TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0002-9947(XX)0000-0

THE JANTZEN SUM FORMULA FOR CYCLOTOMIC q–SCHUR ALGEBRAS

GORDON JAMES AND ANDREW MATHAS

Abstract. The cyclotomic q-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen’s sum formula for the cyclotomic q-Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.

1. Introduction

In [6] Richard Dipper and the authors introduced the cyclotomic q–Schur algebras and showed that they are quasi–hereditary cellular algebras. By definition, a cyclotomic q–Schur algebra is a certain endomorphism algebra attached to an Ariki–Koike algebra in much the same way as the q–Schur algebra [5] is defined as an endomorphism algebra of a particular module for the Iwahori–Hecke algebra of the symmetric group.

One of our motivations for defining the cyclotomic q–Schur algebras was to provide another tool for studying the Ariki–Koike algebras. In this paper we use the cyclotomic q–Schur algebras to prove a version of the Jantzen sum formula for the Ariki–Koike algebras. Most of the argument is devoted to first proving an analogue of Jantzen’s sum formula for the Weyl modules of the cyclotomic q–Schur algebra. The result for the Ariki–Koike algebras is then deduced by a Schur functor argument. As a corollary of these results we obtain criteria for the Weyl modules of the cyclotomic q–Schur algebras, and for certain of the Specht modules of the Ariki–Koike algebras, to be irreducible.

We note that as a special case of our results we obtain, for the first time, an analogue of the Jantzen sum formula for Coxeter groups of type B.

In the case of the q–Schur algebra it is possible to give a geometric proof of Jantzen’s sum formula [1]. As yet, in the cyclotomic case there is no algebra which plays the rôle of the quantum group of type A; consequently, an algebraic approach is necessary. The proof we give generalizes and extends the argument of [13].

2. The cyclotomic q–Schur algebra

We recall some definitions and results from [6].

Fix integers r and n with r ≥ 1 and n ≥ 1 and let R be a commutative ring with 1 and let q, Q_{1}, Q_{2}, …, Q_{r} be elements of R, with q invertible.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. Primary: 16G99 Secondary: 20C20, 20G05.

The authors would like to thank the Isaac Newton Institute for its hospitality. The second author also gratefully acknowledges the support of the Sonderforschungsbereich 343 at the Universität Bielefeld.

The Ariki–Koike algebra ℋ [3] is the associative 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.

Let 𝔖_{n} be the symmetric group on {1, 2, …, n}. Then 𝔖_{n} is generated by s_{1}, s_{2}, …, s_{n−1} where s_{i} = (i, i + 1) for 1 ≤ i < n. If w = s_{i_{1}}s_{i_{2}} … s_{i_{k}} is a reduced expression for w ∈ 𝔖_{n} (that is, k is minimal), we write ℓ(w) = k and define T_{w} = T_{i_{1}}T_{i_{2}} … T_{i_{k}}. Let ℋ (𝔖_{n}) be the subalgebra of ℋ spanned by { T_{w} | w ∈ W }; then ℋ (𝔖_{n}) is the Iwahori–Hecke algebra of 𝔖_{n}.

Define elements L_{1}, L_{2}, …, L_{n} of ℋ by L_{i} = q^{1−i}T_{i−1} … T_{1}T_{0}T_{1} … T_{i−1}. We have the following well–known result (cf. [3, 3.3] and [4, (2.1), (2.2)]). 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).

A composition α = (α_{1}, α_{2}, …) is a finite sequence of non–negative integers; we denote by |α| the sum of the sequence. A multicomposition of n (into r components) is an ordered r–tuple μ = (μ^{(1)}, …, μ^{(r)}) of compositions μ^{(k)} such that ∑_{k=1}^{r} |μ^{(k)}| = n. We call μ^{(k)} the kth component of μ. A partition is a composition whose parts are non–increasing; a multicomposition is a multipartition if all of its components are partitions. Definition 2.2. Suppose that μ is a multicomposition of n and let a = (a_{1}, …, a_{r}) be an r–tuple of integers a_{k} such that 0 ≤ a_{k} ≤ n for all k.

(i) Let u_{a}^{+} = u_{a,1}u_{a,2} … u_{a,r} where u_{a,k} = ∏_{i=1}^{a_{k}}(L_{i} − Q_{k}) for 1 ≤ k ≤ r. (ii) Let x_{μ} = ∑_{w∈𝔖_{μ}} T_{w} where 𝔖_{μ} = 𝔖_{μ^{(1)}} × 𝔖_{μ^{(2)}} × ⋯ × 𝔖_{μ^{(r)}}.

(iii) Let m_{μ} = u_{a}^{+}x_{μ} where a = (a_{1}, …, a_{r}) and a_{k} = ∑_{i=1}^{k−1} |μ^{(i)}|, for 1 ≤ k ≤ r. (iv) Let M^{μ} = m_{μ}ℋ .

Note that u_{a}^{+}x_{μ} = x_{μ}u_{a}^{+} by (2.1)(iv) and that a_{1} = 0 in Definition 2.2(iii).

Given two multicompositions μ and ν write μ ⊵ ν if for all i ≥ 0 and k with 1 ≤ k ≤ r

∑_{j=1}^{k−1} |μ^{(j)}| + ∑_{j=1}^{i} μ_{j}^{(k)} ≥ ∑_{j=1}^{k−1} |ν^{(j)}| + ∑_{j=1}^{i} ν_{j}^{(k)}.

If μ ⊵ ν and μ ≠ ν then we write μ ▷ ν.

Let Λ be a finite subset of the set of all multicompositions of n which have r components such that if μ ∈ Λ and λ ⊵ μ for some multipartition λ then λ ∈ Λ. Let Λ^{+} be the set of multipartitions in Λ.

The main arena for the investigations of this paper is the cyclotomic q–Schur algebra, which we now define. Definition 2.3. [6] The cyclotomic q–Schur algebra is the endomorphism algebra

𝒮 (Λ) = End_{ℋ} ( ⨁_{μ∈Λ} M^{μ}).

CYCLOTOMIC q–SCHUR ALGEBRAS 3

Generally we omit Λ and simply write 𝒮.

In order to describe a basis of 𝒮 and its irreducible representations we next recall the combinatorics of semistandard tableaux from [6].

Suppose that ν is a multicomposition of n and let r = {1, 2, …, r}. The diagram of ν is the set

[ν] = { (i, j, k) ∈ ℕ × ℕ × r | 1 ≤ j ≤ λ_{i}^{(k)} } .

The elements of [ν] are the nodes of ν; more generally, a node is any element of ℕ × ℕ × r.

A ν–tableau T is a mapping from the diagram of ν into ℕ × r; informally, we shall regard T as an ordered r–tuple of labelled diagrams, as in the example below. In particular, we will write T = (T^{(1)}, …, T^{(r)}) and will speak of the components of T and their rows and columns. We say that T is a tableau of type μ if the number of entries in T equal to (i, k) is μ_{i}^{(k)} for all (i, k) ∈ ℕ × r.

Below, and in all later examples, we write i_{k} in place of the ordered pair (i, k). Example 2.4. (i) Suppose that ν is a multicomposition and let T^{ν} be the ν–

tableau of type ν such that T^{ν}(i, j, k) = (i, k) for all (i, j, k) ∈ [ν]. (ii) Let λ = ((3, 2), (2, 1), (1^{2})). Then two λ–tableaux are

T^{λ} = ( \begin{matrix}1_{1} & 1_{1} & 1_{1} \\ 2_{1} & 2_{1} \end{matrix} , 1_{2} 1_{2} 2_{2} , 1_{3} 2_{3} ) and S = ( \begin{matrix}1_{1} & 2_{1} & 1_{2} \\ 2_{2} & 3_{3} \end{matrix} , 2_{2} 1_{3} 2_{3} , 3_{3} 4_{3} ).

Here S is a λ–tableau of type ((1^{2}), (1, 2), (1, 1, 2, 1)).

Given (i, k) and (j, l) in ℕ × r, we say that (i, k) < (j, l) if either k < l, or k = l and i < j. Definition 2.5. [6] A λ–tableau T = (T^{(1)}, …, T^{(r)}) of type μ is semistandard if λ is a multipartition and

(i) the entries in each row of each component T^{(k)} are non–decreasing; and, (ii) the entries in each column of each component T^{(k)} are strictly increasing; and, (iii) if (a, b, c) ∈ [λ] and T(a, b, c) = (i, k) then k ≥ c. Let T_{0}(λ, μ) be the set of semistandard λ–tableaux of type μ.

For example, the λ–tableau T^{λ} defined in Example 2.4(i) is the unique semistandard λ–tableau of type λ. The λ–tableau S in Example 2.4(ii) is also semistandard.

Before we can describe how the semistandard tableaux index a basis for 𝒮 we first need to single out special semistandard tableaux which index a basis of the Ariki–Koike algebra ℋ . Definition 2.6. (i) Let ω = ((0), …, (0), (1^{n})), a multipartition of n. (ii) Let λ be a multipartition. A standard λ–tableau is a semistandard λ–tableau

of type ω. (iii) Let Std(λ) = T_{0}(λ, ω) be the set of standard λ–tableaux.

Let T be a ν–tableau of type ω. Then, for all x ∈ [ν], we have T(x) = (i, r) for some i ∈ {1, 2, …, n}; we identify T with the map 𝔱 determined by T(x) = (𝔱(x), r) for all x ∈ [ν]. Then 𝔱 is a standard tableau if and only if ν is a multipartition and in each component 𝔱^{(k)} the entries are strictly increasing along each row and down each column.

We will always denote tableaux of type ω by lower case letters in order to distinguish them from tableaux of other types.

Given a multicomposition ν, let 𝔱^{ν} be the tableau with the integers 1, 2, …, n entered in order along the rows of [ν]. The symmetric group 𝔖_{n} acts on the set of ν–tableaux of type ω by letter permutations; note that the Young subgroup 𝔖_{ν} is precisely the row stabilizer of 𝔱^{ν}.

If λ is a multipartition and 𝔱 is a standard λ–tableau then define d(𝔱) ∈ 𝔖_{n} by 𝔱 = 𝔱^{λ}d(𝔱). Then d(𝔱) is a (distinguished) right coset representative of 𝔖_{λ} in 𝔖_{n}.

Let ∗: ℋ ⟶ℋ be the R–linear antiautomorphism of ℋ determined by T_{i}^{∗} = T_{i} for all i with 0 ≤ i < n. In particular, T_{w}^{∗} = T_{w^{−1}} for all w ∈ 𝔖_{n}. Definition 2.7. Suppose that λ is a multipartition of n and that 𝔰 and 𝔱 are standard λ–tableaux. Let m_{𝔰𝔱} = T_{d(𝔰)}^{∗}m_{λ}T_{d(𝔱)}.

The proof of the following result can be found in [6, 3.26]. Theorem 2.8. (The Standard Basis Theorem) The Ariki–Koike algebra ℋ is a free R–module with cellular basis

{ m_{𝔰𝔱} | 𝔰 and 𝔱 are standard λ–tableaux for some multipartition λ of n }.

We call this basis the standard basis of ℋ ; it is a cellular basis in the sense of Graham and Lehrer [8]. Note that m_{𝔰𝔱}^{∗} = m_{𝔱𝔰}.

Given a standard λ–tableau 𝔱 and a multicomposition μ let μ(𝔱) be the λ–tableau of type μ obtained by replacing each entry m in 𝔱 by (i, k) if m appears in row i of the kth component of 𝔱^{μ}. For example, T^{λ} = λ(𝔱^{λ}). 2.9. [6, Proposition 6.3] Let μ and ν be multicompositions of n. Then M^{μ} ∩ M^{ν}^{∗} is a free R–module with basis

{ m_{ST} | S ∈ T_{0}(λ, μ) and T ∈ T_{0}(λ, ν) for some multipartition λ of n }

where m_{ST} = ∑_{𝔰,𝔱} m_{𝔰𝔱} and (𝔰, 𝔱) runs over all pairs of standard λ–tableaux such that S = μ(𝔰) and T = ν(𝔱).

