𝔲,𝔳 \\ T̈⋭𝔳 \end{matrix}} r_{𝔲𝔳}n_{α}m_{𝔲𝔳})y_{i}

= (r_{ST}n_{α}m_{Ṡ𝔱} + ∑_{\begin{matrix}𝔲,𝔳 \\ 𝔱⋭𝔳 \end{matrix}} r′_{𝔲𝔳}n_{α}m_{𝔲𝔳}) ∏_{s=1}^{comp_{𝔱^{μ}} (i)}(L_{1} − Q_{s})

= r_{ST} ∏_{s=1}^{comp_{𝔱^{μ}} (i)}(Q_{c} − Q_{s}) · n_{α}m_{Ṡ𝔱} + ∑_{\begin{matrix}𝔲,𝔳 \\ 𝔱⋭𝔳 \end{matrix}} r′′_{𝔲𝔳}n_{α}m_{𝔲𝔳}y_{i}

for some r′_{𝔲𝔳}, r′′_{𝔲𝔳} ∈ R. As Q_{1}, …, Q_{r} are distinct, the coefficient of n_{α}m_{Ṡ𝔱} in xy_{i} is non–zero because c > comp_{𝔱^{μ}}(i) — and r_{ST} ≠ 0. (Note also that n_{α}m_{Ṡ𝔱} ≠ 0 by Corollary 5.10.) Therefore, xy_{i} ≠ 0; however, m_{μ}y_{i} = 0 and x ∈ N(α) ∩ M(μ)^{∗}, so this contradicts our assumption that x ≠ 0. Consequently, x = 0 and the Theorem follows.

6.10 Remarks If Q_{s} = Q_{t}, where s ≠ t, then the rank of N(α) ∩ M(μ)^{∗} can be larger than that predicted by Theorem 6.9. For example, suppose that Q_{1} = Q_{2}, when r = n = 2, and take α = μ = ((1), (1)). Then n_{α} = m_{μ} = L_{1} − Q_{2} and, by (2.6),

n_{α}ℋ ∩ ℋ m_{μ} = M(μ) ∩ M(μ)^{∗}

is the free R–module with basis { m_{ST} | S, T ∈ T_{μ}^{rs}(Λ^{+}) }; this is an R–module of rank 3. In contrast, if Q_{1} ≠ Q_{2} then by Theorem 6.9

n_{α}ℋ ∩ ℋ m_{μ} = Rn_{α}T_{1}m_{μ} = R(L_{1} − Q_{1})T_{1}(L_{1} − Q_{2});

this time the intersection has rank 1. (By direct a calculation, (L_{1} − Q_{1})T_{1}(L_{1} − Q_{2}) is an element of M(μ) ∩ M(μ)^{∗} if and only if Q_{1} = Q_{2}.)

If r = 1 then from the definitions, N(α) = M(α) when q^{2} = −1. Using this observation it is easy to find examples where the conclusions of Theorem 6.9 fail when q^{2} = −1.

Shoji [21] has shown that if the parameters Q_{1}, …, Q_{r} are distinct then M(μ) is an induced module (more accurately, he has shown that there exists an induced module which has the same image as M(μ) in the Grothendieck group of ℋ ). Using Shoji’s work it might be possible to prove an analogue of Frobenius reciprocity for the modules M(μ) and N(α); this would give a better explanation as to why the rank of Hom_{ℋ} (M(μ), N(α)) is independent of the choice of parameters Q_{1}, …, Q_{r} in Theorem 6.9.

Now we reap some consequences of Theorem 6.9. 6.11 Corollary Suppose that ω ∈ Λ, Q_{1}, …, Q_{r} are distinct and that q^{2} ≠ −1. Then E(α) = Ê(α) for all α ∈ Λ. Proof Combine Theorem 6.9 and Lemma 6.8 and compare with Proposition 6.4.

6.12 Corollary Suppose that Q_{1}, …, Q_{r} are distinct and that q^{2} ≠ −1 and let α and μ be multicompositions of n. Then N(α) ∩ M(μ)^{∗} = n_{α}ℋ m_{μ}. Proof Certainly, n_{α}ℋ m_{μ} ⊆ N(α) ∩ M(μ)^{∗}. Conversely, by Theorem 6.9 a basis of N(α) ∩ M(μ)^{∗} is given by the elements n_{α}m_{ṠT}, where S ∈ T_{α}^{cs}(Λ^{+}) and T ∈ T_{μ}^{rs}(Λ^{+}).

