THE REPRESENTATION TYPE OF HECKE ALGEBRAS OF TYPE B

SUSUMU ARIKI AND ANDREW MATHAS

Abstract. This paper determines the representation type of the Iwahori-Hecke algebras of type B when q ≠ ±1. In particular, we show that a single parameter non–semisimple Iwahori–Hecke algebra of type B has finite representation type if and only if q is a simple root of the Poincaré polynomial, confirming a conjecture of Uno’s [18].

1. Introduction

In this paper we determine the representation type of the Hecke algebras of type B. Previously, Uno [18] determined the representation type of the (one parameter) Iwahori–Hecke algebras for the rank 2 Coxeter groups and the Coxeter groups of type A. We build upon Uno’s work to study the Hecke algebras of type B; in particular, we settle Uno’s conjecture in this case.

Let R be an integral domain and suppose that q and Q are invertible elements of R. The Iwahori–Hecke algebra ℋ = ℋ_{q,Q}(B_{n}) of type B_{n} is the unital associative R–algebra with generators T_{0}, T_{1}, …, T_{n−1} and relations

(T_{0} + 1)(T_{0} − Q) = 0, (T_{i} + 1)(T_{i} − q) = 0 for 1 ≤ i ≤ n − 1,
T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0},  T_{i+1}T_{i}T_{i+1} = T_{i}T_{i+1}T_{i} for 1 ≤ i ≤ n − 2,
T_{i}T_{j} = T_{j}T_{i} for 0 ≤ i < j − 1 ≤ n − 2.

We will determine the representation type of ℋ .

Let ℋ_{q}(A_{n−1}) be the subalgebra of ℋ generated by T_{1}, …, T_{n−1}; then ℋ_{q}(A_{n−1}) is isomorphic to the Iwahori–Hecke algebra of the symmetric group of degree n.

Let e ∈ {1, 2, 3, …, ∞} be the multiplicative order of q in R. 1 (Uno) [18, Proposition 3.7,Theorem 3.8] Suppose that R is a field and that q ≠ 1. Then ℋ_{q}(A_{n−1}) is of finite representation type if and only if n < 2e.

Note that although Uno stated the theorem only in the case where R is the field of complex numbers; this is not essential in his proof. We also remark that K. Erdmann and D. K. Nakano [10, Theorem 1.2] have determined the representation type of all of the blocks of ℋ_{q}(A_{n−1}); so (1) also follows from their result.

The following reduction theorem, together with (1), will allow us to assume that −Q is a power of q. 2 (Dipper–James) [7, Theorem 4.17] Suppose that Q ≠ −q^{f} for any f ∈ ℤ. Then ℋ_{q,Q}(B_{n}) is Morita equivalent to

⨁_{m=0}^{n} ℋ_{q}(A_{m−1}) ⊗ ℋ_{q}(A_{n−m−1}).

Key words and phrases. Hecke algebras, representation type.

Combining the last two results we obtain.

Corollary 3. Suppose that R is a field and that Q ≠ −q^{f} for any f ∈ ℤ. Then ℋ_{q,Q}(B_{n}) is of finite representation type if and only if n < 2e.

It remains to determine the representation type of ℋ when Q = −q^{f} for some f ∈ ℤ. When Q = −q^{f} the relation for T_{0} becomes (T_{0} + 1)(T_{0} + q^{f}) = 0. If e is finite we may assume that 0 ≤ f < e. It is convenient to renormalize T_{0} as −T_{0}, when 0 ≤ f ≤ \frac{e}{2}, and as −q^{−f}T_{0}, when \frac{e}{2} < f < e; in this way, the relation for T_{0} becomes (T_{0} − 1)(T_{0} − q^{f}) = 0 where 0 ≤ f ≤ \frac{e}{2} whenever e is finite.

Henceforth we assume that q is a primitive e^{th} root of unity in R and that T_{0} satisfies the relation (T_{0} − 1)(T_{0} − q^{f}) = 0 where 0 ≤ f ≤ \frac{e}{2}. As the R–algebra ℋ now only depends on q we now write ℋ = ℋ_{q}(B_{n}), or ℋ_{R,q}(B_{n}) when we wish to emphasize the choice of R.

The main result of this paper is the following. We will consider the cases q = ±1 (that is, e = 1 and e = 2) separately in [4].

Theorem 4. Suppose that R is a field and that e ≥ 3 and 0 ≤ f ≤ \frac{e}{2}. Then ℋ_{q}(B_{n}) is of finite representation type if and only if

n < min{e, 2f + 4}.

Uno [18] asked whether the representation type of a non–semisimple single parameter Iwahori–Hecke algebra is finite if and only if q is a simple root of the Poincaré polynomial of the corresponding finite Coxeter group. With the assumptions currently in place, the one parameter Hecke algebra of type B corresponds to e being even and f = \frac{e}{2} − 1; so our result gives an affirmative answer to Uno’s question in type B. In fact, if e is even and ℋ is not semisimple then Theorem 4 says that ℋ is of finite representation type if and only if \frac{e}{2} = f + 1 ≤ n < e; this is if and only if q is a simple root of the Poincaré polynomial (q is a root of the factor x^{e} − 1 = 0). If ℋ is a non–semisimple one parameter Hecke algebra of type B with e odd then, by Corollary 3, ℋ is of finite representation type if and only if e ≤ n < 2e; again, this is if and only if q is a simple root of the Poincaré polynomial (this time q is a root of x^{2e} − 1 = 0).

The proof of Theorem 4 will occupy all of this paper. In sections 2 and 3 we recall the results that we need from the representation theory of algebras and from the representation theory of ℋ ; section 4 shows that ℋ has infinite representation type when n ≥ e; section 5 shows that ℋ has infinite representation type when n ≥ 2f + 4; finally, section 6 shows that ℋ has finite representation type in the remaining cases.

2. Preliminaries on representation type

An algebra A has finite representation type if there are only a finite number of isomorphism classes of indecomposable A–modules; otherwise, A has infinite representation type. This section summarizes the results that we need on the representation type of algebras. More details can be found in the books of Auslander, Reiten and Smalø [5] and Benson [6].

Suppose that K is a field. We always assume that K is a splitting field for A. The following two results are well–known. Throughout the paper, all modules are right modules.

Lemma 5. Let A be a finite dimensional K–algebra.

(i) Suppose that I is a two–sided ideal of A such that A/I has infinite representation type. Then A is of infinite representation type.

(ii) Suppose that B is a direct summand of A as a (B, B)–bimodule. Then (a) If B is of infinite representation type then so is A.

(b) If A is of finite representation type then so is B.

Lemma 6. Let A be a finite dimensional K–algebra and let P_{1}, … P_{l} be the complete set of projective indecomposable A–modules, up to isomorphism. Then

(i) A is Morita equivalent to End_{A}(P_{1} ⊕ ⋯ ⊕ P_{l}).

(ii) if End_{A}(P_{i}) has infinite representation type for some i then A has infinite

representation type. (iii) for each i the algebra End_{A}(P_{i}) has finite representation type if and only if End_{A}(P_{i}) ≅ K[x]/⟨x^{m}⟩ for some integer m ≥ 0 (which depends on i).

For any A–module M let Rad M be the Jacobson radical of M. Let D_{1}, …, D_{l} be a complete set of isomorphism classes of simple A–modules and let P_{1}, …, P_{l} be the corresponding projective indecomposables.

In the theory of algebras Dynkin diagrams are valued graphs, with the underlying graph being the usual Dynkin diagram; see, for example, [5, VII.3, p241]. If A is a symmetric algebra, then the separation diagram of A is the valued graph with vertices {1, …, l, 1′, …, l′} and edges i^{(a,b)} j′ where a = [Rad P_{i}/ Rad^{2} P_{i} : S_{j}] and b = [Rad P_{j}/ Rad^{2} P_{j} : S_{i}].

The following result is fundamental, and may be derived from the theory of hereditary algebras.

Theorem 7 (Gabriel). Suppose that A is an indecomposable algebra such that Rad^{2} A = 0. Then A is of finite representation type if and only if the separation diagram of A is a disjoint union of Dynkin diagrams of finite type as a valued graph.

The Auslander–Reiten quiver of A is the directed graph with vertices the indecomposable A–modules and edges the irreducible morphisms between the indecomposables (a map φ: M ⟶ N is irreducible if φ has no left or right inverse and whenever φ factorizes as φ = θψ then either θ has a right inverse or ψ has a left inverse).

Theorem 8 (Auslander). Let A be an indecomposable algebra and suppose that the Auslander–Reiten quiver of A has a connected component which has a finite number of vertices. Then A is of finite representation type.

Uno used Auslander–Reiten sequences and Theorem 8 to prove the following.

Theorem 9 (Uno [18, Theorem 3.6]). Suppose that A is a symmetric indecomposable algebra and that the decomposition matrix of A can be written in the form

( \begin{matrix}1 & 0 & ⋯ & 0 \\ 1 & 1 & ⋯ & 0 \\ ⋮ & ... & ... & ⋮ \\ 0 & ⋯ & 1 & 1 \\ 0 & ⋯ & 0 & 1 \end{matrix} ) .

Then A is of finite representation type.

It turns out that in the cases where ℋ has finite representation type all of the non–semisimple blocks of ℋ satisfy the hypotheses of Theorem 9; hence they are of finite type. Hence, in principle, we can compute all of the indecomposable modules using Auslander–Reiten sequences when ℋ has finite type.

We remark that Uno’s paper does not actually contain the statement above. However, the result can be extracted from his paper because the assumptions of Theorem 9 appear as [18, Theorem 3.4] and these are all that are used in the proof of his Theorem 3.6.

3. Results from the representation theory of ℋ_{q}(B_{n})

We now turn to the representation theory of ℋ_{q}(B_{n}).