Note that m_{λ} = m_{𝔱^{λ}}_{𝔱^{λ}} = m_{T^{λ}}_{𝔱^{λ}} = m_{𝔱^{λ}}_{T^{λ}} = m_{T^{λ}}_{T^{λ}}.

In particular, (2.9) shows that the maps φ_{ST} below are well–defined elements of 𝒮. Definition 2.10. Let λ be a multipartition of n and let μ and ν be multicompositions of n. Suppose that S ∈ T_{0}(λ, μ) and T ∈ T_{0}(λ, ν). Then φ_{ST} ∈ 𝒮 is the ℋ –homomorphism such that

φ_{ST}(m_{α}h) = δ_{αν}m_{ST}h

for all α ∈ Λ and all h ∈ ℋ . Theorem 2.11. (The Semistandard Basis Theorem [6, 6.12]) The cyclotomic q–Schur algebra 𝒮 is free as an R–module with cellular basis

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

We call the basis {φ_{ST}} the semistandard basis of 𝒮. Because it is cellular, the R–linear map ∗: 𝒮 ⟶ 𝒮 determined by φ_{ST}^{∗} = φ_{TS} is an antiautomorphism of 𝒮 (see [6, 6.9]).

For each multipartition λ in Λ^{+} let 𝒮̄^{λ} be the R–submodule of 𝒮 with basis

{ φ_{UV} | U ∈ T_{0}(α, μ), V ∈ T_{0}(α, ν) for some μ, ν ∈ Λ and α ∈ Λ^{+} with α ▷ λ}.

By [6, 6.11], 𝒮̄^{λ} is a two–sided ideal of 𝒮.

Recall the λ–tableau T^{λ} from Example 2.4(i). It is easy to see from the definitions that φ_{T^{λ}}_{T^{λ}} restricts to the identity map on M^{λ}. Definition 2.12. Let λ ∈ Λ^{+}. The Weyl module W^{λ} is the submodule of 𝒮 /𝒮̄^{λ} given by W^{λ} = (𝒮̄^{λ} + φ_{T^{λ}}_{T^{λ}})𝒮.

We remark that in [6] we defined the Weyl module W^{λ} to be a left 𝒮 –module; however it is more convenient here to define it as a right module.

By Theorem 2.11 the Weyl module W^{λ} is a free R–module with basis

{ φ_{T} | T ∈ T_{0}(λ, μ), μ ∈ Λ }

where φ_{T} = 𝒮̄^{λ} + φ_{T^{λ}T}. The cellular structure of 𝒮 defines a natural symmetric bilinear form ⟨ , ⟩ on W^{λ} which is determined by the requirement that

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

for all semistandard λ–tableaux S and T. Note that ⟨φ_{S}, φ_{T}⟩ = 0 unless S and T are tableaux of the same type. Also, by Theorem 2.11,

(2.13)  ⟨xφ, y⟩ = ⟨x, yφ^{∗}⟩ for all x, y ∈ W^{λ} and all φ ∈ 𝒮 .

Consequently, rad W^{λ} = { x ∈ W^{λ} | ⟨x, y⟩ = 0 for all y ∈ W^{λ} } is an 𝒮 –submodule of W^{λ}.

Because φ_{T^{λ}}_{T^{λ}} is the identity map on M^{λ}, one sees that ⟨φ_{T^{λ}}, φ_{T^{λ}}⟩ = 1; together with Theorem 2.11 this implies the following result. 2.14. [6, Theorem 6.16] For each λ ∈ Λ^{+} let F^{λ} = W^{λ}/ rad W^{λ}. Then F^{λ} ≠ 0; moreover, if R is a field then F^{λ} is absolutely irreducible and { F^{λ} | λ ∈ Λ^{+} } is a complete set of non–isomorphic irreducible right 𝒮 –modules.

In addition to the simple 𝒮 –modules we will be concerned with the simple ℋ – modules. Define N̄^{λ} to be the R–submodule of ℋ with basis

{ m_{𝔰𝔱} | 𝔰 and 𝔱 are standard μ–tableaux for some multipartition μ of n with μ ▷ λ }.

(cf. the definition of 𝒮̄^{λ} ). It follows from Theorem 2.8 that N̄^{λ} is a two–sided ideal of ℋ . Definition 2.15. Let λ be a multipartition. The Specht module S^{λ} is the right ℋ –module (N̄^{λ} + m_{λ})ℋ , a submodule of ℋ /N̄^{λ}.

For 𝔱 ∈ Std(λ) let m_{𝔱} = N̄^{λ} + m_{𝔱^{λ}𝔱}. Then, by Theorem 2.8, S^{λ} is free as an R–module with basis { m_{𝔱} | 𝔱 ∈ Std(λ) }.

Define a bilinear form on S^{λ} by requiring that ⟨m_{𝔰}, m_{𝔱}⟩m_{λ} ≡ m_{𝔱^{λ}𝔰}m_{𝔱𝔱^{λ}} mod N̄^{λ}. As before, rad S^{λ} is an ℋ –module and we set D^{λ} = S^{λ}/ rad S^{λ}. 2.16. [6, Theorem 3.30] Suppose that R is a field. Then

{ D^{λ} ≠ 0 | λ is a multipartition of n }

is a complete set of non–isomorphic irreducible ℋ –modules. Moreover, each D^{λ} is either absolutely irreducible or zero.

We remark that the bilinear forms defined on the modules W^{λ} and S^{λ}, and the results relating to them, fit into the general framework of cellular algebras, as devised by Graham and Lehrer [8].

Now suppose that ω ∈ Λ^{+}; equivalently, assume that Λ^{+} is the set of all multipartitions of n. Then φ_{T^{ω}T^{ω}} is idempotent in 𝒮; indeed, if ν is a multicomposition in Λ then φ_{T^{ν}T^{ν}}, the identity map on M^{ν}, is idempotent and the identity element of 𝒮 is ∑_{ν∈Λ} φ_{T^{ν}T^{ν}}.

By identifying h ∈ ℋ with the homomorphism ρ_{h} ∈ Hom_{ℋ} (ℋ , ℋ ) which is given by ρ_{h}(h′) = hh′ for all h′ ∈ ℋ , we see that ℋ ≅ φ_{T^{ω}T^{ω}}𝒮 φ_{T^{ω}T^{ω}}. Let (W^{ν} : F^{λ}) denote the composition multiplicity of the simple module F^{λ} in W^{ν} and similarly for (S^{ν} : D^{λ}). Then, by general arguments (see, for example, [9, §6]), we have the following.

Proposition 2.17. Assume that every multipartition of n belongs to Λ^{+}. Let U be a right 𝒮 –module. Then Uφ_{T^{ω}T^{ω}} is a right ℋ –module. In particular, if λ is a multipartition of n then W^{λ}φ_{T^{ω}T^{ω}} ≅ S^{λ} and F^{λ}φ_{T^{ω}T^{ω}} ≅ D^{λ}. Furthermore, if R is a field and D^{λ} ≠ (0) then (S^{ν} : D^{λ}) = (W^{ν} : F^{λ}) for all multipartitions ν.

Thus, the decomposition matrix of ℋ embeds into the decomposition matrix of 𝒮.

Finally, we also require a better understanding of the basis elements φ_{T} of the Weyl module W^{λ}. As for the Specht modules, if T is a semistandard λ–tableau of type μ we let m_{T} = N̄^{λ} + m_{T^{λ}T}. We claim that φ_{T} can be identified with the map (also denoted φ_{T})

(2.18)  φ_{T} : M^{μ} ⟶(N̄^{λ} + M^{λ})/N̄^{λ} given by m_{μ}h⟼m_{T}h

for all h ∈ ℋ . In order to see this let M = ⨁_{ν∈Λ} M^{ν} and M̄^{λ} = ⨁_{ν∈Λ} N̄^{λ} ∩ M^{ν}. Then each φ in 𝒮̄^{λ} maps M into M̄^{λ}; so we may regard 𝒮 /𝒮̄^{λ} as a set of maps from M into M/M̄^{λ}. Consequently, φ_{T} = 𝒮̄^{λ} +φ_{T^{λ}T} can be identified with the map from M^{μ} into (M̄^{λ} + M^{λ})/M̄^{λ} which sends m_{μ} to M̄^{λ} + m_{T^{λ}T} (since φ_{T^{λ}T}(m_{μ}) = m_{T^{λ}T}). Then, by the third isomorphism theorem,

(M̄^{λ} + M^{λ})/M̄^{λ} ≅ M^{λ}/(M̄^{λ} ∩ M^{λ}) = M^{λ}/(N̄^{λ} ∩ M^{λ}) ≅ (N̄^{λ} + M^{λ})/N̄^{λ}

justifying our claim.

3. The Gram determinant of W_{μ}^{λ}

Throughout this section, fix a multipartition λ ∈ Λ^{+} and a multicomposition μ ∈ Λ. The μ–weight space of W^{λ} is the R–submodule W_{μ}^{λ} = W^{λ}φ_{T^{μ}T^{μ}}; thus W_{μ}^{λ} is R–free with basis { φ_{T} | T ∈ T_{0}(λ, μ) }.

Definition 3.1. The Gram determinant of W_{μ}^{λ} with respect to the semistandard basis is G_{μ}(λ) = det (⟨φ_{S}, φ_{T}⟩), where S and T run over the elements of T_{0}(λ, μ). If T_{0}(λ, μ) is empty, we set G_{μ}(λ) = 1.

(We determine the sign of G_{μ}(λ) by fixing a total ordering of T_{0}(λ, μ) which is compatible with the partial ordering of Definition 3.6 below.)

The purpose of this section is to compute G_{μ}(λ); we do this by first constructing an orthogonal basis for W_{μ}^{λ} when R = 𝔽(q, Q_{1}, …, Q_{r}) and q, Q_{1}, …, Q_{r} are independent transcendental elements over a field 𝔽.

Definition 3.2. Given i ≥ 1 and k ∈ r let y, y + 1, …, z be the entries in row i of 𝔱^{μ^{(k)}}. Then L_{i,k}^{μ} is the element of Hom_{ℋ} (M^{μ}, M^{μ}) given by

L_{i,k}^{μ}(m_{μ}h) = (L_{y} + L_{y+1} + ⋯ + L_{z})m_{μ}h,

for all h ∈ ℋ .

The homomorphism L_{i,k}^{μ} maps into M^{μ} because (2.1)(ii) and (iii) imply the following result.

Lemma 3.3. Suppose that (i, k) ∈ ℕ × r and let y, y + 1, …, z be the entries in row i of 𝔱^{μ^{(k)}}. Then L_{y} + L_{y+1} + ⋯ + L_{z} commutes with every element of ℋ (𝔖_{μ}). In particular, (L_{y} + L_{y+1} + ⋯ + L_{z})m_{μ} = m_{μ}(L_{y} + L_{y+1} + ⋯ + L_{z}).

Note that L_{i,k}^{μ} = 0 if μ_{i}^{(k)} = 0. Furthermore, using (2.1) again, the homomorphisms L_{i,k}^{μ} and L_{j,l}^{μ} commute for all (i, k) and (j, l) in ℕ × r.

Below, often without mention, we will identify an element h of ℋ with the homomorphism ρ_{h} ∈ Hom_{ℋ} (ℋ , ℋ ) given by ρ_{h}(h′) = hh′ for all h′ ∈ ℋ . Under this identification we have that L_{i} = L_{i,r}^{ω} for i = 1, 2, …, n.

Definition 3.4. (i) Let x = (a, b, c) ∈ [λ]. Then the residue of x is

res(x) = q^{b−a}Q_{c}.

(ii) Let (i, k) ∈ ℕ × r and suppose that T ∈ T_{0}(λ, μ). Then

res_{T}(i, k) = ∑_{x∈[λ]:T(x)=(i,k)} res(x).

Similarly, given a standard tableau 𝔱 ∈ Std(λ) we write res_{𝔱}(i) = res(x) where x is the unique node in [λ] such that 𝔱(x) = i.

