EQUATING DECOMPOSITION NUMBERS FOR DIFFERENT PRIMES

GORDON JAMES AND ANDREW MATHAS

Abstract. This paper shows that certain decomposition numbers for the Hecke algebras and q–Schur algebras at different roots of unity in characteristic zero are equal. To prove our results we first establish the corresponding theorem for the canonical basis of the level one Fock space and then apply deep results of Ariki and Varagnolo and Vasserot.

1. Introduction. Throughout this note we adopt the standard notation for the modular representation theory of the symmetric groups, as can be found in [?, ?].

Consider the following two submatrices of the p–modular decomposition matrices of the symmetric groups 𝔖_{n}.

\begin{aligned}10, 1 & 1 \\ 9, 2 & 1 & 1 \\ 7, 4 & . & 1 & 1 \\ 6, 5 & . & . & 1 & 1 \\ 7, 2^{2} & 1 & 1 & 1 & . & 1 \\ 4^{2}, 3 & . & . & 1 & 1 & 1 & 1 \\ 6, 2^{2}, 1 & 1 & 1 & 1 & 1 & 1 & . & 1 \\ 4^{2}, 2, 1 & . & 1 & 1 & 2 & 1 & 1 & 1 & 1 \\ & n = 11 and p = 3 \end{aligned}  \begin{aligned}18, 3 & 1 \\ 17, 4 & 1 & 1 \\ 13, 8 & . & 1 & 1 \\ 12, 9 & . & . & 1 & 1 \\ 13, 4^{2} & 1 & 1 & 1 & . & 1 \\ 8^{2}, 5 & . & . & 1 & 1 & 1 & 1 \\ 12, 4^{2}, 1 & 1 & 1 & 1 & 1 & 1 & . & 1 \\ 8^{2}, 4, 1 & . & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ & n = 21 and p = 5 \end{aligned}

The two matrices are identical except for the labelling and the two bold faced entries (omitted entries are zero).

This paper was motivated by our attempts to compute the bold faced entry in the matrix above for n = 21 and p = 5. At the outset we knew that this number was either 1 or 2; however, we were unable to determine which of these possibilities was correct. Lübeck and Müller [?] have shown that this multiplicity is equal to 1 using computer calculations; see [?, §5.3] for details.

Note that the partitions (4^{2}, 2, 1) and (8^{2}, 4, 1) both have p–weight 3 and p–core (p − 2, p − 2) for p = 3 and p = 5 respectively. Martin and Russell [?] have claimed that all of the p–modular decomposition numbers of the symmetric group of p– weight 3 are 0 or 1 when p > 3; unfortunately, their proof contains a gap when dealing with partitions with p–core (p − 2, p − 2). This particular case is still open when p > 5; as a consequence, the claim in [?] that the decomposition numbers are always 0 or 1 for partitions of p–weight 3 when p > 3 is in doubt.

In this paper we prove a general theorem which indicates why the matrices above are very similar. This result is not about the decomposition matrices of the symmetric groups but rather about the closely related decomposition matrices of the Iwahori–Hecke algebras ℋ_{ℂ,q}(𝔖_{n}) of the symmetric group at a complex root of unity. Let q ∈ ℂ be a primitive e^{th} root of unity, where e ≥ 2, and let q′ ∈ ℂ be a primitive e′^{th} root of unity where e′ > e. Our result shows that certain decomposition numbers of ℋ_{ℂ,q}(𝔖_{n}) and ℋ_{ℂ,q′}(𝔖_{m}) are equal for specified m > n. In particular, for the Hecke algebras all of the decomposition multiplicities above for (n, e) = (11, 3) and (m, e′) = (21, 5) are equal (in the Hecke algebra case the multiplicities d_{(4^{2},2,1),(6,5)}, when e = 3, and d_{(8^{2},4,1),(12,9)}, when e = 5, are both equal to 1).

2000 Mathematics Subject Classification. 20C08, 20C30, 33D80.

In section 4 we also show that the corresponding result holds for the q–Schur algebras in characteristic zero.

2. Abacuses and the Iwahori–Hecke algebra. In order to state our results we recall the abacus notation for partitions introduced in [?]. Fix an integer e ≥ 2. An e–abacus is an abacus with e runners, which we label from left to right as ρ_{0}, …, ρ_{e−1}. Number the bead positions on the abacus by 0, 1, 2, … reading from left to right and then top to bottom; so the bead positions on ρ_{r} are numbered r + me for m ≥ 0. We order the beads on a given e–abacus according to their bead positions.

Let λ = (λ_{1}, λ_{2}, …) be a partition. The length of λ is the smallest integer ℓ(λ) such that λ_{i} = 0 for all i > ℓ(λ). If k ≥ ℓ(λ) then λ has a (unique) e–abacus representation with k beads; namely, the e–abacus with beads at positions

λ_{k}, λ_{k−1} + 1, …, λ_{2} + k − 2, λ_{1} + k − 1.

The bead positions on an abacus for λ encode the first column hook lengths, so this gives a natural bijection between the abacuses with k beads and the partitions of length at most k. For our purposes it is important that the m^{th} bead on an abacus for λ corresponds to row l = k − m + 1 of λ (row l of λ is empty if l > ℓ(λ)).

In what follows we fix k ≥ 0 and (with few exceptions) consider only abacuses with k beads; or, equivalently, partitions of length at most k.

We want to compare e–abacuses with (e + 1)–abacuses. Fix an integer α with 0 ≤ α ≤ e. If λ is a partition with ℓ(λ) ≤ k then λ can be represented on an e– abacus with k beads. Let λ^{+} be the partition corresponding to the (e + 1)–abacus obtained by inserting an empty runner before ρ_{α} in the e–abacus for λ (if α = e we insert an empty e^{th} runner). Let ρ_{0}^{+}, …, ρ_{e}^{+} be the runners of the (e + 1)–abacus of λ^{+}; then ρ_{r}^{+} = ρ_{r} if r < α, ρ_{α}^{+} is empty, and ρ_{r}^{+} = ρ_{r−1} if r > α.

Although our notation does not reflect this the partition λ^{+} does depend upon both the choice of α and the choice of k.