Let ∗ be the anti–involution of ℋ_{q}(B_{n}) determined by T_{i}^{∗} = T_{i} for 0 ≤ i < n. If M is a right ℋ_{q}(B_{n})-module then Hom_{K}(M, K) is naturally a left ℋ_{q}(B_{n})- module and it becomes a right module by twisting the ℋ_{q}(B_{n})–action by the anti– involution ∗. We call this the dual of M; M is self–dual if it is isomorphic to its dual.

It is well–known that ℋ_{q}(B_{n}) is a symmetric algebra; hence we have the following.

Lemma 10. The algebra ℋ_{q}(B_{n}) is a symmetric algebra. In particular, if D is a simple ℋ_{q}(B_{n})–module and P is its projective cover then P/ Rad P ≅ D, Soc P ≅ D and P is self-dual.

As P is self-dual, the dual of the radical series of P is the socle series of P. We remark that this does not mean that the radical series must be symmetric with respect to its middle layer.

Applying Lemma 5(ii)(a) to the inclusion ℋ_{q}(B_{m}) ↪ ℋ_{q}(B_{n}), for m ≤ n, yields the following.

Corollary 11. Suppose that m ≤ n and that ℋ_{q}(B_{m}) is of infinite representation type. Then ℋ_{q}(B_{n}) is of infinite representation type.

Recall that a partition of n is an non–increasing sequence σ = (σ_{1} ≥ σ_{2} ≥ …) of non–negative integers such that |σ| = n where |σ| = ∑_{i} σ_{i}. A bipartition of n is an ordered pair λ = (λ^{(1)}, λ^{(2)}) of partitions λ^{(1)} and λ^{(2)} such that |λ^{(1)}| + |λ^{(2)}| = n; we write λ ⊢ n and |λ| = n. The set of bipartitions is naturally a poset with partial order ⊵ where λ ⊵ μ if for all k ≥ 1 and l ≥ 1

∑_{i=1}^{k} λ_{i}^{(1)} ≥ ∑_{i=1}^{k} μ_{i}^{(1)} and |λ^{(1)}| + ∑_{j=1}^{l} λ_{j}^{(2)} ≥ |μ^{(1)}| + ∑_{j=1}^{l} μ_{j}^{(2)}.

If λ ⊵ μ we say that λ dominates μ. If λ ⊵ μ and λ ≠ μ we write λ ▷ μ.

Let 𝒜 = ℤ[t, t^{−1}] where t is an indeterminate. Then Dipper, James and Murphy have shown that there exist a family { S_{𝒜}^{λ} | λ ⊢ n } of free 𝒜 –modules which are equipped with operators T_{0}, T_{1}, …, T_{n−1} ∈ End_{𝒜} (S^{λ}) which satisfy the defining relations of ℋ_{t}(B_{n}). Consequently, S^{λ} = S_{𝒜}^{λ} ⊗_{𝒜} K is a ℋ_{q}(B_{n})–module, where ℋ_{q}(B_{n}) is the Hecke algebra defined over the field K with q ∈ K and we consider K as an 𝒜 –module by letting t act on K as multiplication by q.

The module S^{λ} is a Specht module of ℋ_{q}(B_{n}). It comes equipped with a symmetric bilinear form ⟨ , ⟩ such that ⟨uT_{i}, v⟩ = ⟨u, vT_{i}⟩ for 0 ≤ i < n and all u, v ∈ S^{λ}.

Consequently, the module

rad S^{λ} = { u ∈ S^{λ} | ⟨u, v⟩ = 0 for all v ∈ S^{λ} }

is an ℋ –submodule of S^{λ}. Set D^{λ} = S^{λ}/ rad S^{λ}.

The modules S^{λ} enjoy the following properties. Theorem 12(i) is proved by modifying the proof of [9, Theorem 6.1]. The others are stated in [9, Theorem 6.5] and [9, Theorem 6.6].

Theorem 12 (Dipper–James–Murphy [9]). Suppose that K is a field.

(i) Any ℋ_{q}(B_{n})–submodule of S^{λ} contains rad S^{λ} or is contained in rad S^{λ}. In particular, the module D^{λ} is either 0 or an absolutely irreducible self– dual ℋ_{q}(B_{n})–module.

(ii) { D^{μ} | D^{μ} ≠ 0 } is a complete set of pairwise non–isomorphic irreducible

ℋ_{q}(B_{n})–modules. (iii) If D^{μ} ≠ 0 then the decomposition multiplicity [S^{λ} : D^{μ}] ≠ 0 only if λ ⊵ μ;

further, [S^{μ} : D^{μ}] = 1.

In particular, if D^{μ} ≠ 0 then D^{μ} is the unique head of S^{μ} and Rad S^{μ} = rad S^{μ}; consequently, S^{μ} is indecomposable and if S^{μ} ≠ D^{μ} then S^{μ} has Loewy length at least 2. If D^{μ} ≠ 0 let P^{μ} be the corresponding principal indecomposable ℋ – module; in other words, P^{μ} is the projective cover of D^{μ}. Let d_{λμ} = [S^{λ} : D^{μ}] be the multiplicity of D^{μ} as a composition factor of S^{λ}.

It is implicit in the work of Dipper, James and Murphy that ℋ_{q}(B_{n}) is a cellular algebra in the sense of Graham and Lehrer [12] (compare [8]). Consequently, the theory of cellular algebras gives us the following result.

Corollary 13. Let P be a projective ℋ –module.

(i) Then P has a Specht filtration; thus, there exist bipartitions ν_{1}, …, ν_{k} and a filtration P = P^{k} > P^{k−1} > ⋯ > P^{1} > 0 such that P^{i}/P^{i−1} ≅ S^{ν_{i}}, for 1 < i ≤ k, and i < j whenever ν_{i} ▷ ν_{j}.

(ii) Suppose that P = P^{μ} for some bipartition μ with D^{μ} ≠ 0. Then the Specht filtration can be chosen so that d_{λμ} = # { 1 ≤ i ≤ k | ν_{i} = λ }. In particular, if λ is maximal in the dominance ordering such that d_{λμ} ≠ 0 then P^{μ} has a submodule isomorphic to S^{λ}.

Proof. These results are implicit in the work of Graham–Lehrer [12] (and slightly more explicit in [16, Lemma 2.19]). The existence of the filtration is exactly [12, Lemma 2.9(ii)]; that we can order the bipartitions ν_{i} by dominance follows from the choice of Φ_{0} ⊂ Φ_{1} ⊂ ⋯ ⊂ Φ_{d} as a maximal chain of ideals in the proof of this result. Part (ii) follows by combining Lemma 2.10(i) and Theorem 3.7(ii) of [12]. □

We remark that some care must be taken when working with Specht filtrations because Specht modules indexed by different bipartitions can be isomorphic when ℋ is not semisimple. This technicality can be avoided by working with a modular system and lifting the projective module to the discrete valuation ring where the Specht filtration is unambiguously defined.

In principle, Theorem 12(ii) produces all of the irreducible ℋ –modules; however, determining when D^{λ} is non–zero is still a difficult problem. The non–zero D^{λ} have now been classified by the first author; to describe this result we need some more nomenclature.

The diagram [λ] of a bipartition λ = (λ^{(1)}, λ^{(2)}) is the set of nodes

[λ] = { (i, j, k) | 1 ≤ j ≤ λ_{i}^{(k)} for i ≥ 1 and k = 1, 2 } ,

which we will think of as being an ordered pair of arrays of boxes in the plane.

Given two nodes x = (i, j, k) and y = (i′, j′, k′) we say that y is below x if either k = k′ and i < i′, or k < k′. Further, x ∈ [λ] is removable if [λ] \ {x} is the diagram of a bipartition; similarly, y ∉ [λ] is addable if [λ] ∪ {y} is the diagram of a bipartition. The content of x is c(x) = j − i + (k − 1)f and the residue of x is res(x) = c(x) (mod e) — recall that q is an e^{th} root of unity and Q = −q^{f}. If r = res(x) we call x an r–node.

An r–node x is normal if whenever y is a removable r–node below x then there are more removable r–nodes between x and y than there are addable r–nodes, and there are at least as many removable r–nodes below x as addable r–nodes below x. In addition, x is good if there are no normal r–nodes above x. Here, 0 ≤ r < e.

Finally, a bipartition μ is Kleshchev if either μ = ((0), (0)) or μ contains a good node x such that [μ] \ {x} is the diagram of a Kleshchev bipartition.

Theorem 14 (Ariki [2]). Suppose that μ is a bipartition of n. Then D^{μ} ≠ 0 if and only if μ is a Kleshchev bipartition.

The proof of this result builds on the next theorem which reveals the deep connections between the representation theory of ℋ_{q}(B_{n}) and the representation theory of the Kac–Moody algebra U(𝔰̂𝔩_{e}) of type A_{e−1}^{(1)} in characteristic zero. Let Λ_{0}, …, Λ_{e−1} be the fundamental weights of U(𝔰̂𝔩_{e}) and for each dominant weight Λ let L(Λ) be the corresponding integral highest weight module.

Let ℋ_{q}(B_{n})–mod be the category of finite dimensional right ℋ_{q}(B_{n})–modules and ℋ_{q}(B_{n})–proj be the category of finite dimensional projective ℋ_{q}(B_{n})–modules. Finally, let K_{0}(C) be the Grothendieck group of the category C.

Theorem 15 (Ariki [1]). For i = 0, 1, …, e − 1 there exist exact functors

e_{i}, f_{i}: ℋ_{q}(B_{n})–mod⟶ℋ_{q}(B_{n±1})–mod

such that the operators induced by these and suitably defined operators h_{i} for i = 0, 1, …, e − 1 and d give K_{0} = ⨁_{n≥0} K_{0}(ℋ_{q}(B_{n})–proj) ⊗_{ℤ} ℚ the structure of a U(𝔰̂𝔩_{e})–module. Furthermore, K_{0} ≅ L(Λ_{0} + Λ_{f}) as an U(𝔰̂𝔩_{e})–module and if K is a field of characteristic zero then the principal indecomposable ℋ_{q}(B_{n})–modules correspond to elements of the Lusztig–Kashiwara canonical basis of L(Λ_{0} + Λ_{f}) under this isomorphism.