(iii) Let a = (a_{1}, …, a_{r}), with 0 ≤ a_{k} ≤ n for all k, and suppose that 𝔱 ∈ Std(λ). Then

res_{𝔱}(a) = ∏_{k=1}^{r} ∏_{i=1}^{a_{k}} ( res_{𝔱}(i) − Q_{k})

(cf. Definition 2.2(i)). Note that if L_{i,k}^{μ} = 0 then res_{T}(i, k) = 0 for all T ∈ T_{0}(λ, μ).

Example 3.5. Let λ = ((3, 1), (1)), μ = ((2), (2, 1)), and let

𝔱 = ( 1 2 4 5 , 3 ) and T = ( 1_{1} 1_{1} 1_{2} 2_{2} , 1_{2} ).

Then 𝔱 ∈ Std(λ), T ∈ T_{0}(λ, μ) and T = μ(𝔱). The residues in the diagram of λ are

( ^{Q_{1}  qQ_{1}  q^{2}Q_{1}} ; q^{−1}Q_{1} ,  Q_{2} ) .

Thus, res_{T}(1_{1}) = res_{𝔱}(1) + res_{𝔱}(2) = Q_{1} + qQ_{1}, res_{T}(1_{2}) = res_{𝔱}(4) + res_{𝔱}(3) = q^{2}Q_{1} + Q_{2} and res_{T}(2_{2}) = res_{𝔱}(5) = q^{−1}Q_{1}. Let a = (0, 2); then m_{μ} = u_{a}^{+}x_{μ} and res_{𝔱}(a) = (Q_{1} − Q_{2})(qQ_{1} − Q_{2}). Definition 3.6. (i) Suppose that T ∈ T_{0}(λ, μ) and (i, k) ∈ ℕ×r. Let T_{i,k} denote

the subtableau of T consisting of all entries (j, l) ≤ (i, k), and let T_{i,k}^{#} be the multipartition whose diagram is determined by T_{i,k}.

(ii) Given S and T ∈ T_{0}(λ, μ) write S ⊵ T if S_{i,k}^{#} ⊵ T_{i,k}^{#} for all (i, k) ∈ ℕ × r. (iii) If S ⊵ T and S ≠ T, we write S ▷ T. Proposition 3.7. Suppose that 𝔱 is a standard λ–tableau and let i be an integer with 1 ≤ i ≤ n. Then for each 𝔰 ∈ Std(λ) there exists a_{𝔰} ∈ R such that

m_{𝔱}L_{i} = res_{𝔱}(i)m_{𝔱} + ∑_{𝔰▷𝔱} a_{𝔰}m_{𝔰}.

Proof. First consider the case where 𝔱 = 𝔱^{λ}; then m_{𝔱} = N̄^{λ} + m_{λ}. Suppose that i appears in row a and column b of the cth component of 𝔱^{λ} and let j be the smallest integer appearing in 𝔱^{λ}^{(c)}; then j ≤ i. Write m_{λ} = x_{λ}u_{a}^{+} as in Definition 2.2 and let T_{j,i} = T_{j}T_{j+1} … T_{i−1} (set T_{j,i} = 1 if j = i) and T_{i,j} = T_{j,i}^{∗}.

Working modulo N̄^{λ} and using (2.1)(ii) and (2.1)(iii), we find that

m_{𝔱^{λ}}L_{i} ≡ m_{λ}L_{i} = x_{λ}u_{a}^{+}L_{i} = q^{j−i}x_{λ}u_{a}^{+}T_{i,j}L_{j}T_{j,i} = q^{j−i}x_{λ}T_{i,j}u_{a}^{+}L_{j}T_{j,i}
= q^{j−i}Q_{c}x_{λ}T_{i,j}u_{a}^{+}T_{j,i} + q^{j−i}x_{λ}T_{i,j}u_{a}^{+}(L_{j} − Q_{c})T_{j,i}
= q^{j−i}Q_{c}u_{a}^{+}x_{λ}T_{i,j}T_{j,i} + q^{j−i}x_{λ}T_{i,j}u_{b}^{+}T_{j,i},

where b = (a_{1}, …, a_{c−1}, a_{c} + 1, a_{c+1}, …, a_{r}). Therefore,

m_{𝔱^{λ}}L_{i} ≡ q^{j−i}Q_{c}u_{a}^{+}x_{λ}T_{i,j}T_{j,i} mod N̄^{λ}

because x_{λ}T_{i,j}u_{b}^{+} ∈ ℋ m_{λ}ℋ ∩ ℋ u_{b}^{+}ℋ ⊆ N̄^{λ} by Theorem 2.8. Now q^{j−i}T_{i,j}T_{j,i} is a q–Murphy operator in the Iwahori–Hecke algebra of the symmetric group on {j, j + 1, …, i}; consequently, by [17, Theorem 4.6],

m_{𝔱^{λ}}L_{i} ≡ q^{b−a}Q_{c}u_{a}^{+}x_{λ} = res_{𝔱^{λ}}(i)u_{a}^{+}x_{λ} ≡ res_{𝔱^{λ}}(i)m_{𝔱^{λ}} mod N̄^{λ}.

This completes the proof when 𝔱 = 𝔱^{λ}. If 𝔱 ≠ 𝔱^{λ} then there exists an integer k such that 𝔰 = 𝔱(k, k + 1) ▷ 𝔱 (and 1 ≤ k < n). Then m_{𝔱} = m_{𝔰}T_{k} and the result follows by the argument of [4, Theorem 3.15].

Recalling the definitions of u_{a}^{+} and res_{𝔱}(a) from (2.2) and (3.4) respectively, we obtain the next result.

Corollary 3.8. Let 𝔱 be a standard λ–tableau. Then for each 𝔰 ∈ Std(λ) there exists a_{𝔰} ∈ R such that

m_{𝔱}u_{a}^{+} = res_{𝔱}(a)m_{𝔱} + ∑_{𝔰▷𝔱} a_{𝔰}m_{𝔰}.

Lemma 3.9. Let T ∈ T_{0}(λ, μ). There exist unique standard λ–tableaux first(T) and last(T) such that

(i) μ(first(T)) = μ(last(T)) = T; and, (ii) first(T) ⊵ 𝔱 ⊵ last(T) for all 𝔱 ∈ Std(λ) such that μ(𝔱) = T.

Furthermore, if d = d(first(T)) and m_{λ} = u_{b}^{+}x_{λ} then

m_{T^{λ}T} = ∑_{w∈𝔖_{λ}d^{−1}𝔖_{μ}} u_{b}^{+}T_{w}

and 𝔖_{λ} ∩ d𝔖_{μ}d^{−1} = 𝔖_{ν_{T}} for some multicomposition ν_{T} of n.

Proof. Parts (i) and (ii) follow easily from the definitions; see [6, 4.7]. The final statements are a consequence of the definition of m_{T^{λ}T} and well–known properties of distinguished double coset representatives (see, for example, [4, 1.6]).

We remark that if 𝔱 is a standard λ–tableau then 𝔱 = first(𝔱) = last(𝔱) and ν_{𝔱} = ω.

Theorem 3.10. Suppose that T is a semistandard λ–tableau of type μ and let (i, k) ∈ ℕ × r. Then for each S ∈ T_{0}(λ, μ) there exists a_{S} ∈ R such that

φ_{T}L_{i,k}^{μ} = res_{T}(i, k)φ_{T} + ∑_{S▷T} a_{S}φ_{S}.

Proof. Recalling our conventions for φ_{T} from (2.18), and using Lemma 3.3, we have

φ_{T}L_{i,k}^{μ}(m_{μ}) = φ_{T}(m_{μ}(L_{y} + L_{y+1} + ⋯ + L_{z})) = m_{T}(L_{y} + L_{y+1} + ⋯ + L_{z})

where y, y + 1, …, z are the entries in row i of 𝔱^{μ^{(k)}}. Let 𝔱 = first(T); then, by Lemma 3.9, m_{T} = m_{𝔱}h for some h ∈ ℋ (𝔖_{μ}). Therefore, by Proposition 3.7,

φ_{T}L_{i,k}^{μ}(m_{μ}) = m_{𝔱}(L_{y} + L_{y+1} + ⋯ + L_{z})h

= res_{T}(i, k)m_{𝔱}h + ∑_{𝔰▷𝔱} a_{𝔰}m_{𝔰}h

= res_{T}(i, k)m_{T} + ∑_{S∈T_{0}(λ,μ)} a_{S}m_{S}

for some a_{𝔰}, a_{S} ∈ R, by (2.9). Since 𝔱 = first(T), we deduce that a_{S} = 0 unless S ▷ T. Therefore,

φ_{T}L_{i,k}^{μ} = res_{T}(i, k)φ_{T} + ∑_{S▷T} a_{S}φ_{S}

as required.

Until further notice we assume that R is the rational function field 𝔽(q, Q_{1}, …, Q_{r}) for some field 𝔽. We will compute the Gram determinant of W_{μ}^{λ} as an element of this field and derive the general case from this. Definition 3.11. (cf. [13, 3.18]) Let T ∈ T_{0}(λ, μ).

(i) Let E_{T} = ∏_{(i,k)∈ℕ×r} ∏_{res_{S}(i,k)≠res_{T}(i,k)^{S∈T_{0}(λ,μ)}} \frac{L_{i,k}^{μ} − res_{S}(i, k)}{res_{T}(i, k) − res_{S}(i, k)}.

(ii) Let ψ_{T} = φ_{T}E_{T}.

In the above definition we adopt the convention that empty products are 1. In particular, only finitely many terms are non–trivial in the definition of E_{T} because the second product is empty whenever L_{i,k}^{μ} = 0. We also do not need to specify the order of the terms in the product since all of the terms commute.

The main reason why we have assumed that R = 𝔽(q, Q_{1}, …, Q_{r}) is because of the following crucial Lemma. The Lemma is false for general R. Lemma 3.12. Suppose that R = 𝔽(q, Q_{1}, …, Q_{r}) and let S and T be distinct semistandard tableaux in T_{0}(λ, μ). Then res_{S}(i, k) ≠ res_{T}(i, k) for some (i, k) ∈ ℕ × r.

Proof. Since S ≠ T we may choose (i, k) ∈ ℕ × r minimal such that S_{i,k} ≠ T_{i,k}. Then res_{S}(i, k) ≠ res_{T}(i, k).

Standard arguments using Theorem 3.10 and Lemma 3.12 now prove the following (cf. [16, (3.4)–(3.11)] or [15, Prop. 3.35]).

Theorem 3.13. Suppose that R = 𝔽(q, Q_{1}, Q_{2}, …, Q_{r}) and let S, T ∈ T_{0}(λ, μ) and (i, k) ∈ ℕ × r. Then the following hold.

(i) ψ_{T}L_{i,k}^{μ} = res_{T}(i, k)ψ_{T}. (ii) E_{T}L_{i,k}^{μ} = res_{T}(i, k)E_{T}. (iii) ψ_{T} = φ_{T} + ∑_{S▷T} a_{S}φ_{S} for some a_{S} ∈ R. (iv) φ_{S}E_{T} = 0 if S ▷ T.

(v) ψ_{S}E_{T} = δ_{ST}ψ_{T}. (vi) { ψ_{T} | T ∈ T_{0}(λ, μ) } is an orthogonal basis of W_{μ}^{λ}.

Corollary 3.14. Let 𝔱 ∈ Std(λ) and suppose that there exists 𝔰 ∈ Std(λ) such that 𝔱 = 𝔰(i, i + 1) and 𝔰 ▷ 𝔱 for some i with 1 ≤ i < n. Then ψ_{𝔱} = ψ_{𝔰}(T_{i} + α) where α = \frac{(q−1) res_{𝔱}(i)}{res_{𝔰}(i)−res_{𝔱}(i)}.