2.1. Example. Suppose that e = 3, k = 4 and α = 2. Let λ = (4^{2}, 3). Then the abacuses (with 4 beads) for λ, λ^{+} and λ^{++} = (λ^{+})^{+} are as follows.

\begin{matrix}\begin{aligned}& e = 3 \\ 2 & 0 & 1 \\ • & · & · \\ · & • & · \\ • & • & · \\ · & · & · \end{aligned} & \begin{aligned}& e′ = 4 \\ 0 & 1 & 2 & 3 \\ • & · & · & · \\ · & • & · & · \\ • & • & · & · \\ · & · & · & · \end{aligned} & \begin{aligned}& e′′ = 5 \\ 1 & 2 & 3 & 4 & 0 \\ • & · & · & · & · \\ · & • & · & · & · \\ • & • & · & · & · \\ · & · & · & · & · \end{aligned} \\ λ = (4^{2}, 3) & λ^{+} = (6^{2}, 4) & λ^{++} = (8^{2}, 5) \end{matrix}

We have labeled the runners by their residues which will be introduced below.

The reader is invited to check that the partitions which label the decomposition matrix for n = 21 and p = 5 in the introduction are precisely the partitions λ^{++} as λ runs over the corresponding partitions of 11. We emphasize that the empty runner can be inserted anywhere in the abacus.

We are now almost ready to describe our main result. As in the introduction let q be a primitive e^{th} root of unity in ℂ and let ℋ_{ℂ,q}(𝔖_{n}) be the Iwahori–Hecke algebra of the symmetric group 𝔖_{n} defined over the complex numbers with parameter q; see, for example, [?].

For each partition λ of n Dipper and the first author [?] defined a right ℋ_{ℂ,q}(𝔖_{n})– module S_{q}^{λ}, called a Specht module. There is a natural bilinear form ⟨ , ⟩ on S_{q}^{λ} and Rad S_{q}^{λ} = { x ∈ S_{q}^{λ} | ⟨x, y⟩ = 0 for all y ∈ S_{q}^{λ} } is an ℋ_{ℂ,q}(𝔖_{n})–submodule of S_{q}^{λ}; set D_{q}^{λ} = S_{q}^{λ}/ Rad S_{q}^{λ}.

Recall that a partition λ is e–regular if no e non–zero parts of λ are equal. Dipper and James showed that D_{q}^{λ} is an absolutely irreducible ℋ_{ℂ,q}(𝔖_{n})–module whenever λ is e–regular and, further, that every irreducible ℋ_{ℂ,q}(𝔖_{n})–module arises uniquely in this way. (Conversely, if λ is not e–regular then D_{q}^{λ} = 0.)

If μ is e–regular let [S_{q}^{λ} : D_{q}^{μ}] be the multiplicity of the simple module D_{q}^{μ} as a composition factor of S_{q}^{λ}.

Let q′ be a primitive (e + 1)^{st} root of unity in ℂ. Then we also have Hecke algebras ℋ_{ℂ,q′}(𝔖_{m}) and modules S_{q′}^{ν} and D_{q′}^{ν}. We shall prove the following.

2.2. Theorem. Suppose that λ and μ be partitions of n of length at most k and suppose that μ is e–regular. Then [S_{q}^{λ} : D_{q}^{μ}] = [S_{q′}^{λ^{+}} : D_{q′}^{μ^{+}}].

It may not be clear to the reader that this result is really saying that the decomposition matrices of the blocks containing S_{q}^{λ} and S_{q′}^{λ^{+}} are equal on those rows indexed by partitions of length at most k, when we order the rows of these matrices first by length and then by dominance. This follows from Lemma ?? below.

It is tempting to speculate that there is some form of category equivalence underpinning this result. However, in general, there are a different number of simple modules in the blocks for λ and λ^{+}, so these blocks are certainly not Morita equivalent.

Rather than prove our comparison theorem for decomposition numbers directly we prove a stronger result relating the LLT–polynomials [?]. We now recall the notation needed to describe this.

3. The Fock space and U_{v}(𝔰̂𝔩_{e}). Let v be an indeterminate over ℂ. The Fock space is the infinite rank free ℂ[v, v^{−1}]–module ℱ = ⨁_{n≥0} ⨁_{λ⊢n} ℂ[v, v^{−1}]λ. The Fock space has a natural structure as a module for the affine quantum group U_{v}(𝔰̂𝔩_{e}); we will describe how the positive part U_{v}^{+}(𝔰̂𝔩_{e}) of U_{v}(𝔰̂𝔩_{e}) acts on ℱ since this is all we shall need (full details can be found in [?, ?]).

The diagram of a partition λ is the set [λ] = { (c, d) ∈ ℕ^{2} | d ≤ λ_{c} }. A node is any ordered pair (c, d) ∈ ℕ^{2}; in particular, all of the elements of [λ] are nodes. The e–residue of the node x = (c, d) is res_{e}(x) = d − c (mod e); x is an i–node if res_{e}(x) = i.

A node x is an addable node of λ if [λ] ∪ {x} is the diagram of a partition (and x ∉ [λ]); similarly, x is removable if [λ] \ {x} is the diagram of a partition (and x ∈ [λ]). For i = 0, …, e − 1 let A_{i}(λ) be the set of addable i–nodes for λ and R_{i}(λ) be the set of removable i–nodes. Given two nodes x = (c, d) and y = (a, b) say that y is above x if c > a; if y is above x we write y ≻ x.

In order to define the action of U_{v}^{+}(𝔰̂𝔩_{e}) on ℱ for i = 0, … e − 1 write λ ⟶^{i} ν if ν is a partition of n + 1 and [ν] = [λ] ∪ {x} for some addable i–node x. Finally, we set N_{i}(λ, ν) = #A_{i}(λ, ν) − #R_{i}(λ, ν) where A_{i}(λ, ν) = { y ∈ A_{i}(λ) | y ≻ x } and R_{i}(λ, ν) = { y ∈ R_{i}(λ) | y ≻ x }.

Let F_{0}, …, F_{e−1} be the Chevalley generators of U_{v}^{+}(𝔰̂𝔩_{e}). Then the action of U_{v}^{+}(𝔰̂𝔩_{e}) on ℱ is determined by

(3.1)  F_{i}λ = ∑_{λ ⟶^{i} ν} v^{N_{i}(λ,ν)}ν,

for 0, 1, …, e − 1.