A modular system with parameters is a modular system (K, O, k) such that K is a field of characteristic zero and O is a discrete valuation ring with residue field k, together with parameters t ∈ O ↪ K and q ∈ k such that t and q have the same multiplicative order in K and k respectively and t maps to q under the canonical map O → k. Let ℋ_{k} = ℋ_{k,q}(B_{n}) and ℋ_{R} = ℋ_{R,t}(B_{n}), for R ∈ {K, O}, be the corresponding Hecke algebras. Then ℋ_{R} ≅ ℋ_{O} ⊗_{O} R for R ∈ {K, k}.

Let Λ = Λ_{0} + Λ_{f} and fix a highest weight vector v_{Λ} in L(Λ).

If μ is a Kleshchev bipartition then we write P_{R}^{μ} for the principal indecomposable ℋ_{R}–module for R ∈ {K, O, k}. So if O is a complete discrete valuation ring then P_{k}^{μ} = P_{O} ⊗_{O} k and P_{K}^{μ} is a direct summand of P_{O} ⊗_{O} K. By abuse of notation, we also let [P_{K}^{μ}] denote both the equivalence class of P_{K}^{μ} in K_{0} and the corresponding canonical basis element of L(Λ).

Corollary 16. Let (K, O, k) be a modular system with parameters. Suppose that μ is a Kleshchev bipartition such that [P_{K}^{μ}] = f_{i_{1}}^{(m_{1})} … f_{i_{l}}^{(m_{l})}v_{Λ} for some m_{1}, …, m_{l} and i_{1}, …, i_{l}. Then the decomposition map sends [P_{K}^{μ}] to [P_{k}^{μ}].

Proof. Let O ⊂ Õ be an embedding into a complete discrete valuation ring Õ where Õ has residue field k and K ⊂ K̃ is a field extension such that Õ ⊂ K̃. Since D^{μ} is absolutely irreducible, P μ ; K̃ = P_{K}^{μ} ⊗ K̃. Hence it is enough to prove the statement under the assumption that O is a complete discrete valuation ring.

Set N = m_{1}! … m_{l}! and let M_{O} = f_{i_{1}}^{m_{1}} … f_{i_{l}}^{m_{l}}1_{ℋ_{O}} where 1_{ℋ_{O}} is the trivial ℋ_{O,t}(B_{0})–module. Then M_{O} is a projective ℋ_{O}–module by the definition of the f_{i}. Therefore, by Corollary 13 applied to ℋ_{O}, the module M_{O} has a Specht filtration. Let M_{K} = M_{O} ⊗ K; then, by assumption, [M_{K}] = N[P_{K}^{μ}]; therefore, [M_{K}] = N[S_{K}^{μ}] + ∑_{λ▷μ} Nd_{λμ}[S_{K}^{λ}], where d_{λμ} = [S_{K}^{λ} : D_{K}^{μ}]. Consequently, there is a surjective homomorphism M_{O} → S_{O}^{μ}^{⊕N}; tensoring with k gives a surjective homomorphism M_{k} →^{φ} D_{k}^{μ}^{⊕N} (as D^{μ} is the head of S^{μ}). Since M_{k} is a projective ℋ_{k}–module there exists a map ψ which makes the following diagram commute.

\begin{aligned}& M_{k} \\ P_{k}^{μ}^{⊕N} & D_{k}^{μ}^{⊕N} & 0_{φ}_{ψ} \end{aligned}

Now, φ is surjective so ψ must also be surjective. On the other hand, P_{k}^{μ} is a projective ℋ_{k}–module so we also have a commutative diagram

\begin{matrix}P_{k}^{μ} & ⊕N \\ M_{k} & P_{k}^{μ}^{⊕N} & 0_{ψ} \end{matrix}

Hence, P_{k}^{μ}^{⊕N} is a direct summand of M_{k}. Note that

dim_{k} P_{k}^{μ}^{⊕N} = dim_{K}(P_{O}^{μ} ⊗ K)^{⊕N} ≥ dim_{K} P_{K}^{μ}^{⊕N}
= dim_{K} M_{K} = dim_{k} M_{k} ≥ dim_{k} P_{k}^{μ}^{⊕N};

thus, M_{k} = P_{k}^{μ}^{⊕N}. However, M_{K} = P_{K}^{μ}^{⊕N}; so we have shown that the modular reduction of P_{K}^{μ}^{⊕N} is P_{k}^{μ}^{⊕N}, as required. □

In order to apply the last two results we need to set up the machinery for computing the canonical basis elements [P_{K}^{λ}]. Let v be an indeterminant over ℤ and let 𝒜 = ℤ[v, v^{−1}]. The Fock space is the infinite dimensional 𝒜 –module

F_{𝒜} = ⨁_{n≥0} ⨁_{λ⊢n} 𝒜 λ.

Let U_{𝒜} (𝔰̂𝔩_{e}) be Lusztig’s 𝒜 –form of the quantum group of U(𝔰̂𝔩_{e}). Then there are v–analogues E_{i} and F_{i} of the operators e_{i} and f_{i} which act on F_{𝒜} and give it the structure of a U_{𝒜} (𝔰̂𝔩_{e})–module; an explicit description of E_{i} and F_{i} is given below. The U_{𝒜} (𝔰̂𝔩_{e})–submodule of F_{𝒜} generated by the bipartition ((0), (0)) is isomorphic to L_{𝒜} (Λ), the 𝒜 –form of L(Λ). (Recall that Λ = Λ_{0} + Λ_{f}.)

Identifying L_{𝒜} (Λ) with U_{𝒜} (𝔰̂𝔩_{e}) · ((0), (0)), Theorem 15 can be reinterpreted as saying that if μ is a Kleshchev bipartition then there exist polynomials d_{λμ}(v) such that

(17)  [P_{K}^{μ}] = μ + ∑_{λ⊢n} d_{λμ}(v)λ, where d_{λμ}(v) ∈ vℤ[v],

and d_{λμ} = d_{λμ}(1). Uglov [17] has given an explicit algorithm for computing the canonical basis elements [P_{K}^{λ}]. Uglov actually works with a different Fock space; however, we can compute the canonical basis of inside our Fock space by modifying his algorithm. For the applications we have in mind it is enough to know that if F_{i_{1}}^{(m_{1})} … F_{i_{l}}^{(m_{l})} · ((0), (0)) can be written in the form of the right hand side of (17) then it is an element of the canonical basis of L_{𝒜} (Λ); hence, in such situations we may apply Corollary 16. We now recall from [3] how U_{𝒜} (𝔰̂𝔩_{e}) acts on F_{𝒜} .

Suppose that λ is a bipartition of n − 1 and that μ is a bipartition of n. We write λ ⟶^{i} μ if [μ] \ [λ] = {x} and res(x) = q^{i}. For 0 ≤ i < e let

N_{i}^{r}(ν, λ) = # { ν ⟶^{i} α | α ▷ λ } − # { β ⟶^{i} λ | ν ▷ β } ,
N_{i}^{l}(λ, μ) = # { λ ⟶^{i} α | μ ▷ α } − # { β ⟶^{i} μ | β ▷ λ } .

By [3, Prop. 2.6] the action of E_{i} and F_{i} on F_{𝒜} is determined by

E_{i}λ = ∑_{ν⟶^{i}λ} v^{−N_{i}^{r}(ν,λ)}ν,  and  F_{i}λ = ∑_{λ⟶^{i}μ} v^{N_{i}^{l}(λ,μ)}μ.

We will only need the formula giving the action of F_{i}. There are similar formulae for the action of the remaining generators of U_{𝒜} (𝔰̂𝔩_{e}).

The final tool that we shall need is the analogue of the Jantzen sum formula for Hecke algebras of type B over an arbitrary field K. The setup is a little technical; we include it for completeness. Let 𝔭 be the maximal ideal of K[t] generated by t−q and let O = K[t]_{𝔭}, where t is an indeterminate over K; then K ≅ O/𝔭 (so O is a localized ring and, in particular, a discrete valuation ring). Let ℋ_{O} be the Hecke algebra over O with parameters t and −q^{f}(t−q+1)^{n}; then ℋ_{O} ⊗K(t) is semisimple and ℋ_{K} is the reduction of ℋ_{O} modulo 𝔭. Let ν_{𝔭} be the 𝔭–adic valuation on O.

Previously we defined the residue res(x) of a node x = (i, j, k). Define the O– residue of x = (i, j, k) to be res_{O}(x) = t^{j−i}q^{(k−1)f}(t − q + 1)^{(k−1)n}. The relationship between these two definitions is that res_{O}(x) ⊗ 1_{K} = q^{res(x)}.

Let λ be a bipartition and for each node x = (i, j, k) ∈ [λ] let r_{x} be the corresponding rim hook (so r_{x} is a rim hook in [λ^{(k)}]); the point is that [λ] \ r_{x} is the diagram of a bipartition. Let ℓℓ(r_{x}) be the leg length of r_{x} and define res_{O}(r_{x}) = res_{O}(f_{x}) where f_{x} is the foot node of r_{x}. The definitions of these terms can be found, for example, in [16].

Suppose that λ and μ are bipartitions of n. If λ ▷̸ μ let g_{λμ} = 1; otherwise set

g_{λμ} = ∏_{x∈[λ]} ∏_{[μ]\r_{y}=[λ]\r_{x}^{y∈[μ]}} ( res_{O}(r_{x}) − res_{O}(r_{y}))^{ε_{xy}},

where ε_{xy} = (−1)^{ℓℓ(r_{x})+ℓℓ(r_{y})}. The g_{λμ} are not as complicated as their definition suggests; they have a nice combinatorial interpretation, see [14, Example 3.39].