Proof. By definition ψ_{𝔱} = φ_{𝔱}E_{𝔱} and φ_{𝔱} = φ_{𝔰}T_{i}. Furthermore, by assumption res_{𝔰}(i) = res_{𝔱}(i + 1) and res_{𝔰}(i + 1) = res_{𝔱}(i); consequently, E_{𝔰} + E_{𝔱} is symmetric in L_{i} and L_{i+1} and so T_{i} commutes with E_{𝔰} + E_{𝔱} by (2.1). Therefore,

ψ_{𝔱}( res_{𝔰}(i) − res_{𝔱}(i)) = φ_{𝔱}E_{𝔱}(L_{i+1} − res_{𝔱}(i)),  by (3.13)(i),
= φ_{𝔱}(E_{𝔰} + E_{𝔱})(L_{i+1} − res_{𝔱}(i)),  by (3.13)(ii),
= φ_{𝔰}T_{i}(E_{𝔰} + E_{𝔱})(L_{i+1} − res_{𝔱}(i))
= φ_{𝔰}(E_{𝔰} + E_{𝔱})T_{i}(L_{i+1} − res_{𝔱}(i)),  by (2.1)(iii),
= ψ_{𝔰}T_{i}(L_{i+1} − res_{𝔱}(i)),  by (3.13)(iv).

Now T_{i}L_{i+1} = (q − 1)L_{i+1} + L_{i}T_{i}, so

ψ_{𝔱}( res_{𝔰}(i) − res_{𝔱}(i)) = (q − 1) res_{𝔱}(i)ψ_{𝔰} + ( res_{𝔰}(i) − res_{𝔱}(i))ψ_{𝔰}T_{i}

and the result follows.

Our next aim is to compute the inner products ⟨ψ_{T}, ψ_{T}⟩. This will require a considerable amount of combinatorial machinery.

Recall the definition of the tableau T_{i,k} from (3.6)(i). We say that a node y ∉ [T_{i,k}^{#}] is an addable node of T_{i,k} if [T_{i,k}^{#}] ∪ {y} is the diagram of a multipartition. Similarly, a node y ∈ [T_{i,k}^{#}] is removable from T_{i,k} if [T_{i,k}^{#}] \ {y} is the diagram of a multipartition.

Given nodes x = (i, j, k) and y = (a, b, c) define x < y if k < c or k = c and j > b. Definition 3.15. (cf. [13, 2.8] and [7, 4.1]) Let T ∈ T_{0}(λ, μ) and for x ∈ [λ] suppose that T(x) = (i, k) and let (j, l) ∈ ℕ × r be maximal such that (j, l) < (i, k).

(i) Let A_{T}(x) = ∏_{y} ( res(x)−res(y)) where the product is over the addable nodes y of T_{i,k} such that x < y and an (i, k) can be added to T_{i,k} at y to give a semistandard tableau.

(ii) Let R_{T}(x) = ∏_{y} ( res(x) − res(y)) where the product is over the removable nodes y of T_{j,l} such that x < y and (i, k) does not appear in the column of T containing y.

(iii) Let γ_{T} = ∏_{x∈[λ]} \frac{A_{T}(x)}{R_{T}(x)}.

Example 3.16. As in Example 3.5, let λ = ((3, 1), (1)) and μ = ((2), (2, 1)). Then T_{0}(λ, μ) consists of the three tableaux

\begin{matrix}T_{1} = ( 1_{1} 1_{1} 1_{2} 1_{2} , 2_{2} ), T_{2} = ( 1_{1} 1_{1} 1_{2} 2_{2} , 1_{2} ) \\ and T_{3} = ( 1_{1} 1_{1} 2_{2} 1_{2} , 1_{2} ). \end{matrix}

Recall that we are currently assuming that q, Q_{1}, …, Q_{r} are indeterminates. We find that

γ_{T_{1}} = \{\frac{q^{2}Q_{1} − Q_{1}}{q^{2}Q_{1} − qQ_{1}} \underline{q^{2}Q_{1} − Q_{2}} \}^{\{\underline{q^{−1}Q_{1} − Q_{2}} \}} ,
γ_{T_{2}} = \{\frac{q^{2}Q_{1} − q^{−1}Q_{1}}{q^{2}Q_{1} − qQ_{1}} \underline{q^{2}Q_{1} − qQ_{2}} \} \{\underline{q^{−1}Q_{1} − qQ_{2}} \frac{q^{−1}Q_{1} − q^{−1}Q_{2}}{q^{−1}Q_{1} − Q_{2}} \} ,

and

γ_{T_{3}} = \{\frac{q^{2}Q_{1} − Q_{1}}{q^{2}Q_{1} − qQ_{1}} \frac{q^{2}Q_{1} − q^{−2}Q_{1}}{q^{2}Q_{1} − q^{−1}Q_{1}} \underline{q^{2}Q_{1} − qQ_{2}} \frac{q^{2}Q_{1} − q^{−1}Q_{2}}{q^{2}Q_{1} − Q_{2}} \}^{\{\underline{q^{−1}Q_{1} − qQ_{2}} \}} .

Inspection shows that γ_{T_{1}}γ_{T_{2}}γ_{T_{3}} ∈ 𝔽[q, q^{−1}, Q_{1}, …, Q_{r}]; cf. Corollary 3.29. Lemma 3.17. Let 𝔱 ∈ Std(λ) and suppose that there exists a tableau 𝔰 ∈ Std(λ) such that 𝔰 ▷ 𝔱 and 𝔱 = 𝔰(i, i + 1) for some i with 1 ≤ i < n. Then

γ_{𝔱} = \frac{( res_{𝔰}(i) − q res_{𝔱}(i))(q res_{𝔰}(i) − res_{𝔱}(i))}{( res_{𝔰}(i) − res_{𝔱}(i))^{2}}γ_{𝔰}.

Proof. The proof of this result is straightforward and essentially identical to [13, 2.11]. We leave the details to the reader.

If m is an integer, let [m]_{q} = 1 + q + ⋯ + q^{m−1} and {m}_{q} = [1]_{q}[2]_{q} … [m]_{q}. For a multicomposition ν, let {ν}_{q} = ∏_{k∈r} ∏_{i≥1}{ν_{i}^{(k)}}_{q}.

Given two rational functions f and g in R we write f ≃ g if f = q^{z}g for some integer z. Since q^{z} is always a unit in the rings we consider there is no loss in restricting our attention to R/≃. We extend this relation to elements of ℋ and 𝒮 in the obvious way. Lemma 3.18. Suppose that m_{λ} = u_{b}^{+}x_{λ}. Then γ_{𝔱^{λ}} ≃ res_{𝔱^{λ}}(b){λ}_{q}. Proof. Remove n from 𝔱^{λ} and apply induction (cf. [13, 2.10]).

Proposition 3.19. Suppose that 𝔱 is a standard λ–tableau. Then ⟨ψ_{𝔱}, ψ_{𝔱}⟩ ≃ γ_{𝔱}. Proof. First suppose that 𝔱 = 𝔱^{λ} and write m_{λ} = u_{b}^{+}x_{λ} as in Definition 2.2. Then ψ_{𝔱} = φ_{𝔱^{λ}} by Theorem 3.13(iii), so ⟨ψ_{𝔱}, ψ_{𝔱}⟩ = ⟨φ_{𝔱^{λ}}, φ_{𝔱^{λ}}⟩. By definition, ⟨φ_{𝔱^{λ}}, φ_{𝔱^{λ}}⟩φ_{T^{λ}}_{T^{λ}} ≡ φ_{T^{λ}}_{𝔱^{λ}}φ_{𝔱^{λ}}_{T^{λ}} mod 𝒮̄^{λ} and

φ_{T^{λ}}_{𝔱^{λ}}φ_{𝔱^{λ}}_{T^{λ}}(m_{λ}) = φ_{T^{λ}}_{𝔱^{λ}}(m_{λ}) = m_{λ}m_{λ} = m_{λ}x_{λ}u_{b}^{+} = {λ}_{q}m_{λ}u_{b}^{+} ≡ γ_{𝔱^{λ}}m_{λ} mod N̄^{λ}

by Corollary 3.8 and Lemma 3.18. Thus, ⟨φ_{𝔱^{λ}}, φ_{𝔱^{λ}}⟩ ≃ γ_{𝔱^{λ}} as required.

Now assume that 𝔱 ≠ 𝔱^{λ}. Then there exists a standard λ–tableau 𝔰 such that 𝔰 ▷ 𝔱 and 𝔰 = 𝔱(i, i + 1) where 1 ≤ i < n. By Corollary 3.14, ψ_{𝔱} = ψ_{𝔰}(T_{i} + α) where α = \frac{(q−1) res_{𝔱}(i)}{res_{𝔰}(i)−res_{𝔱}(i)}. Therefore, using the facts that T_{i}^{2} = (q − 1)T_{i} + q and that ψ_{𝔱} and ψ_{𝔰} are orthogonal, we see that

⟨ψ_{𝔱}, ψ_{𝔱}⟩ = ⟨ψ_{𝔰}T_{i} + αψ_{𝔰}, ψ_{𝔰}T_{i} + αψ_{𝔰}⟩
= ⟨ψ_{𝔰}T_{i}, ψ_{𝔰}T_{i}⟩ + 2α⟨ψ_{𝔰}T_{i}, ψ_{𝔰}⟩ + α^{2}⟨ψ_{𝔰}, ψ_{𝔰}⟩
= ⟨ψ_{𝔰}T_{i}^{2}, ψ_{𝔰}⟩ + 2α⟨ψ_{𝔰}T_{i}, ψ_{𝔰}⟩ + α^{2}⟨ψ_{𝔰}, ψ_{𝔰}⟩
= (2α + q − 1)⟨ψ_{𝔰}T_{i}, ψ_{𝔰}⟩ + (α^{2} + q)⟨ψ_{𝔰}, ψ_{𝔰}⟩
= (2α + q − 1)⟨ψ_{𝔱} − αψ_{𝔰}, ψ_{𝔰}⟩ + (α^{2} + q)⟨ψ_{𝔰}, ψ_{𝔰}⟩
= (q + α)(1 − α)⟨ψ_{𝔰}, ψ_{𝔰}⟩.

The reader will easily verify that (q − α)(1 + α) = \frac{(res_{𝔰}(i)−q res_{𝔱}(i))(q res_{𝔰}(i)−res_{𝔱}(i))}{(res_{𝔰}(i)−res_{𝔱}(i))^{2}}; so induction and Lemma 3.17 complete the proof.

Remark 3.20. The Proposition is really a statement about the orthogonal basis of the Specht module S^{λ}. Indeed, if we let f_{𝔱} = ψ_{𝔱}(m_{λ}) then { f_{𝔱} | 𝔱 ∈ Std(λ) } is an orthogonal basis of S^{λ} (see Proposition 2.17) and ⟨f_{𝔱}, f_{𝔱}⟩ ≃ γ_{𝔱} where ⟨ , ⟩ now denotes the standard inner product on S^{λ}. It follows that, up to a power of q, the determinant of the Gram matrix of S^{λ} is ∏_{𝔱∈Std(λ)} γ_{𝔱}.

In order to compute the inner products ⟨ψ_{T}, ψ_{T}⟩, for arbitrary semistandard tableaux T, we compare the homomorphisms ψ_{𝔱} and ψ_{T}. Lemma 3.21. Suppose that T ∈ T_{0}(λ, μ) and let 𝔱 = last(T). Then there exist b_{𝔰} ∈ R such that

ψ_{T}φ_{T^{μ}T^{ω}} = ψ_{𝔱} + ∑_{𝔰▷𝔱} b_{𝔰}ψ_{𝔰}.

Proof. By Theorem 3.13(iii), there exist a_{S} ∈ R such that

ψ_{T}φ_{T^{μ}T^{ω}}(1) = ψ_{T}(m_{μ}) = φ_{T}(m_{μ}) + ∑_{S▷T} a_{S}φ_{S}(m_{μ})

= m_{T} + ∑_{S▷T} a_{S}m_{S}

= m_{𝔱} + ∑_{𝔰▷𝔱} a_{𝔰}m_{𝔰}

for some a_{𝔰} ∈ R since 𝔱 = last(T). The Lemma now follows from parts (iii) and (vi) of Theorem 3.13.