As n_{α}m_{ṠT} ∈ n_{α}ℋ m_{μ}, by (2.5), we also have the opposite inclusion.

Notice that Theorem 6.9 and Corollary 6.12 imply that if μ is a multipartition then

n_{μ}ℋ ∩ ℋ m_{μ′} = n_{μ}ℋ m_{μ′} = Rn_{μ}m_{𝔱_{μ′} 𝔱^{μ′}} = Rn_{μ}T_{w_{μ}}m_{μ′}.

Du and Rui [11] have shown that n_{μ}ℋ m_{μ′} = Rn_{μ}T_{w_{μ}}m_{μ′} (without assuming that the parameters Q_{s} are distinct). This is interesting because the element n_{μ}T_{w_{μ}}m_{μ′} generates the Specht module S(μ′); see Remark 5.12.

6.13 Corollary Suppose that Q_{1}, …, Q_{r} are distinct, q^{2} ≠ −1 and that

Hom_{ℋ} (M(μ), N(α)) ≠ 0

for some multicompositions α and μ. Then α′ ⊵ μ.

Proof By Lemma 6.8 N(α) ∩ M(μ)^{∗} ≠ 0 since Hom_{ℋ} (M(μ), N(α)) ≠ 0. Therefore, we can find a multipartition λ and tableaux S ∈ T_{α}^{cs}(λ) and T ∈ T_{μ}^{rs}(λ) such that n_{α}m_{ṠT} is a non–zero element of N(α) ∩ M(μ)^{∗}. Hence, α′ ⊵ λ ⊵ μ, so α′ ⊵ μ as required.

Similarly, under the same assumptions, we have that Hom_{ℋ} (N(α), M(μ)) ≠ 0 only if α′ ⊵ μ.

6.14 Corollary Suppose that Q_{1}, …, Q_{r} are distinct and that q^{2} ≠ −1 and let α and μ be multicompositions of n. Then

{ n_{SṪ}m_{μ} | S ∈ T_{α}^{rs}(λ), T ∈ T_{μ}^{cs}(λ) for someλ ⊢ n }

is a basis of N(α) ∩ M(μ)^{∗}.

Proof By Theorem 6.9 the R–module N(α) ∩ M(μ)^{∗} is stable under specialization (or, if you prefer, base change); therefore, it is enough to consider the case ℋ = ℋ_{Z}. Applying the involutions ′ and ∗ to the basis of N(μ) ∩ M(α)^{∗} given by Theorem 6.9 yields the result. Alternatively, this can be proved by modifying the argument of Theorem 6.9.

6.15 Definition Suppose that λ is a multipartition of n and that α and μ are two multicompositions of n. For tableaux A ∈ T_{α}^{rs}(λ) and B ∈ T_{μ}^{cs}(λ) let θ′_{ST} be the homomorphism in E(α) determined by θ′_{AB}(m_{ν}h) = δ_{νμ}n_{AḂ}m_{μ}h, for all h ∈ ℋ and all ν ∈ Λ.

6.16 Proposition Suppose that ω ∈ Λ and that α is a multicomposition of n. Then E(α) is free as an R–module with basis { θ′_{AB} | A ∈ T_{α}^{rs}(λ) and B ∈ T_{Λ}^{cs}(λ) for someλ ∈ Λ^{+} } .

Proof First suppose that R = Z = ℤ[q, q^{−1}, Q_{1}, …, Q_{r}]. In this case, E(α) = Ê(α) by Corollary 6.11, so the Proposition follows from Corollary 6.14 and Lemma 6.8. Furthermore, by Proposition 6.4, { θ_{ST} | S ∈ T_{α}^{cs}(λ) and T ∈ T_{Λ}^{rs}(λ) for some λ ∈ Λ^{+} } is also a basis for E(α) (over Z); therefore, the entries in the transition matrix between these two bases belong to Z and this matrix is invertible over Z.

Now consider the general case. By Proposition 6.4 the module E(α) is stable under base change and {θ_{ST}} is a basis of (Eα) for any ring R. By the last paragraph the transition between the bases {θ_{ST}} and {θ′_{AB}} over Z can be specialized to give an invertible matrix over R. Hence, the {θ′_{AB}} are also a basis of E(α) over R.