Finally, let S_{O}^{λ} and S_{K}^{λ} be the Specht modules for ℋ_{O} and ℋ_{K} respectively. For each i ≥ 0 define S_{O}^{λ}(i) = { u ∈ S_{O}^{λ} | ⟨u, v⟩ ∈ 𝔭^{i} for all v ∈ S_{O}^{λ} }. Then the Jantzen filtration of S_{K}^{λ} is the filtration

S_{K}^{λ} = S_{K}^{λ}(0) ≥ S_{K}^{λ}(1) ≥ S_{K}^{λ}(2) ≥ …

where S_{K}^{λ}(i) = (S_{O}^{λ}(i) + 𝔭S_{O}^{λ})/𝔭S_{O}^{λ}. In particular, note that rad S_{K}^{λ} = S_{K}^{λ}(1). We can now state the analogue of Jantzen’s sum formula for ℋ_{K}.

Theorem 18 (James–Mathas [14, Theorem 4.6]). Let λ be a bipartition of n. Then

∑_{i>0}[S_{K}^{λ}(i)] = ∑_{μ:λ▷μ} ν_{𝔭}(g_{λμ})[S_{K}^{μ}].

in the Grothendieck group K_{0}(ℋ_{K}–mod) of ℋ_{K}.

In general, if λ ▷ μ then ν_{𝔭}(g_{λμ}) is non–zero only if it is possible to remove a rim hook r_{x} from λ and reattach it to μ without changing the residue res(r_{x}) of the foot node. In fact, we will only apply this result when n < e; in this situation we have ν_{𝔭}(g_{λμ}) ∈ {0, 1} so the technicalities above can be ignored.

All of the composition factors of a Specht module belong to the same block (for example, because ℋ is cellular); we abuse notation and say that λ is contained in the block B if S^{λ} is contained in B. Say that two bipartitions λ and μ are linked by hooks if there is a sequence of bipartitions λ = ν_{1}, …, ν_{l} = μ such that, for each i, [ν_{i+1}] \ r_{y_{i}} = [ν_{i}] \ r_{x_{i}} and res(r_{x_{i}}) = res(r_{y_{i}}) for some nodes x_{i} ∈ [ν_{i+1}] and y_{i} ∈ [ν_{i}].

Proposition 19. Suppose that S^{λ} and S^{μ} are in the same block. Then λ and μ are linked by hooks.

Proof. By definition, S^{λ} and S^{μ} are in the same block if and only if there exists a sequence of bipartitions μ = ν_{1}, …, ν_{l} = λ such that S^{ν_{i}} and S^{ν_{i+1}} have a common composition factor. Thus, it is enough to prove that if D^{μ} ≠ 0 appears in S^{λ} then λ and μ are linked by hooks. If λ ≠ μ then λ ▷ μ by Theorem 12(iii). The sum formula implies that D^{μ} appears in S^{ν} for some ν such that λ ▷ ν and λ and ν are linked by hooks. By induction on dominance ν and μ are linked by hooks so we are done. □

If λ is a bipartition let res(λ) be the multiset { res(x) | x ∈ [λ] }. Then as a corollary of the Proposition we have the following (there is an easier proof).

Corollary 20 (Dipper–James–Murphy [9, Corollary 8.7]). Suppose that S^{λ} and S^{μ} are in the same block. Then res(λ) = res(μ) (as multisets).

By the Corollary we can define the residue of a block B to be the multiset res(B) = res(λ) where λ is any bipartition contained in B.

In fact, Grojnowski [13] has recently shown that the converse of Corollary 20 is true; so, two Specht modules S^{λ} and S^{μ} belong to the same block if and only if res(λ) = res(μ). We will not need this stronger result.

4. The representation type when n = e

In this section we will prove the following result. Recall that we are assuming that e ≥ 3.

Theorem 21. Suppose n ≥ e. Then ℋ has infinite representation type.

Proof. By Corollary 11 we may assume that n = e. Further, by Lemma 5 it is enough to show that one block of ℋ has infinite representation type; we will show that the block B with residues {0, 1, …, e − 1} has infinite representation type.

There will be several cases to consider. To begin suppose that f ≠ 0. Because all of the residues in B are distinct a bipartition λ = (λ^{(1)}, λ^{(2)}) appears in B only if λ^{(1)} and λ^{(2)} are both hook partitions; that is, λ = ((a, 1^{b}), (c, 1^{d})) for some a, b, c, d. Define bipartitions

λ_{k} = ((0), (k, 1^{e−k})),  for 1 ≤ k ≤ e,
μ_{k} = ((k, 1^{e−k}), (0)),  for 1 ≤ k ≤ e,
λ_{k,l} = ((f − l, 1^{e−f−k}), (k, 1^{l})),  for 1 ≤ k ≤ e − f and 0 ≤ l < f.

It is easily checked that all of these bipartitions belong to B. Certainly, the two sets of bipartitions {λ_{k}} and {μ_{k}} are disjoint; the restrictions on k and l ensure that λ_{k,l} ≠ λ_{m} and λ_{k,l} ≠ μ_{m} for any m. Consequently, this is a complete list of the bipartitions which appear in B, with no repeats.

Proposition 22. Suppose that n = e and that 1 ≤ f ≤ \frac{e}{2}.

(i) The complete set of Kleshchev bipartitions in B is

{ λ_{k} | 1 ≤ k < e } ∪ { λ_{k,l} | 1 ≤ k ≤ e − f and 0 ≤ l < f } .

(ii) For 1 ≤ k < e we have

[P^{λ_{k}}] = [S^{λ_{k}}] + [S^{λ_{k+1}}] + \begin{cases}[S^{λ_{k,f−1}}] + [S^{λ_{k+1,f−1}}], & if k < e − f, \\ [S^{λ_{e−f,f−1}}], & if k = e − f, \\ [S^{λ_{e−f,e−k}}] + [S^{λ_{e−f,e−k−1}}], & if k > e − f. \end{cases}

(iii) For 1 ≤ k ≤ e − f and 0 ≤ l < f we have

[P^{λ_{k,l}}] = [S^{λ_{k,l}}] + \begin{cases}[S^{λ_{k−1,0}}] + [S^{μ_{f+k−1}}] + [S^{μ_{f+k}}], & if k ≠ 1 and l = 0, \\ [S^{μ_{f}}] + [S^{μ_{f+1}}], & if k = 1 and l = 0, \\ [S^{λ_{k,l−1}}] + [S^{λ_{k−1,l}}] + [S^{λ_{k−1,l−1}}], & if k ≠ 1 and l ≠ 0, \\ [S^{λ_{1,l−1}}] + [S^{μ_{f−l}}] + [S^{μ_{f−l+1}}], & if k = 1 and l ≠ 0. \end{cases}

Proof. It is easy to see that the bipartitions λ_{e} and μ_{k}, for 1 ≤ k ≤ e, are not Kleshchev. Next suppose that 1 ≤ k ≤ e − f. Then a straightforward computation shows that

F_{f+k} … F_{e−1}F_{0}F_{f+k−1} … F_{f+1}F_{1} … F_{f}((0), (0))

= F_{f+k} … F_{e−1}F_{0}((0), (k, 1^{f−1}))

= \begin{cases}λ_{k} + vλ_{k+1} + vλ_{k+1,f−1} + v^{2}λ_{k,f−1}, & if 1 ≤ k < e − f, \\ λ_{e−f} + vλ_{e−f+1} + v^{2}λ_{e−f,f−1}, & if k = e − f. \end{cases}

Note that ((0), (k, 1^{f−1})) has two addable 0–nodes, an addable f–node and an addable f + k–node which is a 0–node if k = e − f, and when we add an addable r–node, there is no removable r–node.

This shows that λ_{k} is Kleshchev for 1 ≤ k ≤ e − f; the formula for [P^{λ_{k}}] now follows from Corollary 16 (and the remarks after (17)).

The remaining cases are similar: for λ_{k} with e − f < k < e we have f ≥ 2 and we compute

F_{k−e+f} … F_{f−1}F_{k−e+f−1} … F_{1}F_{0}F_{e−1} … F_{f+1}F_{f}((0), (0))
= F_{k−e+f} … F_{f−1}F_{k−e+f−1} … F_{1}F_{0}((0), (e − f))
= F_{k−e+f} … F_{f−1}[((0), (k)) + v((k − e + f), (e − f))]
= F_{k−e+f}[((0), (k, 1^{e−k−1})) + v((k − e + f), (e − f, 1^{e−k−1}))]
= λ_{k} + vλ_{k+1} + vλ_{e−f,e−k} + v^{2}λ_{e−f,e−k−1};

for λ_{k,l} with 1 ≤ k ≤ e − f and l = 0 compute

F_{f+k−1} … F_{f}F_{f+k} … F_{e−1}F_{f−1} … F_{0}((0), (0))

= F_{f+k−1} … F_{f}F_{f+k} … F_{e−1}((f), (0)) = F_{f+k−1} … F_{f}((f, 1^{e−f−k}), (0))

= \begin{cases}λ_{k,0} + vλ_{k−1,0} + vμ_{f+k−1} + v^{2}μ_{f+k}, & if k ≠ 1, \\ λ_{1,0} + vμ_{f} + v^{2}μ_{f+1}, & if k = 1. \end{cases}

for λ_{k,l} with 0 < l < f and 2 ≤ k ≤ e − f compute

F_{f−l} … F_{f−1}F_{f+k−1} … F_{f}F_{f+k} … F_{e−1}F_{f−l−1} … F_{0}((0), (0))
= F_{f−l} … F_{f−1}F_{f+k−1} … F_{f}((f − l, 1^{e−f−k}), (0))
= F_{f−l}[((f − l, 1^{e−f−k}), (k, 1^{l−1}))
+v((f − l, 1^{e−f−k+1}), (k − 1, 1^{l−1}))];

and, finally, for λ_{1,l} with 0 < l < f compute

F_{f−l} … F_{f−1}F_{f}F_{f+1} … F_{e−1}F_{f−l−1} … F_{0}((0), (0))
= F_{f−l} … F_{f−1}F_{f}((f − l, 1^{e−f−1}), (0))
= F_{f−l} … F_{f−1}[((f − l, 1^{e−f−1}), (1)) + v((f − l, 1^{e−f}), (0))]
= F_{f−l}[((f − l, 1^{e−f−1}), (1^{l})) + v((f − l, 1^{e−f+l−1}), (0))].