Given a semistandard tableau T recall the multicomposition ν_{T} from Lemma 3.9. Lemma 3.22. Suppose that 𝔱 is a standard λ–tableau and let T = μ(𝔱). Write m_{μ} = u_{a}^{+}x_{μ}. Then there exist c_{S} ∈ R such that

φ_{𝔱}φ_{T^{ω}T^{μ}} = c_{T}φ_{T} + ∑_{S▷T} c_{S}φ_{S}

and where c_{T} ≃ res_{𝔱}(a){ν_{T}}_{q} if T is semistandard and c_{T} = 0 otherwise. Proof. We have

φ_{𝔱}φ_{T^{ω}T^{μ}}(m_{μ}) = φ_{𝔱}(m_{μ}) = φ_{𝔱}(1)m_{μ} = m_{𝔱}u_{a}^{+}x_{μ} = res_{𝔱}(a)m_{𝔱}x_{μ} + ∑_{𝔰▷𝔱} a_{𝔰}m_{𝔰}x_{μ}

for some a_{𝔰} ∈ R by Corollary 3.8. If T is semistandard then m_{T^{λ}𝔱}x_{μ} ≃ {ν_{T}}_{q}m_{T^{λ}T} by Lemma 3.9; cf. [13, 3.9]. On the other hand, if T is not semistandard then m_{T^{λ}𝔱}x_{μ} is a linear combination of terms m_{T^{λ}S} where S ▷ T by (2.9). Either way, if we define c_{T} as in the statement of the Lemma then

φ_{𝔱}φ_{T^{ω}T^{μ}}(m_{μ}) = c_{T}m_{T} + ∑_{S▷T} c_{S}m_{S}

for some c_{S} ∈ R since φ_{T^{λ}𝔱}φ_{T^{ω}T^{μ}}(m_{μ}) ∈ M^{λ} ∩ M^{μ}^{∗}. Hence, φ_{𝔱}φ_{T^{ω}T^{μ}} = c_{T}φ_{T} + ∑_{S▷T} c_{S}φ_{S} as required.

Definition 3.23. Suppose that 𝔱 is a standard λ–tableau such that T = μ(𝔱) is semistandard. We define

P_{𝔱}^{μ} = { (x, y) | x < y and 𝔱(x) < 𝔱(y) and T(x) = T(y) }

and

π_{𝔱}^{μ} = res_{𝔱}(a) ∏_{(x,y)∈P_{𝔱}^{μ}} \frac{q res(x) − res(y)}{res(x) − res(y)}

where m_{μ} = u_{a}^{+}x_{μ}. Example 3.24. Let λ, μ and T be as in Example 3.5 and let 𝔱_{1} = first(T) and 𝔱_{2} = last(T). Then

𝔱_{1} = ( 1 2 3 5 , 4 ), 𝔱_{2} = ( 1 2 4 5 , 3 ),

res_{𝔱_{1}}(a) = res_{𝔱_{2}}(a) = (Q_{1} − Q_{2})(qQ_{1} − Q_{2}) and

π_{𝔱_{1}}^{μ} = res_{𝔱_{1}}(a)\frac{q^{2}Q_{1} − Q_{1}}{qQ_{1} − Q_{1}} and π_{𝔱_{2}}^{μ} = res_{𝔱_{2}}(a)\frac{q^{2}Q_{1} − Q_{1}}{qQ_{1} − Q_{1}} \frac{q^{3}Q_{1} − Q_{2}}{q^{2}Q_{1} − Q_{2}} .

Recall that q, Q_{1}, …, Q_{r} are currently indeterminates. The reader may check that γ_{𝔱_{2}} ≃ π_{𝔱_{2}}^{μ}γ_{T} (use Example 3.16 where T = T_{2}). Lemma 3.25. Let T be a semistandard λ–tableau of type μ and write m_{μ} = u_{a}^{+}x_{μ}.

(i) If 𝔱 = first(T) then π_{𝔱}^{μ} = res_{𝔱}(a){ν_{T}}_{q}. (ii) If 𝔱 = last(T) then γ_{𝔱} ≃ π_{𝔱}^{μ}γ_{T}.

Proof. (i) Since 𝔱 = first(T), the elements of P_{𝔱}^{μ} are ordered pairs (x, y) such that x and y are in the same row of [λ] and T(x) = T(y). Therefore, the nodes in P_{𝔱}^{μ} contribute a factor of {ν_{T}}_{q} to π_{𝔱}^{μ} (cf. [13, 2.15]).

(ii) This is a routine exercise in induction (cf. [13, 2.16]).

Lemma 3.26. Let 𝔱 ∈ Std(λ) and suppose that there exists an integer i with 1 ≤ i < n and 𝔰 ∈ Std(λ) such that 𝔰 = 𝔱(i, i + 1), 𝔰 ▷ 𝔱 and μ(𝔰) = μ(𝔱) ∈ T_{0}(λ, μ). Then

π_{𝔱}^{μ} = \frac{q res_{𝔰}(i) − res_{𝔱}(i)}{res_{𝔰}(i) − res_{𝔱}(i)}π_{𝔰}^{μ}.

Proof. Let x and y be the nodes in the diagram of λ such that 𝔱(x) = i and 𝔱(y) = i + 1. Then 𝔰(x) = i + 1 and 𝔰(y) = i and, since 𝔰 ▷ 𝔱, we have x < y. Therefore, P_{𝔱}^{μ} = P_{𝔰}^{μ} ∪ {(x, y)} and the result follows.

Lemma 3.27. Suppose that 𝔱 is a standard λ–tableau such that T = μ(𝔱) is semistandard. Then ψ_{𝔱}φ_{T^{ω}T^{μ}} ≃ π_{𝔱}^{μ}ψ_{T}. Proof. We first show that ψ_{𝔱}φ_{T^{ω}T^{μ}} is an R–multiple of ψ_{T}. Let (i, k) ∈ ℕ × r and suppose that y, y + 1, …, z are the entries in row i of 𝔱^{μ^{(k)}}. Then, by Lemma 3.3 and Theorem 3.13(i),

ψ_{𝔱}φ_{T^{ω}T^{μ}}L_{i,k}^{μ} = ψ_{𝔱}(L_{y} + ⋯ + L_{z})φ_{T^{ω}T^{μ}} = res_{T}(i, k)ψ_{𝔱}φ_{T^{ω}T^{μ}}.

Hence, by Lemma 3.12, if S ∈ T_{0}(λ, μ) and S ≠ T then ψ_{𝔱}φ_{T^{ω}T^{μ}}E_{S} = 0. However, by Theorem 3.13(vi), ψ_{𝔱}φ_{T^{ω}T^{μ}} = ∑_{S∈T_{0}(λ,μ)} a_{𝔱S}ψ_{S} for some a_{𝔱S} ∈ R; so ψ_{𝔱}φ_{T^{ω}T^{μ}} = a_{𝔱T}ψ_{T} by Theorem 3.13(v). Hence, ψ_{𝔱}φ_{T^{ω}T^{μ}} is an R–multiple of ψ_{T} as claimed and it remains to show that a_{𝔱T} ≃ π_{𝔱}^{μ} for all 𝔱.

Suppose first that 𝔱 = first(T). By Theorem 3.13(iii) there exist b_{𝔰} ∈ R such that ψ_{𝔱} = φ_{𝔱} + ∑_{𝔰▷𝔱} b_{𝔰}φ_{𝔰}. Write m_{μ} = u_{a}^{+}x_{μ}. Then, by Lemma 3.22 there exist c_{S} ∈ R such that

ψ_{𝔱}φ_{T^{ω}T^{μ}} = φ_{𝔱}φ_{T^{ω}T^{μ}} + ∑_{𝔰▷𝔱} b_{𝔰}φ_{𝔰}φ_{T^{ω}T^{μ}} ≃ res_{𝔱}(a){ν_{T}}_{q}φ_{T} + ∑_{S▷T} c_{S}φ_{S}.

However, π_{𝔱}^{μ} ≃ res_{𝔱}(a){ν_{T}}_{q} by Lemma 3.25(i); so ψ_{𝔱}φ_{T^{ω}T^{μ}} ≃ π_{𝔱}^{μ}ψ_{T} by Theorem 3.13(iii) as required.

Finally, suppose that 𝔱 ≠ first(T). Then by Lemma 3.9 there exists a standard λ–tableau 𝔰 such that 𝔰 = 𝔱(i, i + 1), 𝔰 ▷ 𝔱 and μ(𝔰) = μ(𝔱). Now φ_{𝔱} = φ_{𝔰}T_{i} and φ_{T^{ω}T^{μ}}T_{i} = qφ_{T^{ω}T^{μ}} since (i, i + 1) ∈ 𝔖_{μ}. Once again by Corollary 3.14, ψ_{𝔱} = ψ_{𝔰}(T_{i} + α) where α = \frac{(q−1) res_{𝔱}(i)}{res_{𝔰}(i)−res_{𝔱}(i)}. Therefore,

ψ_{𝔱}φ_{T^{ω}T^{μ}} = ψ_{𝔰}(T_{i} + α)φ_{T^{ω}T^{μ}} = (q + α)ψ_{𝔰}φ_{T^{ω}T^{μ}} ≃ (\frac{q res_{𝔰}(i) − res_{𝔱}(i)}{res_{𝔰}(i) − res_{𝔱}(i)})π_{𝔰}^{μ}ψ_{T}

since ψ_{𝔰}φ_{T^{ω}T^{μ}} ≃ π_{𝔰}^{μ}ψ_{T} by induction. To complete the proof it remains to apply Lemma 3.26.

Theorem 3.28. Let T be a semistandard λ–tableau of type μ. Then ⟨ψ_{T}, ψ_{T}⟩ ≃ γ_{T}. Proof. Let 𝔱 = last(T). By Lemma 3.27, π_{𝔱}^{μ}ψ_{T} ≃ ψ_{𝔱}φ_{T^{ω}T^{μ}} and, by Lemma 3.21, there exist b_{𝔰} ∈ R such that ψ_{T}φ_{T^{μ}T^{ω}} = ψ_{𝔱} + ∑_{𝔰▷𝔱} b_{𝔰}ψ_{𝔰}. Therefore, using (2.13) and Theorem 3.13(vi),

π_{𝔱}^{μ}⟨ψ_{T}, ψ_{T}⟩ ≃ ⟨ψ_{T}, ψ_{𝔱}φ_{T^{ω}T^{μ}}⟩ = ⟨ψ_{T}φ_{T^{ω}T^{μ}}^{∗}, ψ_{𝔱}⟩ = ⟨ψ_{T}φ_{T^{μ}T^{ω}}, ψ_{𝔱}⟩

= ⟨ψ_{𝔱}, ψ_{𝔱}⟩ + ∑_{𝔰▷𝔱} b_{𝔰}⟨ψ_{𝔰}, ψ_{𝔱}⟩ = ⟨ψ_{𝔱}, ψ_{𝔱}⟩.

However, ⟨ψ_{𝔱}, ψ_{𝔱}⟩ ≃ γ_{𝔱} by Proposition 3.19 and γ_{𝔱} ≃ π_{𝔱}^{μ}γ_{T} by Lemma 3.25(ii), so the Theorem follows.

Corollary 3.29. Suppose that λ ∈ Λ^{+} and μ ∈ Λ. Then G_{μ}(λ) ≃ ∏_{T∈T_{0}(λ,μ)} γ_{T}.

Proof. By definition, G_{μ}(λ) = det (⟨φ_{S}, φ_{T}⟩), where S and T run over T_{0}(λ, μ). By Theorem 3.13, {ψ_{T}}_{T∈T_{0}(λ,μ)} is also a basis of W_{μ}^{λ} and the transition matrix between this basis and the semistandard basis is unitriangular.

Remark 3.30. By definition, G_{μ}(λ) ∈ 𝔽[q, q^{−1}, Q_{1} …, Q_{r}]; however, a priori, there is no reason why the rational function ∏_{T} γ_{T} should belong to this ring (cf. Example 3.16).

Now that we have computed G_{μ}(λ) we rewrite it in a more usable form. To do this we introduce beta numbers for multipartitions. Definition 3.31. Suppose that ν is a multipartition of n and let c ∈ {1, 2, …, rn} and write c = (r − k)n + j where j ∈ {1, 2 …, n}.