Let Λ_{0}, …, Λ_{e−1} be the fundamental weights of U_{v}(𝔰̂𝔩_{e}) and let L(Λ_{0}) be the irreducible integrable highest weight module of high weight Λ_{0}. Then L(Λ_{0}) ≅ U_{v}(𝔰̂𝔩_{e})∅ = U_{v}^{+}(𝔰̂𝔩_{e})∅ as a U_{v}(𝔰̂𝔩_{e})–module [?], where ∅ ∈ ℱ is the empty partition.

Let be the bar involution on U_{A}(𝔰̂𝔩_{e}), Lusztig’s A–form of U_{v}(𝔰̂𝔩_{e}) (where A = ℤ[v, v^{−1}]). Then Lascoux, Leclerc and Thibon [?] showed that L(Λ_{0}) has a basis { B_{μ} | μ e–regular } which is uniquely determined by the requirements that \overline{B}_{μ} = B_{μ} and

B_{μ} = ∑_{\begin{matrix}λ⊢n \\ μ⊵λ \end{matrix}} b_{λμ}(v)λ

for some polynomials b_{λμ}(v) ∈ ℤ[v] such that b_{μμ}(v) = 1 and b_{λμ}(v) ∈ vℤ[v] whenever λ ≠ μ. This basis is the Kashiwara–Lusztig canonical basis of L(Λ_{0}).

In particular, note that Lascoux, Leclerc and Thibon [?] showed that b_{λμ}(v) = 0 if either |λ| ≠ |μ| or if λ and μ have different e–cores.

We want to compare the actions of U_{v}(𝔰̂𝔩_{e}) and U_{v}(𝔰̂𝔩_{e+1}) on the Fock space ℱ. In order to distinguish between these two algebras let F_{0}^{+}, …, F_{e}^{+} be the Chevalley generators of U_{v}^{+}(𝔰̂𝔩_{e+1}), let Λ_{0}^{+}, …, Λ_{e}^{+} be its fundamental weights, and let ℱ^{+} ≅ ℱ be the Fock space for the U_{v}(𝔰̂𝔩_{e+1})–action. Then L(Λ_{0}^{+}) ≅ U_{v}^{+}(𝔰̂𝔩_{e+1})∅ as a U_{v}(𝔰̂𝔩_{e+1})–module. Given an (e + 1)–regular partition ν let

B_{ν}^{+} = ∑_{ν⊵σ} b_{σν}^{+}(v)σ

be the corresponding canonical basis element of L(Λ_{0}^{+}) ⊆ ℱ^{+}.

We can now state a stronger version of our main result.

3.2. Theorem. Suppose that λ and μ are partitions of n of length at most k and suppose that μ is e–regular. Then b_{λμ}(v) = b_{λ^{+}μ^{+}}^{+}(v).

Ariki [?, Prop. 4.3(2)] has shown that the polynomials b_{λμ}(v) at v = 1 compute the decomposition multiplicities; explicitly, [S_{q}^{λ} : D_{q}^{μ}] = b_{λμ}(1) and [S_{q′}^{λ^{+}} : D_{q′}^{μ^{+}}] = b_{λ^{+}μ^{+}}^{+}(1). Consequently, Theorem ?? implies Theorem ??. The result also hints at additional structure because, conjecturally, the polynomials b_{λμ}(v) and b_{λ^{+}μ^{+}}^{+}(v)

also describe the Jantzen filtrations of S_{q}^{λ} and S_{q′}^{λ^{+}}; see [?, ?].

We begin the proof with the following Lemma which is largely book keeping. For example, the statement of Theorem ?? requires that μ^{+} is (e + 1)–regular.

3.3. Lemma. Let λ and μ be partitions with n of length at most k. Then (i) μ^{+} is (e + 1)–regular; (ii) λ and μ have the same e–core if and only if λ^{+} and μ^{+} have the same

(e + 1)–core; (iii) if λ and μ have the same e–core then |λ^{+}| = |μ^{+}|; and, (iv) λ ⊵ μ if and only if λ^{+} ⊵ μ^{+}

Proof. A partition is (e+1)–regular if and only if its (e+1)–abacus does not contain a string of e + 1 consecutive beads. Hence, μ^{+} is (e + 1)–regular since the runner ρ_{α}^{+} is empty; this proves (i). (In fact, if μ is e–regular then so is μ^{+}.)

Next, recall that the e–abacus for the e–core of λ is obtained by rearranging the beads on each runner of the e–abacus for λ in such a way that no bead has an empty bead position above it. Hence, if κ is the e–core of λ then κ^{+} is the (e + 1)–core of λ^{+}, so (ii) follows.

For (iii) define w by |λ| = |κ| + we; in other words, w is the e–weight of λ. Now, w can be read off the e–abacus for λ by adding up, for each bead β, the number of empty bead positions which are above β and also on the same runner. Consequently, w is also the (e + 1)–weight of λ^{+}; hence, |λ^{+}| = |κ^{+}| + w(e + 1). Lastly, since κ is also the e–core of μ it follows that μ is also a partition of e–weight w and that |μ^{+}| = |κ^{+}| + w(e + 1) = |λ^{+}|, as required.

Finally, let β_{1}^{λ} < ⋯ < β_{k}^{λ} and β_{1}^{μ} < ⋯ < β_{k}^{μ} be the positions of the beads on the e–abacuses for λ and μ respectively. Then it is easy to see that λ ⊵ μ if and only if ∑_{s=t}^{k} β_{s}^{λ} ≥ ∑_{s=t}^{k} β_{s}^{μ} for t = 1, …, k. Rephrasing this condition in terms of the (e + 1)–abacuses of λ^{+} and μ^{+} proves (iv). □

We remark that if λ and μ are partitions of n with different e–cores then, in general, it is not true that |λ^{+}| = |μ^{+}|.

Let ℱ_{>k} be the ℂ[v, v^{−1}]–submodule of ℱ spanned by the partitions of length strictly greater than k. By (??) ℱ_{>k} is a U_{v}^{+}(𝔰̂𝔩_{e})–submodule of ℱ; it is not, however, a U_{v}(𝔰̂𝔩_{e})–submodule. Therefore, ℱ_{k} = ℱ /ℱ_{>k} is a U_{v}^{+}(𝔰̂𝔩_{e})–module. We abuse notation and identify the elements of ℱ with their images in ℱ_{k}; with this understanding, { λ | ℓ(λ) ≤ k } is a basis of ℱ_{k}.