In each case, an application of 16 now completes the proof. □

By Lemma 6(i) in order to prove Theorem 21 it is enough to show that End_{ℋ} (P^{λ_{1}}) is not isomorphic to k[x]/⟨x^{m}⟩ for any m. We need to consider several cases. First we observe that

(23)  [P^{λ_{1}}] = [S^{λ_{1}}] + [S^{λ_{2}}] + [S^{λ_{2,f−1}}] + [S^{λ_{1,f−1}}]

by Proposition 22(i). We will use this to determine the structure of P^{λ_{1}}. Case 1: e ≥ 5 and f ≥ 2. Then e − f ≥ 3 since e − f ≥ \frac{e}{2} = 2.5. By (17) we can use Proposition 22 to compute the decomposition numbers d_{λμ} for the four Specht modules appearing in (23); this gives the following table (omitted entries are zero).

D^{λ_{1}} D^{λ_{2}} D^{λ_{3,f−1}} D^{λ_{2,f−1}} D^{λ_{1,f−1}}
S^{λ_{1}}  1  .  .  .  .
S^{λ_{2}}  1  1  .  .  .
S^{λ_{2,f−1}} 1  1  1  1  .
S^{λ_{1,f−1}} 1  0  0  1  1

Consequently, [P^{λ_{1}}] = 4[D^{λ_{1}}]+2[D^{λ_{2}}]+[D^{λ_{1,f−1}}]+2[D^{λ_{2,f−1}}]+[D^{λ_{3,f−1}}]. By Corollary 13(ii), P^{λ_{1}} has submodule isomorphic to S^{λ_{1,f−1}}. Therefore, S^{λ_{1,f−1}} has both a simple head and simple socle; so, looking at the submatrix of the decomposition matrix above, the Loewy structure of S^{λ_{1,f−1}} is

S^{λ_{1,f−1}} = \begin{matrix}D^{λ_{1,f−1}} \\ D^{λ_{2,f−1}} \\ D^{λ_{1}} \end{matrix} .

We also have

S^{λ_{2}} = \begin{matrix}D^{λ_{2}} \\ D^{λ_{1}} \end{matrix} .

Considering the dual of Rad S^{λ_{1,f−1}} and S^{λ_{2}}, we conclude that D^{λ_{2}} and D^{λ_{2,f−1}} appear in Rad P^{λ_{1}}/ Rad^{2} P^{λ_{1}}. If 0 ⟶ D^{λ_{1}} ⟶ X ⟶ D^{λ_{1}} ⟶ 0 is an exact sequence then, for 1 ≤ i < n, T_{i} −q acts invertibly on X so that T_{i} acts as −1 on X. Similarly, T_{0} also acts as multiplication by a scalar on X since f ≠ 0. Therefore, every such exact sequence splits and so Ext^{1}(D^{λ_{1}}, D^{λ_{1}}) = 0; consequently, D^{λ_{1}} is not a composition factor of Rad P^{λ_{1}}/ Rad^{2} P^{λ_{1}}.

Again, by Corollary 13 P^{λ_{1}} has a Specht filtration so Rad P^{λ_{1}}/S^{λ_{1,f−1}} has a Specht filtration whose successive quotients are S^{λ_{2}} and S^{λ_{2,f−1}}. This means that Rad P^{λ_{1}}/ Rad^{2} P^{λ_{1}} = D^{λ_{2}} ⊕ D^{λ_{2,f−1}}.

We now use the fact that Rad S^{λ_{1,f−1}} = {D^{λ_{2,f−1}} \atop D^{λ_{1}}} to prove that D^{λ_{1}} is contained in Rad S^{λ_{2,f−1}}/ Rad^{2} S^{λ_{2,f−1}}. As Rad S^{λ_{1,f−1}} has a unique head there is a surjection P^{λ_{2,f−1}} ⟶ Rad S^{λ_{1,f−1}}. Note also that P^{λ_{2,f−1}} has a Specht filtration with successive quotients S^{λ_{2,f−1}} and S^{λ_{2,f−2}}, S^{λ_{1,f−1}}, S^{λ_{1,f−2}}. Since S^{λ_{2,f−2}}, S^{λ_{1,f−1}}, S^{λ_{1,f−2}} must map to D^{λ_{1}}, and each of these has unique head which is not isomorphic to D^{λ_{1}}, this surjection induces a map S^{λ_{2,f−1}} ⟶ Rad S^{λ_{1,f−1}}. Therefore, Rad S^{λ_{2,f−1}}/ Rad^{2} S^{λ_{2,f−1}} contains D^{λ_{1}} as a summand. Let U be a module such that Rad^{2} S^{λ_{2,f−1}} ⊆ U ⊆ Rad S^{λ_{2,f−1}} and Rad S^{λ_{2,f−1}}/U ≅ D^{λ_{1}} and set

V = (Rad P^{λ_{1}})/(U + S^{λ_{1,f−1}}).

Then there is a short exact sequence

0 ⟶ {D^{λ_{2,f−1}} \atop D^{λ_{1}}} ⟶ V ⟶ {D^{λ_{2}} \atop D^{λ_{1}}} ⟶ 0

and V/ Rad V ≅ D^{λ_{2}} ⊕ D^{λ_{2,f−1}}. Hence, we also have 0 ⟶ D^{λ_{1}} ⟶ Rad V ⟶ D^{λ_{1}} ⟶ 0 and consequently Rad V = D^{λ_{1}} ⊕ D^{λ_{1}}. Therefore, Rad^{2} P^{λ_{1}}/ Rad^{3} P^{λ_{1}} contains D^{λ_{1}} ⊕D^{λ_{1}} and it follows that P^{λ_{1}}/ Rad^{3} P^{λ_{1}} contains D^{λ_{1}} ⊕D^{λ_{1}} as an ℋ – submodule, that another D^{λ_{1}} appears as the head of P^{λ_{1}}/ Rad^{3} P^{λ_{1}}, and that these are the only ways in which D^{λ_{1}} appear as a composition factor of P^{λ_{1}}/ Rad^{3} P^{λ_{1}}. Therefore, we have that End_{ℋ / Rad^{3} ℋ} (P^{λ_{1}}/ Rad^{3} P^{λ_{1}}) ≅ k̸[x]/⟨x^{m}⟩. Hence, End_{ℋ} (P^{λ_{1}}) is not of finite representation type by Lemma 6, so ℋ has infinite representation type by Lemma 5(ii). Case 2: e = 4 and f = 2. By Proposition 22 and (23), we have the same table as above if we replace λ_{3,f−1} by λ_{3}. Thus

[P^{λ_{1}}] = 4[D^{λ_{1}}] + 2[D^{λ_{2}}] + [D^{λ_{3}}] + 2[D^{λ_{2,1}}] + [D^{λ_{1,1}}].

The argument used in case 1 shows that P^{λ_{1}}/ Rad^{3} P^{λ_{1}} contains D^{λ_{1}} ⊕ D^{λ_{1}} as an ℋ –submodule. Hence, End_{ℋ} (P^{λ_{1}}) is again of infinite type. Case 3: f = 1. Again using Proposition 22 and (23) we find that

[P^{λ_{1}}] = 4[D^{λ_{1}}] + 2[D^{λ_{2}}] + [D^{λ_{1,0}}] + 2[D^{λ_{2,0}}] + δ_{e≥4}[D^{λ_{3,0}}],

where δ_{e≥4} = 1 if e ≥ 4 and δ_{e≥4} = 0 otherwise. Almost the same argument as before again shows that P^{λ_{1}}/ Rad^{3} P^{λ_{1}} contains D^{λ_{1}} ⊕ D^{λ_{1}} as an ℋ –submodule; however, to show that Rad S^{λ_{2,0}}/ Rad^{2} S^{λ_{2,0}} contains D^{λ_{1}} we need to argue in a different way: since [P^{λ_{2,0}}] = [S^{λ_{2,0}}] + [S^{λ_{1,0}}] + [S^{μ_{2}}] + [S^{μ_{3}}] and [S^{μ_{2}}] = [D^{λ_{2,0}}] + [D^{λ_{1,0}}], [S^{μ_{3}}] = [D^{λ_{3,0}}] + [D^{λ_{2,0}}] do not contain D^{λ_{1}}, the surjection P^{λ_{2,0}} ⟶ Rad S^{λ_{1,0}} induces a surjection S^{λ_{2,0}} ⟶ Rad S^{λ_{1,0}}. Thus, once again ℋ is of infinite representation type by Lemma 6 and Lemma 5. Case 4: f = 0. It remains to consider the case f = 0. This case is somewhat degenerate as 1 is the only possible eigenvalue for the action of T_{0} upon a representation. Because f = 0 there are no bipartitions of the form λ_{k,l} in B so Proposition 22 cannot hold in this case. A similar argument shows that we have the following simpler statement when f = 0.

Proposition 24. Suppose that f = 0.

(i) { λ_{k} | 1 ≤ k < e } is the complete set of Kleshchev bipartitions in B. (ii) For 1 ≤ k < e we have [P^{λ_{k}}] = [S^{λ_{k}}] + [S^{λ_{k+1}}] + [S^{μ_{k}}] + [S^{μ_{k+1}}].