(i) Define column c of ν to be column j of ν^{(k)}. (ii) Let s(c) = k. (iii) Let β_{c} = n − j + ν_{j}^{(k)′}. Then the sequence

β = (β_{1}, …, β_{n}, β_{n+1}, …, β_{2n}, …, β_{(r−1)n+1}, …, β_{rn}),

is the sequence of beta numbers for ν.

Note that, in the sense of [11, p. 77], (β_{1} …, β_{n}) are beta numbers for ν^{(r)′}, the partition which is conjugate to ν^{(r)}, and (β_{n+1}, …, β_{2n}) are beta numbers for ν^{(r−1)′} and so on. Definition 3.32. Given a sequence of integers α = (α_{1}, α_{2}, …, α_{rn}) and a multicomposition τ of n, define the integer d_{τ} (α) as follows.

(i) If α_{b} = α_{c} for some b ≠ c with s(b) = s(c), or α_{c} < 0 for any c, then d_{τ} (α) = 0. (ii) If α does not satisfy (i) then there exists a unique multipartition ν and unique

elements w_{1}, …, w_{r} of 𝔖_{n} such that if β is the sequence of beta numbers for ν then

β = (α_{1w_{1}}, …, α_{nw_{1}}, α_{n+1w_{2}}, …, α_{n+nw_{2}}, …, α_{(r−1)n+nw_{r}}).

Define d_{τ} (α) = (−1)^{ℓ(w_{1})+···+ℓ(w_{r})}|T_{0}(ν, τ)|. Given a multipartition ν ∈ Λ^{+} we also define d_{τ} (ν) = |T_{0}(ν, τ)|.

Now fix λ ∈ Λ^{+} and μ ∈ Λ and let β = (β_{1}, …, β_{rn}) be the sequence of beta numbers for λ. Then d_{μ}(β) = |T_{0}(λ, μ)|.

Let (i, k) ∈ ℕ × r be maximal such that μ_{i}^{(k)} ≠ 0 and set z = μ_{i}^{(k)}. Then every semistandard tableau T ∈ T_{0}(λ, μ) has precisely z entries equal to (i, k) and these entries are at the feet of distinct columns of T. Indexing the columns of λ by 1, 2 …, rn as in (3.31)(ii), let the columns which contain an entry (i, k) be labelled by C = {c_{1} < c_{2} < ⋯ < c_{z}} and let λ^{C} be the multipartition of n − z whose sequence of beta numbers is

β^{C} = (β_{1}, …, β_{c_{1}} − 1, …, β_{c_{2}} − 1, …, β_{c_{z}} − 1, …, β_{rn}).

Then the tableau T̄ obtained from T by deleting all of the entries (i, k) is a semistandard λ^{C}–tableau of weight μ̄, where μ̄ is the multicomposition of n − z with μ̄_{j}^{(l)} = μ_{j}^{(l)}, if (j, l) ≠ (i, k), and μ̄_{i}^{(k)} = 0.

Since d_{μ}(β) = |T_{0}(λ, μ)|, by letting T range over the elements of T_{0}(λ, μ), the above argument shows that d_{μ}(β) = ∑_{C} dμ̄(β^{C}) where the sum is over all subsets of {1, 2, …, rn} with z elements. Similarly, with the help of Definition 3.32, for any sequence of integers α = (α_{1}, …, α_{rn}) we have that

(3.33)  d_{μ}(α) = ∑_{|C|=z^{C⊆{1,2,... ,rn}}} dμ̄(α^{C}),

where the sequence α^{C} is defined in the same way as β^{C}.

Now consider the Gram determinant G_{μ}(λ). Applying Corollary 3.29 and Definition 3.15,

G_{μ}(λ) ≃ ∏_{T∈T_{0}(λ,μ)} γ_{T} = ∏_{T∈T_{0}(λ,μ)} ∏_{x∈[λ]} \frac{A_{T}(x)}{R_{T}(x)}

= ∏_{T∈T_{0}(λ,μ)} \{ ∏_{T(x)≠(i,k)^{x∈[λ]}} \frac{A_{T}(x)}{R_{T}(x)} \}\{ ∏_{T(x)=(i,k)^{x∈[λ]}} \frac{A_{T}(x)}{R_{T}(x)} \}

≃ ∏_{|C|=z^{C⊆\{1,2,... ,rn\}}} Gμ̄(λ^{C}) × ∏_{T∈T_{0}(λ,μ)} ∏_{T(x)=(i,k)^{x∈[λ]}} \frac{A_{T}(x)}{R_{T}(x)}.

Let T ∈ T_{0}(λ, μ) and fix a node x ∈ [λ] with T(x) = (i, k). By definition, both R_{T}(x) and A_{T}(x) are products of the form ∏_{y} ( res(x) − res(y)). For each node y appearing in the product R_{T}(x) there exists a unique y′ in A_{T}(x) such that y′ is in the row below y; moreover, this gives a one–to–one correspondence between the factors of R_{T}(x) and the factors of A_{T}(x) which are indexed by nodes which are not in the first row of some component. Given such a triple x, y, y′, suppose that x is in column c of λ and that y and y′ are in columns b and b′ respectively. Then c > b ≥ b′, s(b) = s(b′) and A_{T}(x)/R_{T}(x) contains the factor

\frac{res(x) − res(y′)}{res(x) − res(y)} = \frac{q^{−β_{c}}Q_{s(c)} − q^{−β_{b′} −1}Q_{s(b)}}{q^{−β_{c}}Q_{s(c)} − q^{−β_{b}}Q_{s(b)}} ≃ ∏_{j=b′}^{b} \frac{q^{−β_{c}+1}Q_{s(c)} − q^{−β_{j}}Q_{s(b)}}{q^{−β_{c}}Q_{s(c)} − q^{−β_{j}}Q_{s(b)}} .

In this way, each column b < c which does not does not contain an entry (i, k), and such that n ∤ b, contributes a factor to A_{T}(x)/R_{T}(x). The restriction that b is not divisible by n is due to the fact if n | b and b < c then column b cannot contain a removable node. The columns b < c with n dividing b correspond to the as yet unaccounted for factors of A_{T}(x); thus such columns contribute a factor only to the numerator of γ_{T}. Hence, we obtain the following “branching rule” for G_{μ}(λ).

(3.34) G_{μ}(λ) ≃ ∏_{|C|=z^{C⊆\{1,2,... ,rn\}}} Gμ̄(λ^{C}) ∏_{c∈C} \{\frac{∏_{b∉C^{b<c}}(q^{−β_{c}+1}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})}{∏_{n∤b∉C^{b<c}}(q^{−β_{c}}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})}\}^{dμ̄(β^{C})} .

We are now ready to give our first reformulation of Corollary 3.29. For convenience, we let rn = {1, 2 …, rn} and given integers h ≥ 0 and c ∈ rn define

S(h, c) = { b ∈ rn | s(b) > s(c) or s(b) = s(c) and β_{b} ≥ β_{c} + h } .

Theorem 3.35. Let λ ∈ Λ^{+}, μ ∈ Λ and let β be the sequence of beta numbers for λ. Then

G_{μ}(λ) ≃ ∏_{h≥1} ∏_{c=1}^{rn} ∏_{b∈S(h,c)} (q^{−β_{c}+h}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{d_{μ}(β_{1},... ,β_{b}+h,... ,β_{c}−h,... ,β_{rn})}.

Proof. If n = 0 then G_{μ}(λ) = 1 and, by virtue of Definition 3.32, the right hand side is also 1. Consequently, we may assume by induction that ∏_{C} Gμ̄(λ^{C}) ≃ G_{1}G_{2}G_{3}G_{4} where

G_{1} =∏_{h≥1} ∏_{|C|=z^{C⊆rn}} ∏_{c∉C^{c∈rn}} ∏_{b∉C^{b∈S(h,c)}} (q^{−β_{c}+h}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{dμ̄(β_{1}^{C},... ,β_{b}^{C}+h,... ,β_{c}^{C}−h,... ,β_{rn}^{C})},

G_{2} =∏_{h≥1} ∏_{|C|=z−1^{C⊆rn}} ∏_{c∉C^{c∈rn}} ∏_{b∉C^{b∈S(h,c)}} (q^{−β_{c}+1+h}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{dμ̄(β_{1}^{C},... ,β_{b}^{C}+h,... ,β_{c}^{C}−1−h,... ,β_{rn}^{C})}

=∏_{h≥2} ∏_{|C|=z−1^{C⊆rn}} ∏_{c∉C^{c∈rn}} ∏_{b∉C^{b∈S(h,c)}} (q^{−β_{c}+h}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{dμ̄(β_{1}^{C},... ,β_{b}^{C}+h−1,... ,β_{c}^{C}−h,... ,β_{rn}^{C})},

G_{3} =∏_{h≥1} ∏_{|C|=z−1^{C⊆rn}} ∏_{c∉C^{c∈rn}} ∏_{b∉C^{b∈S(h,c)}} (q^{−β_{c}+h}Q_{s(c)} − q^{−β_{b}+1}Q_{s(b)})^{dμ̄(β_{1}^{C},... ,β_{b}^{C}−1+h,... ,β_{c}^{C}−h,... ,β_{rn}^{C})}

≃∏_{h≥0} ∏_{|C|=z−1^{C⊆rn}} ∏_{c∉C^{c∈rn}} ∏_{b∉C^{b∈S(h,c)}} (q^{−β_{c}+h}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{dμ̄(β_{1}^{C},... ,β_{b}^{C}+h,... ,β_{c}^{C}−h−1,... ,β_{rn}^{C})},

and

G_{4} =∏_{h≥1} ∏_{|C|=z−2^{C⊆rn}} ∏_{c∉C^{c∈rn}} ∏_{b∉C^{b∈S(h,c)}} (q^{−β_{c}+h}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{dμ̄(β_{1}^{C},... ,β_{b}^{C}−1+h,... ,β_{c}^{C}−1+h,... ,β_{rn}^{C})}.

Note that in all of the products above the sets S(h, c) should really depend upon β^{C}; however, the reader may check that the additional factors that this introduces (in G_{3} and G_{4}) all have exponent 0 and hence are trivial.

By our branching rule (3.34), G_{μ}(λ) ≃ B_{2}B_{3} ∏_{C} Gμ̄(λ^{C}) where

B_{2} = ∏_{|C|=z^{C⊆rn}} ∏_{b∉C^{b<c}}(q^{−β_{c}+1}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{dμ̄(β^{C})}

and

B_{3} = ∏_{|C|=z^{C⊆rn}} ∏_{n∤b∉C^{b<c}}(q^{−β_{c}}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{−dμ̄(β^{C})}.

Now if s(b) = s(c) then b < c if and only if β_{b} > β_{c} and this is if and only if β_{b} ≥ β_{c} + 1. Therefore,

B_{2}G_{2} =∏_{h≥1} ∏_{|C|=z−1^{C⊆rn}} ∏_{c∉C^{c∈rn}} ∏_{b∉C^{b∈S(h,c)}} (q^{−β_{c}+h}Q_{s(c)} − q^{−β_{b}}Q_{s(b)})^{dμ̄(β_{1}^{C},... ,β_{b}^{C}+h−1,... ,β_{c}^{C}−h...β_{rn}^{C})},

Similarly, we find that

B_{3}G_{3} ≃∏_{h≥1} ∏_{|C|=z−1^{C⊆rn}} ∏_{c∉C^{c∈rn}} ∏_{b∉C^{b∈S(h,c)}} (q^{−β_{c}+h}Q_{s(c)}−q^{−β_{b}}Q_{s(b)})^{dμ̄(β_{1}^{C},... ,β_{b}^{C}+h,... ,β_{c}^{C}−h−1,... ,β_{rn}^{C})}.

Note that the restriction that n ∤ b in B_{3} is not important because if n | b and b ∈ S(0, c) then β_{b} = 0 and so β_{c} − 1 < 0 and dμ̄(β^{C}) = 0; consequently, neither B_{3} nor G_{3} has a non–trivial factor indexed by column b.

Now G_{μ}(λ) ≃ G_{1}(B_{2}G_{2})(B_{3}G_{3})G_{4}; so using (3.33) we deduce the result.