As in the previous section, if E is an 𝒮 (Λ)–module then its contragredient dual E^{⊛} is the dual space Hom_{R}(E, R) equipped with the contragredient action: (fφ)(x) = f(xφ^{∗}), for f ∈ E^{⊛}, φ ∈ 𝒮 (Λ) and x ∈ E. Again, we say that E is self–dual if E ≅ E^{⊛}.

If E is an 𝒮 (Λ)–module and μ ∈ Λ let E_{μ} = Eφ_{μ} be the μ–weight space of E.

6.17 Theorem Suppose that Q_{1}, …, Q_{r} are invertible elements of R and that ω ∈ Λ and let α be a multicomposition. Then E(α) is self–dual.

Proof Define a bilinear map { , }_{α}: E(α) × E(α)⟶R by

\{θ_{ST}, θ′_{AB}\}_{α} = \{ \begin{matrix}⟨m_{ṠT}, n_{AḂ}⟩, & if Type(T) = Type(B), \\ 0, & otherwise. \end{matrix}

where θ_{ST} and θ′_{AB} run over the bases Proposition 6.4 and Proposition 6.16, respectively. By definition, the different weight spaces E_{μ}, μ ∈ Λ, of E(α) are orthogonal with respect to { , }_{α}. Suppose then that Type(T) = Type(B) and let λ = Shape(S). Then, as in the proof of Proposition 5.13,

\{θ_{ST}, θ′_{AB}\}_{α} = \{ \begin{matrix}Q_{λ}, & if (A′, B′) = (S, T), \\ 0, & if (A′, B′) ⋭ (S, T), \end{matrix}

where λ = Shape(S). Hence, { , }_{α} is a non–degenerate bilinear form on E(α).

Once again, the harder task is to prove that { , }_{α} is associative. Choose tableaux S, T, A and B as above. Suppose that φ ∈ Hom_{ℋ} (M(ν), M(μ)) for some ν, μ ∈ Λ. Then θ_{ST}φ ∈ E_{ν} and θ′_{AB}φ^{∗} ∈ E_{μ}; therefore, if μ ≠ ν then

{θ_{ST}φ, θ_{AB}}_{α} = 0 = {θ_{ST}, θ_{AB}φ^{∗}}_{α}

because different weight spaces are orthogonal with respect to { , }_{α}. Suppose then that μ = ν. By considering weight spaces we may also assume that Type(T) = μ = Type(B). Write φ(m_{μ}) = m_{μ}h for some h ∈ ℋ ; then (θ_{ST}φ)(m_{μ}h) = n_{α}m_{ṠT}h, so θ_{ST}φ is determined by n_{α}m_{ṠT}h. Similarly, since φ(m_{μ}) ∈ m_{μ}ℋ ∩ ℋ m_{μ} we can also write φ(m_{μ}) = h̃m_{μ}. Therefore, φ^{∗}(m_{μ}) = m_{μ}h̃^{∗} = (h̃m_{μ})^{∗} = (m_{μ}h)^{∗} = h^{∗}m_{μ}; consequently, (θ′_{AB}φ∗)(m_{μ}) = θ′_{AB}(m_{μ})h̃^{∗} = n_{AḂ}m_{μ}h̃^{∗} = n_{AḂ}h^{∗}m_{μ}. The bilinear form ⟨ , ⟩ on ℋ is associative so, as in the proof of Proposition 5.13, we have

{θ_{ST}φ, θ_{AB}}_{α} = ⟨m_{ṠT}h, n_{AḂ}⟩ = ⟨m_{ṠT}, n_{AḂ}h^{∗}⟩ = {θ_{ST}, θ_{AB}φ^{∗}}_{α}.

Therefore, { , }_{α} is associative and the proof is complete.

Combining the last two results we obtain our main theorem.

6.18 Theorem Suppose that R is a field, ω ∈ Λ and that Q_{1}, …, Q_{r} are invertible elements of R.

(i) If λ ∈ Λ^{+} then

E(λ) ≅ T (λ′) ⊕ ⨁_{λ′▷μ} T (μ)^{e_{λμ}}

for some non–negative integers e_{λμ}. (ii) The tilting modules of 𝒮 (Λ) are the indecomposable direct summands of the

modules { E(λ) | λ ∈ Λ^{+} } .