To apply this result we do not have to work as hard as in the previous cases. First observe that for 1 ≤ k < e the module D^{λ_{k}} can be constructed by letting T_{0} act as 1 on the simple ℋ (A_{e})–module D^{(k,1^{e−k})} (by induction both modules have the same dimension; namely, \binom{e−1}{k−1}). Let M^{λ_{k}} be the ℋ –module obtained by

letting T_{0} act as ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) on the ℋ (A_{e})–module D^{(k,1^{e−k})} ⊕ D^{(k,1^{e−k})}. Then M^{λ_{k}}

cannot be semisimple as an ℋ –module because T_{0} does not act as a scalar. On the other hand, the socle of M^{λ_{k}} is simple (being isomorphic to D^{λ_{k}}); hence, M^{λ_{k}} is indecomposable. This implies that [Rad P^{λ_{k}}/ Rad^{2} P^{λ_{k}} : D^{λ_{k}}] ≠ 0.

Now [S^{λ_{k}}] = [D^{λ_{k}}] + [D^{λ_{k−1}}] for 1 ≤ k < e by Proposition 24. Hence, S^{λ_{k}} is indecomposable and [Rad P^{λ_{k}}/ Rad^{2} P^{λ_{k}} : D^{λ_{k−1}}] ≠ 0 for 2 ≤ k < e. As D^{λ_{k}} is self–dual, by taking duals we also have

[Rad P^{λ_{k}}/ Rad^{2} P^{λ_{k}} : D^{λ_{k+1}}] ≠ 0,

for 1 ≤ k < e − 1. Combining these facts we conclude that

Rad P^{λ_{1}}/ Rad^{2} P^{λ_{1}} ⊃ D^{λ_{1}} ⊕ D^{λ_{2}},
Rad P^{λ_{k}}/ Rad^{2} P^{λ_{k}} ⊃ D^{λ_{k−1}} ⊕ D^{λ_{k}} ⊕ D^{λ_{k+1}}, for 2 ≤ k < e − 1,

and that Rad P^{λ_{e−1}}/ Rad^{2} P^{λ_{e−1}} ⊃ D^{λ_{e−1}} ⊕ D^{λ_{e−2}}. Therefore, the separation diagram of B contains

⋯

Consequently, B has infinite representation type by Theorem 7.

This completes the proof of Theorem 21. □

5. The representation type when n ≥ 2f + 4

Recall that we are assuming that e ≥ 3 and 0 ≤ f ≤ \frac{e}{2}. This section is devoted to the proof of the following result.

Theorem 25. Suppose n ≥ 2f + 4. Then ℋ has infinite representation type.

Proof. By Corollary 11 it is enough to show that ℋ_{n} has infinite representation type when n = 2f + 4. Also, by the last section we may assume that e > n. Let B be the block B with residues {−1, 0, 0, 1, 1, …, f, f, f + 1}; so B contains the bipartition ((2, 2), (2^{f}))). We will show that B has infinite representation type. To do this we need to consider two cases separately.

Case 1: Suppose that f = 0 (and e > n = 4). We consider the block B with residues {−1, 0, 0, 1}. It is easy to see that there are precisely six bipartitions in this block; namely, λ_{1} = ((0), (2^{2})), λ_{2} = ((1), (2, 1)), λ_{3} = ((1^{2}), (2)), λ_{4} = ((2), (1^{2})), λ_{5} = ((2, 1), (1)) and λ_{6} = ((2^{2}), (0)). Furthermore, of these bipartitions only λ_{1} and λ_{2} are Kleshchev. Since

F_{0}F_{1}F_{e−1}F_{0}((0), (0)) = F_{0}[((0), (2, 1)) + v((2, 1), (0))]
= λ_{1} + vλ_{2} + vλ_{5} + v^{2}λ_{6}

and

F_{1}F_{e−1}F_{0}^{(2)}((0), (0)) = F_{1}F_{e−1}((1), (1)) = F_{1}[((1), (1^{2})) + v((1^{2}), (1))]
= λ_{2} + vλ_{4} + vλ_{3} + v^{2}λ_{5},

by Corollary 16 the transpose of the decomposition matrix of B is

S^{λ_{1}} S^{λ_{2}} S^{λ_{3}} S^{λ_{4}} S^{λ_{5}} S^{λ_{6}}
D^{λ_{1}} 1  1  0  0  1  1
D^{λ_{2}} 0  1  1  1  1  0

Let μ_{1} = ((0), (2, 1)) and μ_{2} = ((1), (1^{2})). We first observe that P^{λ_{2}} is the induced module P^{μ_{2}} ↑^{B}; this follows from a calculation in the Grothendieck group. Similarly, we also have that P^{λ_{1}} = P^{μ_{1}} ↑^{B}. Further, P^{μ_{i}} is uniserial of length 2 and [P^{μ_{i}}] = 2[D^{μ_{i}}], for i = 1, 2. Next note that D^{μ_{i}} = S^{μ_{i}}, for i = 1, 2; hence, the branching rules for Specht modules imply that [D^{μ_{1}} ↑^{B}] = 2[D^{λ_{1}}] + [D^{λ_{2}}] and [D^{μ_{2}} ↑^{B}] = [D^{λ_{1}}] + 2[D^{λ_{2}}].

Next note that there are surjective homomorphisms P^{λ_{i}} ⟶ D^{μ_{i}} ↑^{B} for i = 1, 2. As D^{μ_{i}} ↑^{B} is self–dual, D^{μ_{i}} ↑^{B} is a uniserial module whose top and bottom is isomorphic to D^{λ_{i}}. To conclude, each P^{λ_{i}} has a submodule M_{i} such that both P^{λ_{i}}/M_{i} and M_{i} are isomorphic to the uniserial module D^{μ_{i}} ↑^{B}.

Now we prove that Ext^{1}(D^{λ_{2}}, D^{λ_{2}}) ≠ 0. As Ext^{1}(D^{λ_{1}}, D^{λ_{1}}) ≠ 0, we know that the radical series of P^{λ_{1}} has the form

\begin{matrix}D^{λ_{1}} \\ D^{λ_{1}} ⊕ D^{λ_{2}} \\ D^{λ_{1}} ⊕ D^{λ_{2}} \\ D^{λ_{1}} \end{matrix}

Assume that Ext^{1}(D^{λ_{2}}, D^{λ_{2}}) = 0. Then P^{λ_{2}} must have the form

\begin{matrix}D^{λ_{2}} \\ D^{λ_{1}} \\ D^{λ_{2}} ⊕ D^{λ_{2}} \\ D^{λ_{1}} \\ D^{λ_{2}} \end{matrix}

Consider a homomorphism P^{λ_{2}} ⟶ D^{λ_{2}} ⊂ Rad P^{λ_{1}}/ Rad^{2} P^{λ_{1}}. The assumption that Rad P^{λ_{2}}/ Rad^{2} P^{λ_{2}} = D^{λ_{1}} together with the radical structure of P^{λ_{1}} imply that this map lifts to a homomorphism P^{λ_{2}} ⟶ P^{λ_{1}}, whose image is of the form

\begin{matrix}D^{λ_{2}} \\ D^{λ_{1}} \\ D^{λ_{1}} \end{matrix}

This contradicts our assumption that Rad^{2} P^{λ_{2}}/ Rad^{3} P^{λ_{2}} = D^{λ_{2}} ⊕ D^{λ_{2}}.

Hence, Ext^{1}(D^{λ_{2}}, D^{λ_{2}}0 and Ext^{1}(D^{λ_{1}}, D^{λ_{1}}) are both non–zero. Therefore, Rad P^{λ_{1}}/ Rad^{2} P^{λ_{1}} and Rad P^{λ_{2}}/ Rad^{2} P^{λ_{2}} both contain D^{λ_{1}} ⊕ D^{λ_{2}} as an ℋ –submodule. Therefore, the separation diagram of B is . Consequently, B is of infinite type, by Gabriel’s theorem, and ℋ has infinite representation type by Lemma 5(ii). Case 2: Suppose that 0 < f ≤ \frac{e}{2}. Now we are considering the block B of ℋ with residues {−1, 0, 0, 1, 1, …, f, f, f + 1}. Notice that f + 2 is a residue in B only if f + 1 ≥ e − 1 and −2 is a residue in B only if e − 2 ≤ f + 1; however, e > 6 because e > n = 2f + 4 and f ≥ 1; so f + 2 and −2 cannot be residues in B. Consequently, all of the bipartitions in B have the form ((a, b), (2^{k}1^{l})).

Define bipartitions λ_{1} = ((0), (2^{f+2})), λ_{2} = ((1), (2^{f+1}, 1)), λ_{3} = ((1^{2}), (2^{f+1})), λ_{4} = ((2), (2^{f}, 1^{2})) and λ_{5} = ((2, 1), (2^{f}, 1)). We will show that B has infinite representation type by computing End_{ℋ} (P^{λ_{2}}).

Lemma 26. The bipartitions λ_{1}, λ_{2}, λ_{3}, λ_{4} and λ_{5} are all Kleshchev. Furthermore, [P^{λ_{2}}] = [S^{λ_{2}}] + [S^{λ_{3}}] + [S^{λ_{4}}] + [S^{λ_{5}}] and the first five rows of the decomposition matrix of B are as follows (all omitted entries are zero):

D^{λ_{1}} D^{λ_{2}} D^{λ_{3}} D^{λ_{4}} D^{λ_{5}}
S^{λ_{1}} 1  .  .  .  .
S^{λ_{2}} 1  1  .  .  .
S^{λ_{3}} 0  1  1  .  .
S^{λ_{4}} 0  1  0  1  .
S^{λ_{5}} 1  1  1  1  1

Proof. We use Corollary 16 to compute [P^{λ_{i}}], for i = 1, …, 5. We find that

F_{1} … F_{f+1}F_{e−1}F_{0}^{(2)}F_{1} … F_{f}((0), (0)) = F_{1} … F_{f+1}F_{e−1}F_{0}^{(2)}((0), (1^{f}))

= F_{1} … F_{f+1}[((1), (1^{f+2})) + v((1^{2}), (1^{f+1}))]
= F_{1}[((1), (2^{f}, 1^{2})) + v((1^{2}), (2^{f}, 1))]
= ((1), (2^{f+1}, 1)) + v((2), (2^{f}, 1^{2})) + v((1^{2}), (2^{f+1}))
+v^{2}((2, 1), (2^{f}, 1)).