Finally, we reinterpret Theorem 3.35 in terms of moving nodes in the diagram of λ. For each node x = (i, j, k) in [λ] let r_{x} denote the corresponding rim hook (in λ^{(k)}); see [10, §18]. Let ℓℓ(r_{x}) = λ_{i}^{(k)′} −i be the leg length of r_{x} and res(r_{x}) = res(f_{x}) where f_{x} is the foot node of r_{x} (that is, f_{x} is the last node in the column of λ^{(k)} which contains x).

Definition 3.36. Suppose that λ and ν are multipartitions in Λ^{+}. If λ ▷̸ ν let g_{λν} = 1; otherwise let g_{λν} be the element of ℚ(q, Q_{1}, …, Q_{r}) given by

g_{λν} = ∏_{x∈[λ]} ∏_{[ν]\r_{y}=[λ]\r_{x}^{y∈[ν]}} ( res(r_{x}) − res(r_{y}))^{ε_{xy}},

where ε_{xy} = (−1)^{ℓℓ(r_{x})+ℓℓ(r_{y})}.

Remark 3.37. These functions are not as complicated as their definition suggests. First, note that g_{λν} = 1 unless λ^{(k)} = ν^{(k)} for all but at most two k ∈ r. If g_{λν} ≠ 1 and λ^{(k)} ≠ ν^{(k)} and λ^{(l)} ≠ ν^{(l)} for k ≠ l, then g_{λν} ≃ (q^{d}Q_{k} − Q_{l})^{±1} for some integer d with −n < d < n. If λ and ν differ only on the kth component then g_{λν} ≃ (q^{a}Q_{k} − Q_{k})/(q^{b}Q_{k} − Q_{k}) ≃ [a]_{q}/[b]_{q} for some integers a and b in {1, 2, …, n}; cf. [13, 2.30].

Given an integral domain R containing parameters q̂, Q̂_{1}, …, Q̂_{r} we can consider g_{λν} as an element of the field of fractions of R by evaluating the indeterminates in g_{λν} appropriately and replacing 1 by the identity of R.

Corollary 3.38. Let R be an integral domain. Then

G_{μ}(λ) ≃ ∏_{ν∈Λ^{+}}(g_{λν})^{d_{μ}(ν)},

considered as an element of R.

Proof. By general arguments, if R is an integral domain then the Gram determinant G_{μ}(λ) can be computed by evaluating the polynomial ∏_{T} γ_{T} at appropriate values of the indeterminates. Thus it suffices to consider the case where R = 𝔽[q, q^{−1}, Q_{1}, …, Q_{r}].

Consider the expression we have obtained for G_{μ}(λ) in Theorem 3.35. As in the proof of [13, 2.30], the only time that the exponent d_{μ}(β_{1}, …, β_{b} + h, …, β_{c} − h, …, β_{rn}) is non–zero is when a rim hook with foot node in column c of λ can be moved to a rim hook with foot node in column b. Hence G_{μ}(λ) can be expressed as in the statement of the Corollary.

This formula for G_{μ}(λ) is the nicest of the three we have obtained. It is not hard to apply; for each node x ∈ [λ] we have to move the rim hook r_{x} down in the diagram in all possible ways.

Example 3.39. Consider once more the case where λ = ((3, 1), (1)) and μ = ((2), (2, 1)). In the diagrams below, we label the nodes x and y, their rim hooks, and we have circled the corresponding foot nodes. We list the factors res(r_{x}) − res(r_{y}) rather than the functions (g_{λν})^{d_{μ}(ν)}. In the last column we have given the

CYCLOTOMIC q–SCHUR ALGEBRAS 19

numbers d_{μ}(ν) = |T_{0}(ν, μ)|.

(  x ◯ ,  ) ⟶ ( y ◯ ,  )  (q^{2}Q_{1} − Q_{1})^{d_{μ}((2^{2}),(1))}  2
( y ◯ ,  )  (q^{2}Q_{1} − q^{−2}Q_{1})^{d_{μ}((2,1^{2}),(1))} 1
(  , y ◯ )  (q^{2}Q_{1} − qQ_{2})^{d_{μ}((2,1),(2))}  2
(  , y ◯ )  (q^{2}Q_{1} − q^{−1}Q_{2})^{d_{μ}((2,1),(1^{2}))} 1
( x ◯ × ,  ) ⟶ ( y ; × ◯ ,  )  (qQ_{1} − Q_{1})^{−d_{μ}((2^{2}),(1))}  2
( y ; × ◯ ,  )  (qQ_{1} − q^{−3}Q_{1})^{−d_{μ}((1^{4}),(1))}  0
( , y ◯ × )  (qQ_{1} − qQ_{2})^{d_{μ}((1^{2}),(3))}  0
( , y ; × ◯ )  (qQ_{1} − q^{−2}Q_{2})^{−d_{μ}((1^{2}),(1^{3}))} 0
( x × × ; × ◯ ,  ) ⟶ ( y × ; × ; × ◯ ,  )  (q^{−1}Q_{1} − q^{−2}Q_{1})^{−d_{μ}((2,1^{2}),(1))} 1
( y ; × ; × ; × ◯ ,  )  (q^{−1}Q_{1} − q^{−3}Q_{1})^{d_{μ}((1^{4}),(1))} 0
( ∅ , y ◯ × × × ) (q^{−1}Q_{1} − qQ_{2})^{−d_{μ}((0),(5))}  0
( ∅ , \begin{matrix}y & × & × \\ × ◯ & × \end{matrix} )  (q^{−1}Q_{1} − q^{−1}Q_{2})^{d_{μ}((0),(3,2))} 0
( ∅ , y × ; × × ; × ◯ )  (q^{−1}Q_{1} − q^{−2}Q_{2})^{−d_{μ}((0),(2^{2},1))} 0
( ∅ , y ; × ; × ; × ◯ )  (q^{−1}Q_{1} − q^{−4}Q_{2})^{d_{μ}((0),(1^{5}))} 0
( x ◯  ,  ) ⟶ (  , y ◯ ) (q^{−1}Q_{1} − qQ_{2})^{d_{μ}((3),(2))}  2
(  , y ◯ )  (q^{−1}Q_{1} − q^{−1}Q_{2})^{d_{μ}((3),(1^{2}))} 1

If it is raining, the reader might like to check that the answer this gives for G_{μ}(λ) agrees with that of Example 3.16 (it does).

Motivated by the theory of Coxeter groups, we make our next definition. Definition 3.40. The Poincaré polynomial of the generic Ariki–Koike algebra ℋ is the element of ℤ[q, q^{−1}, Q_{1}, …, Q_{r}] given by

P (q; Q_{1}, …, Q_{r}) = {n}_{q} ∏_{1≤k<l≤r} ∏_{−n<d<n}(q^{d}Q_{k} − Q_{l}).

If R is an integral domain let P_{R}(q; Q_{1}, …, Q_{r}) denote the corresponding specialization of P (q; Q_{1}, …, Q_{r}).

Note that it is possible that P_{R}(q; Q_{1}, …, Q_{r}) ≠ 0 even if Q_{k} = 0 for some k ∈ r. It is for this reason that we had to introduce the rational functions g_{λν} above. Corollary 3.41. Suppose that R is an integral domain such that P_{R}(q; Q_{1}, …, Q_{r}) is a non–zero element of R. Then G_{μ}(λ) ≠ 0 for all λ ∈ Λ^{+} and all μ ∈ Λ. Proof. By Remark 3.37, for all λ and μ, each factor of the polynomial g_{λν} divides P (q; Q_{1}, …, Q_{r}); hence the result.

Suppose that R is a field and that Λ^{+} is the set of all multipartitions of n. Then it is easy to see that the converse of Corollary 3.41 is true. Furthermore, G_{μ}(λ) ≠ 0 for all λ and μ if and only if each Weyl module is irreducible, so this is equivalent to 𝒮 being semisimple by [8, 3.8]. However, 𝒮 is semisimple if and only if ℋ is semisimple by Proposition 2.17; so we see that ℋ is semisimple if and only if P_{R}(q; Q_{1}, …, Q_{r}) ≠ 0. Thus we recover the main result of [2].

We also note that with a little more care the proof of Lemma 3.12, and hence Theorem 3.13, goes through for any field R provided that q ≠ 1, P_{R}(q; Q_{1}, …, Q_{r}) ≠ 0 and Q_{k} ≠ 0 for all k ∈ r.

4. The sum formula and irreducibility

We now apply the results of the previous section to describe the Jantzen filtration of the Weyl modules W^{λ} and the Specht modules S^{λ}. First we need some preparation.

Throughout this section we assume that R is a principal ideal domain and that 𝔭 is a prime in R. Let 𝔽 = R/𝔭R. Then given an R–module U_{R} its reduction modulo 𝔭 is the 𝔽–module U_{𝔽} = (U_{R} + 𝔭U_{R})/𝔭U_{R} ≅ U_{R} ⊗_{R} 𝔽.

Suppose that U_{R} is a free R–module of finite rank equipped with a symmetric bilinear form ⟨ , ⟩. For each i ≥ 0 let

U_{R}(i) = { u ∈ U_{R} | 𝔭^{i} divides ⟨u, u′⟩ for all u′ ∈ U_{R} } .

The Jantzen filtration of the 𝔽–module U_{𝔽} is

U_{𝔽} = U_{𝔽}(0) ⊇ U_{𝔽}(1) ⊇ ⋯

where U_{𝔽}(i) = U_{R}(i) ⊗_{R} 𝔽 for all i ≥ 0. Since U_{𝔽} is finite dimensional, U_{𝔽}(i) = 0 for all sufficiently large i.

Let e_{1}, e_{2}, …, e_{N} be an R–basis of U_{R} and let G = det (⟨e_{i}, e_{j}⟩) be the determinant of the associated Gram matrix (an element of R).

Let ν_{𝔭}: R^{×} ⟶ℕ be the 𝔭–adic valuation map. Then an argument due to Jantzen proves the following. 4.1. [14, Lemma 3] Suppose that the bilinear form ⟨ , ⟩ is non–degenerate. Then

ν_{𝔭}(G) = ∑_{i>0} dim_{𝔽}U_{𝔽}(i).

Fix q̂, Q̂_{1}, …, Q̂_{r} ∈ R and let q = q̂+ 𝔭R, Q_{1} = Q̂_{1} + 𝔭R, …, Q_{r} = Q̂_{r} + 𝔭R be their canonical images in 𝔽. Let 𝒮_{R} be the associated cyclotomic q̂–Schur algebra over R with parameters q̂, Q̂_{1}, …, Q̂_{r}. Then its reduction modulo 𝔭, 𝒮_{𝔽} = 𝒮_{R}⊗_{R}𝔽, is the cyclotomic q–Schur algebra with parameters q, Q_{1}, …, Q_{r}. If λ ∈ Λ^{+} then W_{R}^{λ} will denote the corresponding Weyl module of 𝒮_{R} and W_{𝔽}^{λ} ≅ W_{R}^{λ} ⊗_{R} 𝔽 the Weyl module of 𝒮_{𝔽}.

As we did in [13, §3], given a right 𝒮_{𝔽}–module U_{𝔽} and integers a_{λ} ∈ ℤ we write

U_{𝔽} ⟷ ∑_{λ∈Λ^{+}} a_{λ}W_{𝔽}^{λ} if

U_{𝔽} ⊕ ⨁_{a_{λ}<0^{λ∈Λ^{+}}}(−a_{λ})W_{𝔽}^{λ} and ⨁_{a_{λ}>0^{λ∈Λ^{+}}} a_{λ}W_{𝔽}^{λ}

have the same composition factors. Using the cellular structure of 𝒮_{𝔽} we obtain the following result. Lemma 4.2. Suppose that U_{𝔽} is an 𝒮_{𝔽}–module such that for all μ ∈ Λ^{+} we have

dim U_{𝔽}φ_{T^{μ}T^{μ}} = ∑_{λ∈Λ^{+}} a_{λ} dim W_{𝔽}^{λ}φ_{T^{μ}T^{μ}}

for some a_{λ} ∈ ℤ. Then U_{𝔽} ⟷ ∑_{λ∈Λ^{+}} a_{λ}W_{𝔽}^{λ}.