Similarly, ℱ_{k}^{+} = ℱ^{+}/ℱ_{>k}^{+} is a U_{v}^{+}(𝔰̂𝔩_{e+1})–module. We want to compare the action of U_{v}^{+}(𝔰̂𝔩_{e}) on ℱ_{k} with the action of U_{v}^{+}(𝔰̂𝔩_{e+1}) on ℱ_{k}^{+}; to do this we reinterpret (??) in terms of abacuses.

Suppose λ is a partition with ℓ(λ) ≤ k and consider the e–abacus of λ. For 0 ≤ r < e define the e–residue of the runner ρ_{r} to be the integer res_{e}(ρ_{r}) determined by the following two conditions.

(i) The e–residue of the runner which holds the last bead is λ_{1} − 1 (mod e). (ii) Modulo e, the e–residues of the runners increase by 1 from left to right.

In Example ?? the runners are labelled by their f–residues for f = 3, 4 and 5 respectively. Similarly, we define the (e + 1)–residues res_{e+1}(ρ_{r}^{+}), for 0 ≤ r ≤ e, of the runners of the (e + 1)–abacus for λ^{+}.

The e–residue of a bead β is defined to be the e–residue of the corresponding runner. As we have seen, the k beads on the e–abacus correspond to the first k rows of λ (in reverse order); it is easy to see that the e–residue of a bead is equal to the e–residue of the node at the end of the corresponding row of λ. In particular, this implies that e–residues of the runners depend only on the e–core of λ.

The operator F_{i} acts on a partition λ by adding nodes of e–residue i. Because the e–residues of the runners correspond to the e–residues of nodes at the end of the rows of λ, this is the same as moving a bead on the e–abacus of λ from the runner with e–residue i − 1 to an adjacent empty position on the runner with e–residue i.

Recall that in the definition of λ^{+} we have fixed an integer α with 0 ≤ α < e. We now introduce a ℂ[v, v^{−1}]–linear map ^{α}F_{i}: ℱ_{k} ⟶ ℱ_{k}^{+} for i = 0, …, e − 1. To define ^{α}F_{i} it is enough to describe ^{α}F_{i}λ for each partition λ with ℓ(λ) ≤ k. As above, let ρ_{0}, …, ρ_{e−1} be the runners of the e–abacus of λ (with k beads) and let ρ_{0}^{+}, …, ρ_{e}^{+} be the runners of the (e + 1)–abacus for λ^{+}. There is a unique r such that i = res_{e}(ρ_{r}) (and 0 ≤ r < e); set j = res_{e+1}(ρ_{r}^{+}). Define

αF_{i}λ = \begin{cases}F_{j}^{+}λ^{+}, & if 0 ≤ r < α, \\ F_{j+1}^{+}F_{j}^{+}λ^{+}, & if r = α, \\ F_{j+1}^{+}λ^{+}, & if α < r < e, \end{cases}

where j + 1 is understood modulo e. Similarly, we define the divided powers ^{α}F_{i}^{(a)} for a ≥ 1; for example, when r = α we set ^{α}F_{i}^{(a)}λ = F_{j+1}^{+(a)}F_{j}^{+(a)}λ^{+}.

For our final piece of notation observe that ℓ(λ^{+}) ≥ ℓ(λ) for any partition λ. Therefore, we have a well–defined ℂ[v, v^{−1}]–linear map Θ : ℱ_{k} ⟶ℱ_{k}^{+} determined by Θ(λ) = λ^{+} for ℓ(λ) ≤ k. As with λ^{+}, we emphasize that Θ depends upon both α and k. The map Θ is injective but not surjective, having image the span of those partitions of length at most k which have an (e + 1)–abacus with k beads and with an empty runner ρ_{α}^{+}.

3.4. Lemma. Suppose that 0 ≤ i < e. Then the following diagram commutes.

\begin{aligned}ℱ_{k} & ℱ_{k} \\ & ℱ_{k}^{+} \end{aligned}F_{i}^{α}F_{i} Θ

Proof. It suffices to verify the lemma for a partition λ ∈ ℱ_{k}. As above, let ρ_{r} be the runner in the e–abacus for λ for which i = res_{e}(ρ_{r}) and set j = res_{e+1}(ρ_{r}^{+}).

First consider F_{i}λ = ∑_{ν} v^{N_{i}(λ,ν)}ν^{+}. Recall that the beads on the e–abacus for λ are naturally indexed by the rows of λ and that the e–residue of a bead is defined to be the e–residue of the node which is at the end of the corresponding row. Therefore, an addable i–node of λ corresponds to a node on runner r − 1 of the e–abacus which can be moved to the adjacent position on runner r (which must therefore be empty). Similarly, a removable i–node corresponds to a node on runner r which can be moved back to the adjacent position on runner r − 1; here, r ± 1 is to be understood modulo e. The addable and removable nodes of λ^{+} have analogous descriptions.

Fix a partition ν with λ ⟶^{i} ν and write [ν] = [λ]∪{x}. Then there exists a node at position β_{x} on the runner r − 1 of the e–abacus for λ which can be moved to the adjacent position on runner r so as to give the e–abacus for ν. Then N_{i}(λ, ν) = A − B, where A = # { y ∈ A_{i}(λ) | y ≻ x } and B = # { y ∈ R_{i}(λ) | y ≻ x }. If y is an addable or removable node of λ then y ≻ x if and only if it corresponds to a bead at position β_{y} with β_{y} > β_{x}. Hence, A is equal to the number of beads on runner ρ_{r−1} which come after β_{x} such that the adjacent position on ρ_{r} is vacant; similarly, B is equal to the number of beads on ρ_{r} which are after β_{x} and for which the adjacent position on runner r − 1 is vacant.

Now consider the (e + 1)–abacuses for λ^{+} and ν^{+}. Assume first that r < α. Then the runners ρ_{r−1} and ρ_{r} for λ are the same as the runners ρ_{r−1}^{+} and ρ_{r}^{+} for λ^{+} and so the last paragraph shows that the addable and removable i–nodes for λ correspond exactly to the addable and removable j–nodes for λ^{+}. Hence, N_{j}(λ^{+}, ν^{+}) = A − B = N_{i}(λ, ν). Similarly, when r > α the addable and removable i–nodes for λ correspond to the addable and removable (j + 1)–nodes for λ^{+} and N_{i}(λ, ν) = N_{j+1}(λ^{+}, ν^{+}).