As in Case 1, by Corollary 16 this shows that λ_{2} is Kleshchev, proves the formula for [P^{λ_{2}}] and thus gives the second column of the decomposition matrix of B. The remaining claims follow from the following calculations, which we leave to the reader.

F_{0} … F_{f+1}F_{e−1}F_{0} … F_{f}((0), (0)) = λ_{1} + vλ_{2} + vλ_{5} + …
F_{1} … F_{f+1}F_{0} … F_{f}F_{e−1}F_{0}((0), (0)) = λ_{3} + vλ_{5} + …
F_{e−1}F_{2} … F_{f+1}F_{0}F_{1}^{(2)}F_{2} … F_{f}F_{0}((0), (0)) = λ_{4} + vλ_{5} + …

Here, “+ … ” indicates a linear combination of more dominant terms and if e = ∞ then we replace F_{e−1} by F_{−1}. □

Consequently, [P^{λ_{2}}] = 4[D^{λ_{2}}] + 2[D^{λ_{1}}] + 2[D^{λ_{3}}] + 2[D^{λ_{4}}] + [D^{λ_{5}}]. Now, by Corollary 13(ii), P^{λ_{2}} has a submodule isomorphic to S^{λ_{5}}. The composition factors D^{λ_{3}} and D^{λ_{4}} of S^{λ_{5}} cannot appear in Rad P^{λ_{2}}/ Rad^{2} P^{λ_{2}} because then D^{λ_{5}} would appear in the head of P^{λ_{2}}. On the other hand, because S^{λ_{2}}, S^{λ_{3}} and S^{λ_{4}} are indecomposable, D^{λ_{1}} ⊕ D^{λ_{3}} ⊕ D^{λ_{4}} appears in Rad P^{λ_{2}}/ Rad^{2} P^{λ_{2}}. Recall, again, that P^{λ_{2}}/S^{λ_{5}} has a Specht filtration with successive quotients S^{λ_{2}}, S^{λ_{3}} and S^{λ_{4}}. Therefore, the ℋ –modules D^{λ_{3}} and D^{λ_{4}} which appear in Rad P^{λ_{2}}/ Rad^{2} P^{λ_{2}} are composition factors of S^{λ_{3}} and S^{λ_{4}}. Since Rad S^{λ_{3}} = Rad S^{λ_{4}} = D^{λ_{2}} and other D^{λ_{2}} are the head and the socle of P^{λ_{2}}, D^{λ_{2}} does not appear in Rad P^{λ_{2}}/ Rad^{2} P^{λ_{2}}. Thus Ext^{1}(D^{λ_{2}}, D^{λ_{2}}) = 0 and we see that P^{λ_{2}}/ Rad^{3} P^{λ_{2}} contains D^{λ_{2}} ⊕ D^{λ_{2}} as an ℋ – submodule by the same argument as before. Consequently, End_{ℋ} (P^{λ_{2}}/ Rad^{3} P^{λ_{2}}) ≅ k[x, y]/⟨x^{2}, xy, y^{2}⟩ and B is of infinite representation type by Lemma 6. Therefore, ℋ has infinite representation type by Lemma 5(ii).

This completes the proof of Theorem 25. □

6. Finite representation type

In this final section we show that ℋ_{q}(B_{n}) is of finite representation type when n < min{e, 2f + 4}. To do this we use a different combinatorial description of bipartitions which was suggested to the first author by Fomin. Recall that we are assuming that q is a primitive e^{th} root of unity in R and that T_{0} satisfies the relation (T_{0} −1)(T_{0} −q^{f}) = 0 where 0 ≤ f ≤ \frac{e}{2}. We note that if K is a field of characteristic zero then there is a different argument by Geck [11, Corollary 9.7].

First, consider a partition λ. The diagram of λ is the set

{ (i, j) ∈ ℕ^{2} | 1 ≤ j ≤ λ_{i} and i ≥ 1 } ,

which we think of as an array of boxes in the plane. Just as we can identify λ with its diagram we can also identify λ with its border

{ (i, j) ∈ ℕ^{2} | λ_{i} < j ≤ λ_{i−1} + 1 }

(we set λ_{0} = ∞). We can think of the border of λ as a (doubly infinite) path from (∞, 1) to (1, ∞). Writing 0 for each vertical edge and 1 for each horizontal edge in the border we identify λ with a doubly infinite sequence of 0’s and 1’s. We call this the path sequence of λ. A path sequence is also called a Maya diagram.

For example, if λ is the partition (4, 2, 1) then by looking at the diagram of λ

× × × × 0 1 1 …
× × 0 1 1
× 0 1
0 1

0

⋮

the path sequence of λ is … 00101011011 . . .. Here the crosses mark the nodes in the diagram of λ and the 0’s and 1’s are the nodes in the border of λ.

In order to keep track of the contents of the nodes we insert a horizontal bar into the path sequence after the node which appears on the diagonal { (i, i) | i ∈ ℕ }. The example above becomes … 00101|011011 . . .. The bar divides the path sequence into two regions which we refer to as the left and right regions of the path.

Note that the number of 1’s in the left region is always the same as the number of 0’s in the right region. Conversely, any sequence

… p_{−2}p_{−1}p_{0}|p_{1}p_{2}p_{3} …

of zeros and ones with P_{−} = ∑_{i≤0} p_{i} < ∞, P_{+} = ∑_{i>0}(1 − p_{i}) < ∞ and P_{−} = P_{+} always corresponds to a partition.

A rim hook in (the diagram of) λ corresponds to a subsequence 1 … 0 in the path sequence of λ; explicitly, if x = (i, j) ∈ [λ] then the rim hook r_{x} corresponds to the border path from (λ′_{j} + 1, j + 1) to (i, λ_{i} + 1). As there is a bijection between the nodes and rim hooks of λ, the number of subsequences of a path sequence of the form 1 … 0 is equal to n. Hence, it follows that removing a rim hook from λ is the same as swapping a 0 and a 1 in the path sequence for λ. Further, the leg length of the hook is equal to the number of zeros in the corresponding 1 … 0 subsequence minus 1. Conversely, wrapping a rim hook onto λ is the same as changing a 0 … 1 subsequence to 1 … 0. These observations will allow us to rephrase the Jantzen sum formula (18) in terms of path sequences.

Given p = … p_{−2}p_{−1}p_{0}|p_{1}p_{2}p_{3} … let \overline{p} = … \overline{p}_{−2}\overline{p}_{−1}\overline{p}_{0}|\overline{p}_{1}\overline{p}_{2}\overline{p}_{3} … be the sequence of left partial sums; that is, \overline{p}_{i} = ∑_{j≤i} p_{i}. For example, if λ = (0) then the path sequence is 0 = … 000|111 … and \overline{0} = … 000|123 . . ..

The content of a node x = (i, j) ∈ ℕ^{2} is c(x) = j − i. For a partition λ let c_{k}(λ) = # { x ∈ [λ] | c(x) = k }. Using the notation of the last paragraph, if p = … p_{−2}p_{−1}p_{0}|p_{1}p_{2}p_{3} … is the path sequence of λ then the content multiplicities c_{k}(λ) are given by the sequence … c_{−2}c_{−1}c_{0}|c_{1}c_{2} ⋯ = \overline{p} − \overline{0}.

We now turn to path sequences for bipartitions. If λ = (λ^{(1)}, λ^{(2)}) is a bipartition then we have a path sequence … p_{−2}p_{−1}p_{0}|p_{1}p_{2} … for λ^{(1)} and a path sequence … s_{−2}s_{−1}s_{0}|s_{1}s_{2} … for λ^{(2)}. Recall that the content of node x = (i, j, k) in [λ] is defined to be c(x) = j − i + (k − 1)f (and res(x) = c(x) (mod e)). In particular, p_{0} corresponds to a node of content 0 and s_{0} corresponds to a node of content f. Accordingly, we shift the nodes in the path sequence for λ^{(2)} by f positions to the right and define the path sequence of the bipartition λ to be the sequence {(p_{i}, s_{i−f})} of ordered pairs which we write with two separation bars as follows

\begin{matrix}… & p_{−1} & p_{0} & p_{1} & … & p_{f} & p_{f+1} & p_{f+2} & … \\ … & s_{−1−f} & s_{−f} & s_{1−f} & … & s_{0} & s_{1} & s_{2} & … \end{matrix} .

We refer the three regions in the path sequence of a bipartition as the left, middle and right regions of the sequence.

For example, if λ = ((4, 2, 1), (2^{2}, 1)) and f = 2 then the contents in λ and its path sequence are as follows.

( 0 1 2 3 ; −1 0 ; −2 , 2 3 1 2 0 ) and \begin{matrix}… 000101 & 01 & 1011 … \\ … 000001 & 01 & 0011 … \end{matrix} .

As before, it is easy to see that we can recover the contents of a bipartition from the partial sums ∑_{j≤i}(p_{j} + s_{j−f}) from the path sequence.

We now develop a calculus with which to analyze path sequences of bipartitions. Let A = {0 \atop 1}, B = {1 \atop 0}, C = {0 \atop 0} and D = {1 \atop 1} be the four possible ordered pairs which can appear in the path sequence of a bipartition. Define a_{l}, a_{m} and a_{r} to be the number of A’s in the left, middle and right regions, respectively, of the path sequence; similarly, we define b_{l}, b_{m}, b_{r}, c_{l}, c_{m}, c_{r}, d_{l}, d_{m} and d_{r}. Notice that c_{l} and d_{r} are both infinite; all of the other quantities are non–negative and finite. 27 Suppose that n ≤ 2f + 3. Then

(i) b_{l} + d_{l} = a_{m} + c_{m} + a_{r} + c_{r}.