Proof By Theorem 6.5, E(λ) has a Δ–filtration; therefore, E(λ) also has a ∇–filtration since E(λ) is self–dual by Theorem 6.17. Hence, E(λ) is a tilting module. Furthermore, by Corollary 6.7, [E(λ):Δ(λ′)] = 1 and if [E(λ):Δ(μ)] > 0 then λ′ ⊵ μ. Therefore, by Ringel’s theorem (6.1), there exist non–negative integers e_{λμ} such that

E(λ) ≅ T (λ′) ⊕ ⨁_{λ′▷μ} T (μ)^{e_{λμ}}.

This proves (i). Part (ii) now follows by induction on the dominance order using (6.1).

7. Ringel duality

We now turn our attention to the Ringel dual of 𝒮 (Λ). By definition, the Ringel dual of 𝒮 (Λ) is the algebra End_{𝒮 (Λ)} (T), where T is any full tilting module for 𝒮 (Λ); thus, the Ringel dual is determined only up to Morita equivalence. By Theorem 6.18 the module

E(Λ) = ⨁_{α∈Λ} E(α)

is a full tilting module for 𝒮 (Λ) when ω ∈ Λ and the parameters Q_{1}, …, Q_{r} are invertible.

Let α and β be two multicompositions. Then for any S ∈ T_{β}^{rs}(λ) and T ∈ T_{α}^{rs}(λ) there is an ℋ –module homomorphism φ′_{ST} : N(α) ⟶ N(β); this induces an 𝒮 (Λ)–module homomorphism Φ′_{ST} : E(α)⟶E(β) given by Φ′_{ST}(θ) = φ′_{ST}θ, for θ ∈ E(α). In fact, we will show that these give all of the 𝒮 (Λ)–module homomorphisms from E(α) to E(β). 7.1 Proposition Suppose that ω ∈ Λ and let α and β be multicompositions of n. Then Hom_{𝒮 (Λ)} (E(α), E(β)) is free as an R–module with basis

{ Φ′_{ST} | S ∈ T_{β}^{rs}(λ), T ∈ T_{α}^{rs}(λ) for someλ ∈ Λ^{+} } .

Proof As indicated above the maps Φ′_{ST} belong to Hom_{𝒮 (Λ)} (E(α), E(β)). Moreover, they are linearly independent because the φ′_{ST} are a basis of Hom_{ℋ} (N(α), N(β)) by Proposition 4.3(i). Thus, it remains to see that these maps span Hom_{𝒮 (Λ)} (E(α), E(β)).

Suppose that Φ ∈ Hom_{𝒮 (Λ)} (E(α), E(β)). If θ ∈ Hom_{ℋ} (M(μ), N(α)) then Φ(θ) belongs to Hom_{ℋ} (M(μ), N(β)); that is, Φ maps weight spaces to weight spaces. Now, by definition, E(α) is generated by θ_{α} so Φ is determined by Φ(θ_{α}). Moreover,

Φ(θ_{α}) ∈ Hom_{ℋ} (ℋ , N(β)) ≅ N(β),

because Φ maps weight spaces to weight spaces. Now Φ commutes with the action of ℋ ≅ Hom_{ℋ} (ℋ , ℋ ) — we identify h ∈ ℋ with left multiplication by h. Therefore, Φ induces a map N(α) ⟶ N(β) via n_{α} = θ_{α}(1) ⟼ Φ(θ_{α})(1). Hence, we can write Φ(θ_{α}) = ∑_{S,T} r_{ST}φ′_{ST} for some r_{ST} ∈ R by Proposition 4.3(i). Therefore, Φ = ∑_{S,T} r_{ST}Φ′_{ST}, completing the proof.

The R–module E(Λ) is in fact an (𝒮′(Λ), 𝒮 (Λ))–bimodule. Moreover, it has the following double centralizer property.

If A is an algebra let A^{op} be the opposite algebra in which the order of multiplication is reversed. 7.2 Corollary Suppose that ω ∈ Λ. Then there are canonical isomorphisms of R–algebras

End_{𝒮 (Λ)} (E(Λ)) ≅ 𝒮′(Λ)^{op} and End_{𝒮 (Λ)′} (E(Λ)) ≅ 𝒮 (Λ).

Proof The first isomorphism, Φ′_{ST} ⟼ φ′_{ST}, is immediate from (the proof of) Proposition 7.1. The second isomorphism follows by symmetry.