Finally, consider the case when r = α. This time runner ρ_{r−1} is equal to ρ_{r−1}^{+} and runner ρ_{r} is equal to ρ_{r+1}^{+}; whereas runner ρ_{r}^{+} = ρ_{α}^{+} of λ^{+} is empty. Therefore, the addable and removable i–nodes of λ again correspond to addable and removable j–nodes of λ^{+} except that this time there are additional addable j–nodes of λ^{+} corresponding to the adjacent pairs of beads on the runners ρ_{r−1} and ρ_{r} of the e–abacus of λ. Let σ be the partition such that λ^{+} ⟶^{j} σ ⟶^{j + 1} ν^{+}. Since ρ_{r}^{+} is empty, λ^{+} has no removable j–nodes. Therefore, if we let l be the number of pairs of adjacent beads on runners ρ_{r−1} and ρ_{r} which are above β_{x} then N_{j}(λ^{+}, σ) = A + l. Next observe that σ has a single addable (j + 1)–node (corresponding to the bead which we just moved), and that the removable (j + 1)–nodes of σ correspond to the removable i–nodes of λ together with the l beads on runner ρ_{r} which we have already paired with an adjacent bead on ρ_{r−1}; therefore, N_{j+1}(σ, ν^{+}) = −(B + l). Consequently, N_{j}(λ^{+}, σ) + N_{j+1}(σ, ν^{+}) = A − B = N_{i}(λ, ν) and so we have

F_{j+1}^{+}F_{j}^{+}λ^{+} = ∑_{ν^{+}} v^{N_{i}(λ,ν)}ν^{+} = Θ(F_{i}λ),

where the sum is over those partitions ν^{+} for which there exists a partition σ such that λ^{+} ⟶^{j} σ ⟶^{j + 1} ν^{+}. Note that there are additional terms in the expansion of F_{j}λ^{+} (corresponding to the pairs of adjacent beads on runners ρ_{r−1} and ρ_{r} of the e–abacus for λ); however, they all disappear when F_{j+1} is applied because these extra partitions do not have any addable (j + 1)–nodes. This completes the proof. □

Now consider L(Λ_{0})_{k} = L(Λ_{0})/(L(Λ_{0}) ∩ ℱ_{k}). If μ is an e–regular partition with ℓ(μ) ≤ k let B̃_{μ} = B_{μ} + ℱ_{>k}. As noted by Goodman and Wenzl [?, Lemma 4.1], the elements { B_{μ} | μ is e–regular and ℓ(μ) ≤ k } give a basis of L(Λ_{0})_{k}.

The bar involution induces a well–defined map on ℱ_{k} via \overline{a + ℱ_{k}} = \overline{a} + ℱ_{k} for all a ∈ ℱ. It is easy to see that B̃_{μ} is the unique element of ℱ_{k} which is bar invariant and of the form μ + ∑_{λ} b_{λμ}(v)λ for some polynomials b_{λμ}(v) ∈ vℤ[v], the sum being over the partitions of length at most k.

Similarly, { B̃_{ν}^{+} | ν is (e + 1)–regular and ℓ(ν) ≤ k }, where B̃_{ν}^{+} = B_{ν}^{+} + ℱ_{>k}^{+}, is a basis of the U_{v}^{+}(𝔰̂𝔩_{e+1})–module L(Λ_{0}^{+})_{k} = L(Λ_{0}^{+})/(L(Λ_{0}^{+}) ∩ ℱ_{k}^{+}).

3.5. Proposition. Suppose that μ is an e–regular partition with at most k rows. Then B̃_{μ^{+}}^{+} = Θ(B̃_{μ}).

Proof. Looking at the definitions, Θ(B̃_{μ}) = μ^{+} plus a qℤ[q]–linear combination of less dominant terms. Therefore, it is enough to show that Θ(B̃_{μ}) is a bar invariant element of ℱ_{k}^{+}.

Let ∅ ∈ ℱ_{k} be the image of the empty partition in ℱ_{k}. Following Lascoux, Leclerc and Thibon [?, Lemma 6.4] let (r_{1}^{a_{1}}, …, r_{s}^{a_{s}}) be the e–residue sequence of μ corresponding to the e–ladders in the diagram of μ. Then A_{μ} = F_{r_{s}}^{(a_{s})} … F_{r_{1}}^{(a_{1})}∅ is a bar invariant element of ℱ_{k} of the form A_{μ} = μ + ∑_{λ} a_{λμ}(v)λ where a_{λμ}(v) ∈ ℤ[v, v^{−1}] and the sum is over partitions λ of length a most k such that μ ▷ λ. Therefore, there exist uniquely determined polynomials α_{σμ}(v) ∈ ℤ[v] such that B̃_{μ} = A_{μ} − ∑_{σ} α_{σμ}(v)B̃_{σ}, where the sum is over e–regular partitions σ such that μ ▷ σ and ℓ(σ) ≤ k.

Now consider the element A_{μ}^{+} = Θ(A_{μ}) = μ + ∑_{λ} a_{λμ}(v)λ^{+} in ℱ_{k}^{+}. By the Lemma, A_{μ}^{+} = ^{α}F_{r_{s}}^{(a_{s})} …^{α}F_{r_{1}}^{(a_{1})}∅; hence, A_{μ}^{+} is bar invariant. By induction on dominance B̃_{σ}^{+} = Θ(B̃_{σ}) for μ ▷ σ. Therefore, the element Θ(B̃_{μ}) = A_{μ}^{+} − ∑_{σ} α_{σμ}(v)B̃_{σ^{+}} is also bar invariant. Consequently, B̃_{μ^{+}}^{+} = Θ(B̃_{μ}) as we wanted to show. □

It is easy to see [?] that the polynomials b_{στ}^{+} (v) are non–zero only if σ and τ have the same (e + 1)–core. Therefore, if μ is an e–regular partition with ℓ(μ) ≤ k then B̃_{μ^{+}}^{+} = ∑_{μ⊵λ} b_{λ^{+}μ^{+}}^{+}(v)λ^{+}; on the other hand, Θ(B_{μ}) = ∑_{μ⊵λ} b_{λμ}(v)λ^{+}, where in both sums ℓ(λ) ≤ k. Hence, Theorem ?? follows from the Proposition.