(ii) b_{r} + c_{r} = a_{l} + d_{l} + a_{m} + d_{m}.
(iii) f = a_{m} + b_{m} + c_{m} + d_{m}.
(iv) 2f + 3 ≥ (b_{l} + d_{l})(a_{m} + c_{m} + a_{r} + c_{r}) + (a_{l} + d_{l} + a_{m} + d_{m})(b_{r} + c_{r})

+ (b_{m} + d_{m})(a_{r} + c_{r}) + (a_{l} + d_{l})(b_{m} + c_{m}) + a_{l}b_{l} + a_{m}b_{m} + a_{r}b_{r}.

Proof. Parts (i) and (ii) both follow from the fact mentioned earlier that the number of ones in a region to the left of a bar is equal to the number of zeros in the region to the right of the same bar. Part (iii) is true by definition. Finally, (iv) follows by counting the rim hooks in λ (recall that the rim hooks in the path sequence for a partition correspond to subsequences of the form 1 … 0); this is the only place where we use the restriction on n. □

We want to understand this system of inequalities when n ≤ 2f + 3. As a first step, adding parts (i) and (ii) and subtracting (iii) we see that

(28)  b_{l} + b_{m} + b_{r} = f + a_{l} + a_{m} + a_{r}.

Careful inspection shows that by combining parts (i), (ii) and (iv) of (27), and omitting some terms for the second inequality, we have

2f + 3 ≥ (a_{m} + c_{m} + a_{r} + c_{r})^{2} + (a_{l} + d_{l} + a_{m} + d_{m})^{2} + (b_{m} + d_{m})a_{r}
+ (b_{m} + d_{m})c_{r} + (a_{l} + d_{l})(b_{m} + c_{m}) + a_{l}b_{l} + a_{m}b_{m} + a_{r}b_{r}
≥ (a_{l} + a_{m} + a_{r})(a_{m} + b_{m} + c_{m} + d_{m}) + a_{l}a_{m} + a_{m}a_{r} + a_{l}^{2} + a_{m}^{2}
+ a_{r}^{2} + a_{l}(b_{l} + d_{l}) + a_{r}(b_{r} + c_{r})
= (a_{l} + a_{m} + a_{r})f + a_{l}a_{m} + a_{m}a_{r} + a_{l}^{2} + a_{m}^{2} + a_{r}^{2}
+ a_{l}(a_{m} + c_{m} + a_{r} + c_{r}) + a_{r}(a_{l} + d_{l} + a_{m} + d_{m});

the last line again uses (i)–(iii) of (27). Finally, throwing away a few more terms and rearranging gives

2f + 3 ≥ (a_{l} + a_{m} + a_{r})(f + a_{l} + a_{m} + a_{r}).

Now, if a_{l} + a_{m} + a_{r} ≥ 2 then the right hand side is greater than or equal to 2f + 4; as this is impossible we must therefore have

(29)  a_{l} + a_{m} + a_{r} ≤ 1.

In particular, note that at most one of a_{l}, a_{m} and a_{r} can be non–zero. Notice that this is the first time that the assumption n ≤ 2f + 3 has really been needed; it is exactly what is required to ensure that a_{l} + a_{m} + a_{r} ≤ 1.

Using similar arguments it is possible to classify the possible path sequences when n ≤ 2f + 3; however, we won’t need this.

Theorem 30. Suppose that n < min{e, 2f +4}. Then ℋ is of finite representation type.

Proof. Fix a block B of ℋ and recall from Corollary 20 that two simple modules belong to the same block only if they have a common multiset of residues. Note that because n < e the number of distinct residues contained in the diagram of λ is strictly less than n; consequently, we can find a k, with 0 ≤ k < e, such that −k − 1 (mod e) is not a residue in B.

Suppose that D^{λ} appears in B. Then the contents of all of the nodes in λ^{(1)} are contained in the interval [−k, e − k − 1]. Case 1: Suppose that e−k ≥ f. As −k−1 (mod e) is not a residue for the block B the contents of the nodes in λ^{(2)} are all contained in the interval [−k, e − k] — note that f ∈ [−k, e − k]. Therefore, the multiset of residues for the block B is the same as the multiset of contents for B. Consequently, we can unwrap a rim hook from λ and wrap it back on again without changing the residue of the foot node only if the resulting bipartition μ has the same multiset of contents as λ.

Recall that unwrapping a rim hook from a partition is the same as swapping the ends of a 1 … 0 subsequence to give 0 … 1 and that wrapping a hook back on changes some 0 … 1 into 1 … 0. Now, the contents of a bipartition λ are determined by the partial sums in the path sequence of λ; because of this, the only way to unwrap a rim hook from λ and then wrap it back on to give a bipartition μ with the same multiset of contents is by interchanging some A and B in the path sequence:

λ = … B … A …  ⟶  μ = … A … B ….

Moreover, λ ▷ μ if and only if A moves to the left. (Note that |μ| = |λ| in this case as the number of 1 … 0 subsequences in the two path sequences is the same.)

If a_{l} + a_{m} + a_{r} = 0 then the path sequence for λ does not contain any A’s so by the sum formula, Theorem 18, and by Proposition 19, the Specht module S^{λ} = D^{λ} = P^{λ} is the only simple module in the block B. In particular, B is semisimple and so of finite type in this case.

If a_{l} + a_{m} + a_{r} ≠ 0 then a_{l} + a_{m} + a_{r} = 1 by (29). In this case by Proposition 19 and (28) the block B contains at most f + 2 bipartitions; namely, the bipartitions λ_{0}, …, λ_{f+1} whose path sequences contain exactly one A and (f +1) B’s and which agree with the path sequence for λ on all of the C’s and D’s. By ordering these bipartitions according to the location of the (unique) A in their path sequence we may assume that λ_{f+1} ▷ ⋯ ▷ λ_{0} (for example, A occupies the leftmost position in the path sequence of λ_{0} and the rightmost position in λ_{f+1}).

For our purposes, it is enough to prove that S^{λ_{0}} = D^{λ_{0}} and [S^{λ_{i}}] = [D^{λ_{i}}] + [D^{λ_{i−1}}] for 0 < i ≤ f; in particular, we do not need to know that D^{λ_{i}} ≠ 0, for 0 ≤ i ≤ f. In fact these modules are always non–zero; we include the proof below because it yields the remarkable fact that when n < min{e, 2f + 4} the number of Specht modules belonging to a block is either 1, f + 2 or e − f + 2.

The removable nodes in a partition correspond to the 10 subsequences in the path sequence. Suppose that 0 ≤ i ≤ f. Then λ_{i} contains a removable node x ∈ [λ_{i}^{(2)}]; furthermore, this node is automatically good because if x is a r–node then there is no addable r–node below x because |λ^{(2)}| < e. Let μ_{i} be the bipartition with [μ_{i}] = [λ_{i}] \ {x}. Then μ_{i} is Kleshchev because either the path sequence for μ_{i} contains an A (so μ_{i} is Kleshchev by induction on n), or D^{μ_{i}} = S^{μ_{i}} ≠ 0 (by the second last paragraph); hence, λ_{i} is Kleshchev. On the other hand, λ_{f+1} is not Kleshchev because either we can apply induction after removing a node from λ_{f+1}^{(1)}, or the path sequence for λ_{f+1} is … BBBA … in which case it is easy to see that λ_{f+1} is not Kleshchev.

Now S^{λ_{0}} = D^{λ_{0}} by Theorem 12(iii) since λ_{i} ▷ λ_{0} for i > 0. To complete the proof we claim that [S^{λ_{i}}] = [D^{λ_{i}}] + [D^{λ_{i−1}}], for i = 1, …, f, and S^{λ_{f+1}} = D^{λ_{f}}. To see this we apply the sum formula. As discussed earlier, the leg length of a hook 1 … 0 in a path sequence is given by the number of 0’s strictly contained in the subsequence. Consequently, when we unwrap the hook B … A from λ_{i} and wrap it back on to give some λ_{l} then, modulo 2, the difference in the leg lengths of the two rim hooks is equal to the number of B’s which are strictly contained in the subsequence for the rim hook. Therefore, by Theorem 18, for i = 1, …, f + 1

∑_{j>0}[S^{λ_{i}}(j)] = [S^{λ_{i−1}}] − [S^{λ_{i−2}}] + ⋯ + (−1)^{i−1}[S^{λ_{0}}].

As we already know that λ_{0}, … λ_{f} are Kleshchev, and that λ_{f+1} is not, our claim now follows by induction on i. Consequently, the decomposition matrix of the block B is

D^{λ_{0}} D^{λ_{1}} … D^{λ_{f}}
S^{λ_{0}}  1
S^{λ_{1}}  1  1
⋮  ... ...
S^{λ_{f}} 1 1
S^{λ_{f+1}}  1

and B has finite representation type by Theorem 9. Case 2: Suppose that 0 < e − k < f. In this case the contents of the nodes in λ^{(2)} are contained in the interval [e − k, 2e − k − 1]. Renormalizing T_{0} as q^{f}T_{0} the relation for T_{0} becomes (T_{0} − 1)(T_{0} − q^{e−f}) = 0. The Specht module S^{λ} is relabelled as S^{(λ^{(2)},λ^{(1)})} and the residues in [λ] are all changed by adding e − f ≡ −f (mod e). Consequently, the residues for λ are all contained in the interval [e − f − k, 2e − f − k − 1]. Therefore, with this renormalization, the multiset of residues for B is the same as the multiset of contents for B. Consequently, we can repeat the argument of Case 1 to deduce that decomposition matrix for B has the form above; so, again, B has finite representation type by Theorem 9. □

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Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502. Japan., E–mail: ariki@kurims.kyoto-u.ac.jp

School of Mathematics and Statistics F07, University of Sydney, Sydney NSW 2006. Australia., E–mail: a.mathas@maths.usyd.edu.au, www: www.maths.usyd.edu.au/u/mathas