Let R_{f} be the field of fractions of R and extend ν_{𝔭} to a map R_{f}^{×} ⟶ ℤ in the natural way. Recall the rational functions g_{λν} ∈ R_{f} and P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) from the end of the previous section. We can now state one of our main results. Theorem 4.3. Let λ ∈ Λ^{+} and suppose that P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) ≠ 0. Then

∑_{i>0} W_{𝔽}^{λ}(i) ⟷ ∑_{λ▷ν^{ν}} ν_{𝔭}(g_{λν})W_{𝔽}^{ν}.

Remark 4.4. Note that we may omit the condition that λ dominates ν from the second sum, since ν_{𝔭}(g_{λν}) = 0 unless λ ▷ ν; however we have included this condition to emphasise that only these multipartitions matter. Proof. Suppose that μ ∈ Λ^{+} with λ ⊵ μ and recall that G_{μ}(λ) is the Gram determinant of the μ–weight space W_{R}^{λ}φ_{T^{μ}T^{μ}} with respect to the semistandard basis. Because P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) ≠ 0 we know by Corollary 3.41 that G_{μ}(λ) ≠ 0 in R. Therefore, we can apply (4.1) to W_{R}^{λ}φ_{T^{μ}T^{μ}} to deduce that

ν_{𝔭}(G_{μ}(λ)) = ∑_{i>0} dim_{𝔽} W_{𝔽}^{λ}φ_{T^{μ}T^{μ}}(i).

Since d_{μ}(λ) = dim_{𝔽} W_{𝔽}^{λ}φ_{T^{μ}T^{μ}}, the result follows by Corollary 3.38 and Lemma 4.2.

As our first application of this result we describe the irreducible Weyl modules. Recall that if λ ∈ Λ^{+} then F_{𝔽}^{λ} = W_{𝔽}^{λ}/ rad W_{𝔽}^{λ}.

Corollary 4.5. Let λ ∈ Λ^{+} and suppose that P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) ≠ 0. Then W_{𝔽}^{λ} is irreducible if and only if ν_{𝔭}(g_{λν}) = 0 for all multipartitions ν ∈ Λ^{+} such that λ ▷ ν.

Proof. Because rad W_{𝔽}^{λ} = W_{𝔽}^{λ}(1), the Weyl module W_{𝔽}^{λ} is irreducible if and only if W_{𝔽}^{λ}(1) = (0). If there exists a multipartition ν ∈ Λ^{+} with ν_{𝔭}(g_{λν}) ≠ 0 then we may assume that ν is minimal in the dominance ordering with this property. Then F_{𝔽}^{ν} is a composition factor of the right hand side of Theorem 4.3 (in particular, we must have ν_{𝔭}(g_{λν}) > 0); consequently, W_{𝔽}^{λ}(1) ≠ (0) and so W_{𝔽}^{λ} is reducible.

Conversely, if ν_{𝔭}(g_{λν}) = 0 for all ν ∈ Λ^{+} such that λ ▷ ν then rad W_{𝔽}^{λ} = (0) by Theorem 4.3; hence, W_{𝔽}^{λ} is irreducible.

We extend the relation ⟷ to ℋ_{𝔽}–modules in the obvious way. By considering the case where Λ^{+} is the set of all multipartitions and using Proposition 2.17 we obtain the following.

Theorem 4.6. Let λ be a multipartition of n and suppose that P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) is a non–zero element of R. Then

∑_{i>0} S_{𝔽}^{λ}(i) ⟷ ∑_{λ▷ν^{ν}} ν_{𝔭}(g_{λν})S_{𝔽}^{ν}.

As in Corollary 4.5, Theorem 4.6 automatically gives sufficient conditions for S_{𝔽}^{λ} = D_{𝔽}^{λ}. In order to obtain necessary conditions we need to work a little harder. Theorem 4.7. Suppose that λ ∈ Λ^{+}.

(i) If S_{𝔽}^{λ} = D_{𝔽}^{λ} then W_{𝔽}^{λ} is irreducible. (ii) Suppose W_{𝔽}^{λ} is irreducible, P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) ≠ 0 and that every multiparti-

tion of n is contained in Λ^{+}. Then S_{𝔽}^{λ} = D_{𝔽}^{λ}. (iii) Suppose that P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) ≠ 0. Then S_{𝔽}^{λ} = D_{𝔽}^{λ} if and only if ν_{𝔭}(g_{λν}) = 0

for all multipartitions ν of n such that λ ▷ ν.

Remark 4.8. Notice that when D_{𝔽}^{λ} ≠ (0) part (iii) gives necessary and sufficient conditions for S_{𝔽}^{λ} to be irreducible; however, it can happen that S_{𝔽}^{λ} is irreducible when D_{𝔽}^{λ} = (0). Even for the symmetric group (that is, the case where r = 1 and q = 1), the complete classification of the irreducible Specht modules S_{𝔽}^{λ} is an open problem (except in characteristic 2; see [12]). If ω ∉ Λ^{+} then there also exist examples where W_{𝔽}^{λ} is irreducible and S_{𝔽}^{λ} ≠ D_{𝔽}^{λ}.

Proof. (i) Suppose that S_{𝔽}^{λ} = D_{𝔽}^{λ}. Then rad S_{𝔽}^{λ} = (0), so the Gram determinant of S_{𝔽}^{λ}, with respect to its standard basis { m_{𝔱} | 𝔱 ∈ Std(λ) } must be a non–zero element of 𝔽. However, by Remark 3.20 and Corollary 3.38 this determinant is equal to

G_{ω}(λ) + 𝔭R ≃ ∏_{ν∈Λ^{+}}(g_{λν})^{d_{ω}(ν)} + 𝔭R.

Hence, S_{𝔽}^{λ} = D_{𝔽}^{λ} if and only if (g_{λν})^{d_{ω}(ν)} ∉ 𝔭R for all multipartitions ν of n. On the other hand the Gram determinant of W_{𝔽}^{λ}, with respect to its semistandard basis, is equal to

∏_{μ∈Λ} G_{μ}(λ) + 𝔭R ≃ ∏_{ν∈Λ^{+}} ∏_{μ∈Λ}(g_{λν})^{d_{μ}(ν)} + 𝔭R.

Therefore, W_{𝔽}^{λ} = F_{𝔽}^{λ} if and only if (g_{λν})^{d_{μ}(ν)} ∉ 𝔭R for all ν ∈ Λ^{+} and all μ ∈ Λ. However, it is easy to see that if ν ∈ Λ^{+} and μ ∈ Λ then d_{μ}(ν) ≠ 0 only if d_{ω}(ν) ≠ 0 so the result follows.

(ii) Suppose that W_{𝔽}^{λ} is irreducible and let ν be a multipartition of n such that D_{𝔽}^{ν} ≠ (0) and (S_{𝔽}^{λ} : D_{𝔽}^{ν}) > 0. By Proposition 2.17, (W_{𝔽}^{λ} : F_{𝔽}^{ν}) = (S_{𝔽}^{λ} : D_{𝔽}^{ν}) > 0. However, W_{𝔽}^{λ} is irreducible, so ν = λ and S_{𝔽}^{λ} = D_{𝔽}^{λ} as claimed.

Part (iii) now follows from parts (i), (ii) and Corollary 4.5.

The last four results apply to fields 𝔽 of the form R/𝔭 where R is a principal ideal domain, 𝔭 is a prime in R and P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) ≠ 0. At first sight the requirement that P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) ≠ 0 appears to be very restrictive; however, given any field 𝔽 containing 0 ≠ q, Q_{1}, …, Q_{r} we can always find suitable R and 𝔭.

Notation 4.9. Fix a field 𝔽 containing elements q, Q_{1}, …, Q_{r}, with q ≠ 0, and let 𝒮_{𝔽} be the cyclotomic q–Schur algebra over 𝔽 with these parameters. Let R = 𝔽[q̂], where q̂ is an indeterminate over 𝔽, and 𝔭 = q̂− q. Define 𝒮_{R} to be the cyclotomic q̂–Schur algebra over R with parameters q̂, Q̂_{1}, …, Q̂_{r} where

Q̂_{k} = \begin{cases}Q_{k}(q̂− q + 1)^{kn}, & if Q_{k} ≠ 0, \\ (q̂− q)^{kn}, & if Q_{k} = 0. \end{cases}

Let ℋ_{R} and ℋ_{𝔽} be the associated Ariki–Koike algebras.

Notice that R is a principal ideal domain, 𝔭 is prime in R and 𝔽 ≅ R/𝔭. Moreover, if π: R ⟶ 𝔽 is the canonical projection then π(q̂) = q and π(Q̂_{k}) = Q_{k} for k = 1, 2 …, r. Therefore, 𝒮_{𝔽} ≅ 𝒮_{R} ⊗_{R} 𝔽 and ℋ_{𝔽} ≅ ℋ_{R} ⊗_{R} 𝔽. Finally, P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) ≠ 0 because by construction every factor of P_{R}(q̂; Q̂_{1}, …, Q̂_{r}) is non–zero. Thus we are in a situation where we can apply Theorem 4.3 and its corollaries. Note that the Jantzen filtrations of the Weyl modules and Specht modules depend upon R and 𝔭 rather than on 𝔽.

In [13, Theorem 4.19] we gave a purely combinatorial classification of those partitions λ such that W_{𝔽}^{λ} is irreducible (this is the case r = 1). We build upon this to give such a criterion for the general case.

In addition to using the notation of (4.9), we write res_{R} and res_{𝔽} for residues in the rings R and 𝔽 respectively. We also write 𝒮_{𝔽}^{(1)}(n) for the q–Schur algebra 𝒮_{𝔽}(Λ) where Λ is the set of all partitions of n and Q_{1} = 1 (that is, the case r = 1).

Theorem 4.10. Let λ = (λ^{(1)}, …, λ^{(r)}) ∈ Λ^{+} be a multipartition of n. Then W_{𝔽}^{λ} is reducible if and only if

(i) for some k ∈ r the 𝒮_{𝔽}^{(1)}(n_{k})–module W_{𝔽}^{λ^{(k)}} is reducible where n_{k} = |λ^{(k)}|; or, (ii) for some ν in Λ^{+} there exist x = (i, j, k) ∈ [λ] and y = (a, b, c) ∈ [ν] such that

c > k, [λ] \ r_{x} = [ν] \ r_{y} and res_{𝔽}(r_{x}) = res_{𝔽}(r_{y}).

Remark 4.11. Less formally, condition (ii) says that it is possible to unwrap a rim hook from [λ] and wrap it back on to a later component without changing the residue of the foot node.

Proof. By Corollary 4.5, W_{𝔽}^{λ} is reducible if and only if there exists a multipartition ν ∈ Λ^{+} such that ν_{𝔭}(g_{λν}) ≠ 0. By Remark 3.37, ν_{𝔭}(g_{λν}) ≠ 0 if and only if λ ▷ ν and there exist x = (i, j, k) ∈ [λ] and y = (a, b, c) ∈ [ν] such that c ≥ k, [λ]\r_{x} = [ν]\r_{y} and 𝔭 divides res_{R}(r_{x}) − res_{R}(r_{y}). Now, 𝔭 = q̂− q, so 𝔭 divides res_{R}(r_{x}) − res_{R}(r_{y}) if and only if res_{𝔽}(r_{x}) = res_{𝔽}(r_{y}). By [13, Theorem 4.15], W_{𝔽}^{λ^{(k)}} is reducible if and only if there exists a partition μ of n_{k} (where n_{k} = |λ^{(k)}|) such that λ^{(k)} dominates μ and there exists x ∈ [λ^{(k)}] and y ∈ [μ] with [λ^{(k)}] \ r_{k} = [μ] \ r_{y} and res_{𝔽}(r_{x}) = res_{𝔽}(r_{y}). The Theorem now follows.

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Imperial College, Queen’s Gate, London SW7 2BZ. U.K.

E-mail address: g.james@ic.ac.uk

University of Sydney, Sydney N.S.W. 2006. Australia.

E-mail address: a.mathas@maths.usyd.edu.au