Goodman and Wenzl [?, Prop. 5.14] have proved a result very similar to Proposition ??; however, it is not the same because we do not require that the weight μ+ρ belong to a proper face of an alcove. Our result is also more explicit. Nonetheless, the origin for these results is that the b_{λμ}(v) are parabolic Kazhdan–Lusztig polynomials for the parabolic subgroup 𝔖_{k} of the extended affine Weyl group 𝔖̂_{k} [?,?,?]; in turn the parabolic Kazhdan–Lusztig polynomials are naturally indexed by the alcoves and, generically, the alcove geometry does not depend on k or e.

4. The q–Schur algebras. In this section we extend Theorem ?? to the decomposition matrices of the q–Schur algebras S_{ℂ,q}(n, n) of Dipper and the first author [?].

The Fock space also admits an action from a Heisenberg algebra H_{e} [?]. The action of H_{e} on ℱ commutes with the action of U_{v}(𝔰̂𝔩_{e}) and it is useful because ℱ is irreducible when considered as a module for the algebra generated by the actions of U_{v}(𝔰̂𝔩_{e}) and H_{e} on ℱ. In addition, the action of H_{e} allowed Leclerc and Thibon [?, §7.9] to extend the bar involution to the whole of ℱ; in turn, this enabled them to extend the canonical basis of L(Λ_{0}) to give a basis { B_{μ} | μ a partition } of ℱ where, the element B_{μ} is again uniquely determined by the two conditions that \overline{B}_{μ} = B_{μ} and

B_{μ} = ∑_{\begin{matrix}λ⊢n \\ μ⊵λ \end{matrix}} b_{λμ}(v)λ

for some polynomials b_{λμ}(v) ∈ ℤ[v] such that b_{μμ}(v) = 1 and b_{λμ}(v) ∈ vℤ[v] whenever λ ≠ μ. We will show that Theorem ?? generalizes to the non–regular case.

As in the previous sections we are only interested in the action of a subalgebra H_{e}^{+} of H_{e}; for the full story see [?, §7.5]. The algebra H_{e}^{+} is generated by elements V_{m} for m ≥ 0; before we can describe how V_{m} acts on ℱ we need some more notation.

An e–ribbon is a connected strip of e–nodes which does not contain a 2×2 square; more precisely, an e–ribbon is a set of e nodes R = {(a_{1}, b_{1}), …, (a_{e}, b_{e})} such that (a_{i+1}, b_{i+1}) is either (a_{i} + 1, b_{i}) or (a_{i}, b_{i} − 1), for i = 1, …, e − 1. The head of R is the node head(R) = (a_{1}, b_{1}) and spin_{e}(R) = # { 1 ≤ i < e | a_{i+1} = a_{i} + 1 } − 1 is the e–spin of R.

If λ and ν are partitions then we write λ ^{m:e} ν if [λ] ⊆ [ν] and [ν] \ [λ] is a disjoint union of m e–ribbons such that the head of each ribbon is either in the first row of λ or is of the form (i, j) where (i − 1, j) ∈ [λ]. Lascoux, Leclerc and Thibon (see [?, §4.1]), call ν/λ an e–ribbon tableau of weight (m) and they note that there is a unique way of writing [ν] \ [λ] as a disjoint union of ribbons; we will see this below when we reinterpret ribbons in terms of abacuses. Finally, if λ ^{m:e} ν then spin_{e}(ν/λ), the e–spin of ν/λ, is the sum of the e–spins of the ribbons in [ν] \ [λ].

For example, if λ = (3) and e = 2 then the partitions ν with λ ^{2:2} ν are

with spins −2, −2 and 2 respectively.

The algebra H_{e}^{+} is the subalgebra of H_{e} generated by elements V_{m} for m ≥ 1. For each m, V_{m} acts on the Fock space as the ℂ[v, v^{−1}]–linear map determined by

V_{m}λ = ∑_{λ ^{m:e} ν}(−q)^{spin_{e}(ν/λ)}ν

for all partitions λ. Observe that ℱ_{>k} is a H_{e}^{+}–module; hence, there is a well– defined action of V_{m} on the quotient space ℱ_{k}.

Similarly, there is an action of the positive Heisenberg algebra H_{e+1}^{+} on the Fock space ℱ^{+} and this induces an action on ℱ_{k}^{+}. We denote the generators of H_{e+1}^{+} by V_{m}^{+} for m ≥ 1.

4.1. Lemma. Suppose that λ and ν are partitions of n of length at most k. (i) If λ ^{m:e} ν then λ and μ have the same e–core. (ii) We have λ ^{m:e} ν if and only if λ^{+} ^{m:e+1} μ^{+}; moreover, if λ ^{m:e} ν

then spin_{e}(ν/λ) = spin_{e+1}(ν^{+}/λ^{+}).

Proof. The Lemma will follow once we reinterpret the condition λ ^{m:e} ν in terms of abacuses. Suppose that λ ^{m:e} ν. Then [ν] \ [λ] is a disjoint union of e– ribbons. Extend the partial order ≻ on the set of nodes to a total order by defining (a, b) ≻ (c, d) if either c > a or c = a and b > d. Totally order the ribbons R_{1}, …, R_{m} in [ν] \ [λ] so that i > j whenever head(R_{i}) ≻ head(R_{j}). Then the condition that the head of R_{i} is of the form (a, b) with either a = 1 or (a−1, b) ∈ [λ] is equivalent to saying that [ν] \ (R_{1} ∪ ⋯ ∪ R_{i}) is the diagram of a partition for i = 1, …, m. Hence, it is enough to treat the case m = 1. So let R = R_{1} where [ν] = [λ] ∪ R.