As a special case we have a description of the Ringel dual of 𝒮 (Λ). 7.3 Corollary Suppose that R is a field, ω ∈ Λ and that Q_{1}, …, Q_{r} are invertible elements of R. Then the Ringel dual of 𝒮 (Λ) is isomorphic to 𝒮′(Λ)^{op}.

Finally, we want to determine the ∇–filtration multiplicities in the tilting modules. We actually don’t need to do any work here because the general theory of tilting modules for quasi–hereditary algebras tells us that [T (λ) : ∇(μ)] = [Δ(μ′), L(λ′)] (see, for example, [10, Appendix]); however, we want to show how this result can be derived using Young modules and Specht filtrations.

Recall that F_{ω}: 𝒮 (Λ)–mod ⟶ℋ –mod; M ⟼Mφ_{ω} is the Schur functor. 7.4 Proposition Assume that R is a field, ω ∈ Λ and that Q_{s} ≠ 0, for 1 ≤ s ≤ r. Suppose that λ ∈ Λ. Then F_{ω}(T (λ′)) ≅ Y′(λ) as ℋ –modules. Proof Applying the definitions F_{ω}(E(λ)) ≅ Hom_{ℋ} (ℋ , N(λ)) ≅ N(λ), where the last isomorphism comes from Lemma 6.8 (or directly). Hence, the Schur functor F_{ω} induces an injective map from End_{𝒮 (Λ)} (E(λ), E(β)) to End_{ℋ} (N(λ), N(β)); by Proposition 4.3(i) and Proposition 7.1 this is an isomorphism. Consequently, if an indecomposable tilting module T (λ′) is a direct summand of E(λ) then F_{ω}(T (λ′)) is an indecomposable direct summand of N(λ). Therefore, by Proposition 4.4, F_{ω}(T (λ′)) ≅ Y′(μ) for some multipartition μ. Now, E(λ) ≅ T (λ′) ⊕ ⨁_{λ′▷μ} T (μ)^{e_{λμ}} by Theorem 6.18(i) and

F_{ω}(E(λ)) ≅ N(λ) ≅ Y′(λ) ⊕_{ν▷λ} Y′(ν)^{c_{λμ}}

by Proposition 4.4(i). Hence, the result follows by induction on the dominance ordering.

7.5 Corollary Suppose that R is a field, ω ∈ Λ and that Q_{1}, …, Q_{r} are non–zero elements of R. Let λ and μ be multipartitions of n. Then [T (λ′) : ∇(μ′)] = [Δ(μ) : L(λ)].

Proof Now F_{ω}(Δ(ν)) ≅ S(ν) by (2.10)(i); therefore, as F_{ω} projects onto the ω–weight space, F_{ω}(∇(ν)) = F_{ω}(Δ(ν)^{⊛}) ≅ S(ν)^{⊛}. Consequently, we have

[T (λ′) : ∇(μ′)] = [F_{ω}(T (λ′)) : F_{ω}(∇(μ′))],
= [Y′(λ) : S(μ′)^{⊛}]  by Proposition 7.4,
= [Y′(λ) : S′(μ)],  by Corollary 5.7,
= [Δ′(μ) : L′(λ)],  by Proposition 4.4(iv),
= [Δ(μ) : L(λ)],

where the last equality follows because the isomorphism 𝒮 (Λ) ≅ 𝒮′(Λ) of Proposition 4.3(iii) identifies the Weyl modules Δ(μ) and Δ′(μ), and the simple modules L(λ) and L′(λ).

To conclude, we remark that in the case of the q–Schur algebras (i.e. when r = 1) our proof of Corollary 7.5 looks quite different to Donkin’s [9]; however, in spirit the two arguments are the same in that they both rely on a duality between the symmetric and exterior powers and on the isomorphism of Proposition 4.3(iii).

Acknowledgements

I would like to thank Steve Donkin for explaining the tilting modules of the q–Schur algebras to me and for answering my numerous questions about them. I am also grateful to the referee for some minor corrections.

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Tilting modules for cyclotomic Schur algebras 29

School of Mathematics and Statistics F07 University of Sydney, Sydney N.S.W. 2006. a.mathas@maths.usyd.edu.au

Australia.  www.maths.usyd.edu.au/u/mathas/