Now the ribbon R is a rim hook and it is well–known that removing a rim hook of length e from ν is the same as moving a bead β on an e–abacus for ν to the (empty) bead position on the same runner which is in the preceding row. Further, by definition, spin_{e}(ν/λ) is the leg length of R minus one and, in terms of the e–abacus, the leg length of R is equal to the number of beads on the abacus which are between the old and new positions of β. Similarly, the condition λ^{+} ^{m:e+1} μ^{+} depends only on the (e + 1)–abacuses of ν^{+} and λ^{+}. As the e and (e + 1) abacuses differ only by the insertion of an empty runner, all of the assertions of the Lemma now follow. □

4.2. Corollary. Suppose that m ≥ 1. Then the following diagram commutes.

\begin{matrix}ℱ_{k} & ℱ_{k} \\ ℱ_{k}^{+} & ℱ_{k}^{+}_{V_{m}}_{V_{m}^{+}}_{Θ}_{Θ} \end{matrix}

Proof. As with Lemma ?? it suffices to check the result for a partition λ of length at most k. By the definitions and the previous Lemma,

Θ(V_{m}λ) = ∑_{λ ^{m:e} ν}(−q)^{spin_{e}(ν/λ)}Θ(ν)

= ∑_{λ^{+} ^{m:e+1} ν^{+}}(−q)^{spin_{e+1}(ν^{+}/λ^{+})}ν^{+}

= V_{m}^{+}λ^{+}.

Therefore, V_{m}^{+}(Θ(λ)) = Θ(V_{m}λ) and we’re done. □

Let μ be a partition. As in the last section let B̃_{μ} and B̃_{μ}^{+} denotes the images of B_{μ} ∈ ℱ and B_{μ}^{+} ∈ ℱ^{+} in ℱ_{k} and ℱ_{k}^{+} respectively. Then a basis of ℱ_{k} is given by the B̃^{μ} as μ runs over all partitions of length at most k and similarly for ℱ_{k}^{+}.

By Proposition ?? we know that B̃_{μ^{+}}^{+} = Θ(B_{μ}) whenever μ is an e–regular partition with ℓ(μ) ≤ k. We can now drop the requirement that μ should be e–regular.

4.3. Proposition. Suppose that μ is a partition with ℓ(μ) ≤ k. Then B̃_{μ^{+}}^{+} = Θ(B̃_{μ}).

Proof. As before, the element Θ(B̃_{μ}) is equal to μ^{+} plus a qℤ[q]–linear combination of less dominant terms, so it is enough to show that Θ(B̃_{μ}) is invariant under the bar involution since B̃_{μ^{+}}^{+} is the unique bar invariant element in ℱ_{k}^{+} of this form.

By [?, Prop. 7.6] the bar involution on ℱ is completely determined by the con-

ditions \overline{∅} = ∅, \overline{F_{i}^{(m)}x} = F_{i}^{(m)}\overline{x} and \overline{V_{m}x} = V_{m}\overline{x}, for all x ∈ ℱ, 0 ≤ i < e and m ≥ 1. For each partition τ = (τ_{1}, …, τ_{s}) let V_{τ} = V_{τ_{1}} … V_{τ_{s}}. Then

ℱ = ⨁_{τ} U_{v}^{+}(𝔰̂𝔩_{e})V_{τ} ∅

is a decomposition of ℱ into a direct sum of irreducible U_{v}(𝔰̂𝔩_{e})–modules (where τ runs over all partitions of all integers); see [?, §7.5]. Moreover, the modules U_{v}^{+}(𝔰̂𝔩_{e})V_{τ} ∅, for different τ, are all isomorphic as U_{v}^{+}(𝔰̂𝔩_{e})–modules. Therefore, there exists a bar invariant basis of ℱ of the form A_{στ} = F_{σ}V_{τ} ∅ where F_{σ} ∈ U_{A}^{+}(𝔰̂𝔩_{e}), σ is an e–regular partition and τ is an arbitrary partition (the elements F_{σ} are defined in terms of e–residue sequences as in the proof of Proposition ??). Consequently, we can write B̃_{μ} = ∑_{σ,τ} a_{στ} (v)A_{στ} for some bar invariant Laurent polynomials a_{στ} (v) ∈ ℤ[v, v^{−1}]. Now, A_{στ}^{+} = Θ(A_{στ}) = Θ(F_{σ}V_{τ} ∅) = ^{α}F_{σ}V_{τ}^{+}∅ by Lemma ?? and Corollary ??; therefore, A_{στ}^{+} is a bar invariant element of ℱ_{k}^{+}. Hence, Θ(B̃_{μ}) = ∑_{σ,τ} a_{στ} (v)A_{στ}^{+} is a bar invariant element of ℱ_{k}^{+}, as we needed to show. □

4.4. Remark. For each composition τ = (τ_{1}, …, τ_{s}) Leclerc and Thibon [?] show that the action of the element V_{τ} = V_{τ_{1}} … V_{τ_{s}} upon ℱ is described by certain polynomials associated with the ribbon tableaux of weight τ. This is completely analogous to the way in which the action of F_{r_{s}}^{(a_{s})} … F_{r_{1}}^{(a_{1})} on ℱ can be described in terms of polynomials associated with standard tableaux.

Comparing the coefficient of λ^{+} in B̃_{μ^{+}}^{+} and Θ(B̃_{μ}) we obtain the following.

4.5. Corollary. Suppose that λ and μ are partitions of length at most k. Then b_{λμ}(v) = b_{λ^{+}μ^{+}}^{+}(v).

In order to compute the polynomials b_{λμ}(v) when μ is not e–regular it is necessary to first invert the “R–matrix” which describes the bar involution on the basis of ℱ given by the set of partitions. Computationally, this is quite time consuming; in comparison the regular case is much easier, being essentially Gaussian elimination. Corollary ?? therefore gives a slightly more efficient way of computing the polynomials b_{λμ}(v) = b_{λ^{+}μ^{+}}^{+}(v) since μ^{+} is an (e + 1)–regular partition by Lemma ??(i).

Next, let W_{q}^{λ} and F_{q}^{μ} be the Weyl modules and simple modules, respectively, for

the q–Schur algebra S_{ℂ,q}(n, n). Similarly, let W_{q′}^{λ^{+}} and F_{q′}^{μ^{+}} be the Weyl modules and simple modules of the q′–Schur algebra S_{ℂ,q′}(n, n). Varagnolo and Vasserot [?] have shown that [W_{q}^{λ} : F_{q}^{μ}] = b_{λμ}(1); similarly, [W_{q′}^{λ^{+}} : F_{q′}^{μ^{+}}] = b_{λ^{+}μ^{+}}^{+}(1).

4.6. Corollary. Suppose that λ and μ are partitions of length at most k. Then

[W_{q}^{λ} : F_{q}^{μ}] = [W_{q′}^{λ^{+}} : F_{q′}^{μ^{+}}] = [S_{q′}^{λ^{+}} : D_{q′}^{μ^{+}}].

Proof. That [W_{q}^{λ} : F_{q}^{μ}] = [W_{q′}^{λ^{+}} : F_{q′}^{μ^{+}}] follows directly from the previous Corollary the and remarks above. For the second claim, observe that the partition μ^{+} is (e+1)–regular by Lemma ??(i); therefore, D_{q′}^{μ^{+}} ≠ 0. Consequently, [W_{q′}^{λ^{+}} : F_{q′}^{μ^{+}}] =

[S_{q′}^{λ^{+}} : D_{q′}^{μ^{+}}] by Schur–Weyl reciprocity. □

Standard Schur functor arguments yield the corresponding statements for the q–Schur algebras S_{ℂ,q}(n, r) and S_{ℂ,q′}(n, r); we leave the details to the reader.

The last result is interesting because it shows that every decomposition number for S_{ℂ,q}(n, n) is also a decomposition number for some Hecke algebra ℋ_{ℂ,q′}(𝔖_{m}). In contrast Erdmann [?] has shown that in a given characteristic knowing all decomposition numbers for the classical Schur algebras (i.e. q = 1) is equivalent to knowing all decomposition numbers for the symmetric groups (for all n and for a fixed p). Leclerc [?] has proved the analogous result relating the decomposition numbers of the q–Schur algebras S_{ℂ,q}(n, n) and the Hecke algebras ℋ_{ℂ,q}(𝔖_{n}) (for all n and for a fixed q). No such result is known in the cross characteristic case (i.e. positive characteristic with q ≠ 1).

Finally, we remark that the full action of U_{v}(𝔰̂𝔩_{e}) on ℱ_{k} and U_{v}(𝔰̂𝔩_{e+1}) on ℱ_{k}^{+} are compatible via the map Θ (in order to make this statement precise ℱ_{k} should be considered as a submodule of ℱ for the action of U_{v}^{−}(𝔰̂𝔩_{e})). This can be proved using similar arguments or, more simply, by invoking [?, Prop. 7.9] which says that the actions of U_{v}^{+}(𝔰̂𝔩_{e}) and U_{v}^{−}(𝔰̂𝔩_{e}) on ℱ are adjoint with respect to a natural scalar product on ℱ. The same argument also proves the corresponding statements for the Heisenberg algebras H_{e} and H_{e+1}.

5. Examples. Below we give part of the “crystallized” decomposition matrices (b_{λμ}(v)) of the Hecke algebras ℋ_{ℂ,q}(𝔖_{n}) for (e, n) = (2, 6), (3, 11), (4, 16) and (5, 21). By our results, taking k = 4 and α = 2, these submatrices are all the same.

⋯ e = 5  e = 4 e = 3 e = 2
18, 3  14, 2 10, 1 6  1
17, 4  13, 3 9, 2  5, 1  v 1
⋯ 13, 8  11, 5 7, 4  4, 2  . v 1
12, 9  9, 7  6, 5  3^{2}  . . v 1
13, 4^{2} 10, 3^{2} 7, 2^{2} 4, 1^{2} v v^{2} v . 1
⋯ 8^{2}, 5 6^{2}, 4 4^{2}, 3 2^{3} . . v^{2} v v 1
12, 4^{2}, 1 9, 3^{2}, 1 6, 2^{2}, 1 3, 1^{3} v^{2} v v^{2} v v . 1
8^{2}, 4, 1 6^{2}, 3, 1 4^{2}, 2, 1 2^{2}, 1^{2} . v^{2} v^{3} v^{2} v^{2} v v 1

Note that when e = 2 some of the rows are indexed by partitions which are not 2–regular; in this case the matrix above is a submatrix of the decomposition matrix for the corresponding q–Schur algebra with e = 2 and n = 6. Setting v = 2 we recover the decomposition matrices of the Iwahori–Hecke algebras corresponding to those of the introduction.

To emphasize the dependence on k we again start with (e, n) = (2, 6) but now take k = 6 (and α = 2); this yields the following matrices.

e = 5  e = 4  e = 3  e = 2
21, 6, 3^{2}  16, 4, 2^{2}  11, 2, 1^{2}  6  1
20, 7, 3^{2}  15, 5, 2^{2}  10, 3, 1^{2}  5, 1  v 1
16, 11, 3^{2}  12, 8, 2^{2}  8, 5, 1^{2}  4, 2  . v 1
16, 7^{2}, 3  12, 5^{2}, 2  8, 3^{2}, 1  4, 1^{2} v v^{2} v 1
15, 12, 3^{2}  11, 9, 2^{2}  7, 6, 1^{2}  3, 3  . . v . 1
15, 7^{2}, 4  11, 5^{2}, 3  7, 3^{2}, 2  3, 1^{3} v^{2} v v^{2} v v 1
11^{2}, 8, 3  8^{2}, 6, 2  5^{2}, 4, 1  2^{3}  . . v^{2} v v . 1
11^{2}, 7, 4  8^{2}, 5, 3  5^{2}, 3, 2  2^{2}, 1^{2} . v^{2} v^{3} v^{2} v^{2} v v 1
11, 7^{2}, 4^{2}  8, 5^{2}, 3^{2}  5, 3^{2}, 2^{2}  2, 1^{4} v^{2} v^{3} . v . v^{2} . v 1
10, 7^{2}, 4^{2}, 1 7, 5^{2}, 3^{2}, 1 4, 3^{2}, 2^{2}, 1 1^{6}  v^{3} . . v^{2} …. v 1

Notice that in the first example the lengths of the partitions in the columns for e = 2 and e = 5 were always the same; this is no longer true in this example. For these partitions, the dominance order is a total order when e ≥ 3; for this reason we have ordered the rows of this matrix according to dominance for the partitions with e ≥ 3; this accounts for the different ordering of the rows for e = 2 in the two examples.

A consequence of Lemma ?? is that all of these matrices are the rows of decomposition matrices of the corresponding blocks which are indexed by partitions with at most k rows (where the partitions indexing the rows are ordered first by length and then by dominance).

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G. J. Department of Mathematics, Imperial College

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

        g.james@ic.ac.uk            www.ma.ic.ac.uk/∼gdj/

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

        Sydney N.S.W. 2006.  a.mathas@maths.usyd.edu.au
        Australia.  www.maths.usyd.edu.au/u/mathas/