REPRESENTATION TYPE OF HECKE ALGEBRAS OF TYPE A

KARIN ERDMANN AND DANIEL K. NAKANO

Abstract. In this paper we provide a complete classification of the representation type for the blocks for the Hecke algebra of type A, stated in terms of combinatorical data. The computation of the complexity of Young modules is a key component in the proof of this classification result.

1 Introduction

1.1 Let R be an integral domain q be a unit in R with q ≠ 1. Let us regard the symmetric group Σ_{r} as the set of bijections of {1, …, r}. The Hecke algebra H_{q}(r) = H_{R,q}(Σ_{r}) is the free R-module with basis {T_{w} : w ∈ Σ_{r}}. The multiplication in the algebra is defined by the rule

T_{w}T_{s} = \begin{cases}T_{ws} & if ℓ(ws) > ℓ(w) \\ qT_{ws} + (q − 1)T_{w} & otherwise \end{cases}

where s = (i, i + 1) ∈ Σ_{r} is a basic transpositions and w ∈ Σ_{r}. The function ℓ : Σ_{r} → ℕ is the usual length function. Here we will assume that R = k is a field; in this case the Hecke algebra H_{q}(r) is a finite-dimensional self-injective algebra. Let l be the multiplicative order of q in k^{∗}, so that either q is a primitive l-th root of unity, or l = ∞.

In [DJ2], Dipper and James gave a necessary and sufficient condition for the semisimplicity of H_{q}(r). For q ≠ 1, this is the case if and only if l > r. So the case when l is finite (and l ≤ r) is most interesting here.

Let λ = (λ_{1}, λ_{2}, …, λ_{t}) be a partition of r. That is, |λ| = ∑_{i=1}^{t} λ_{i} = n and λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{t} > 0. A partition λ is l regular if λ_{i} − λ_{i+1} ≤ l − 1 for 1 ≤ i ≤ t − 1 and λ_{t} ≤ l − 1. It is well-known that the simple modules for the Hecke algebra H_{q}(r) are naturally indexed by l-regular partitions of r. For each partition λ of r there is a q-Specht module of the Hecke algebra H_{q}(r), denoted by S^{λ}. If λ is l-regular then S^{λ} has a unique simple quotient denoted by D^{λ}.

The blocks of the Hecke algebra can be described via the Nakayama Rule, that is D^{λ} and D^{μ} belong to the same l-block if and only if λ and μ have the same l-core. The l-core is partition whose Young diagram is obtained by removing as many rim l-hooks from Young diagram corresponding to the original partition. For details we refer to [Jam], [DJ2] or [JM]. Equivalently, S^{λ} and S^{μ} belong to the same l-block if and only if λ and μ have the same l-core. Let B_{λ} denote the block of H_{q}(r) containing S^{λ}. Given a partition λ of r, the weight w := w(λ) of the block B_{λ} is given by |γ|+lw = r where γ is the l-core corresponding to B_{λ}.

Date: August 2000.

Research of the second author partially supported by NSF grant DMS-9800960.

The block containing the trivial module is called principal block, its core is the partition (t) where 0 ≤ t < l − 1 and t ≡ r (mod l).

A finite-dimensional algebra A has finite representation type if and only if there are only finitely many indecomposable modules in mod(A). Otherwise, A has infinite representation type. Algebras of infinite representation type fall into two mutually exclusive categoriesones which are either tame or wild. An algebra A has tame representation type if for each dimension d there are finitely many one parameter families of modules which contain all d-dimensional indecomposable modules. In this situation there is a good chance one may be able to classify all indecomposable representations. On the other hand, algebras of wild representation type are those algebras whose representation theory is as complicated as that of the free associative algebra k⟨x, y⟩.

1.2 Main result: The main purpose of this paper is to classify the representation type of the blocks of the Hecke algebra H_{q}(r). This classification is given in terms of the “weight” of the corresponding partition λ for the block B_{λ}.

Theorem . Let B_{λ} be a block for H_{q}(r).

(a) B_{λ} is semisimple if and only if w(λ) = 0;

(b) B_{λ} has finite representation type (and is not semisimple) if and only if w(λ) = 1;

(c) B_{λ} has tame representation type if and only if w(λ) = 2 and q is a primitive 2nd root of unity;

(d) B_{λ} has wild representation type if and only if q is a primitive l-th root of unity, w(λ) ≥ 2 and l ≥ 3, or l = 2 and w(λ) ≥ 3.

For blocks of group algebras of finite groups, the representation type may be characterized in terms of the defect groups of the block; see the introduction in [Erd] for a discussion. The defect groups for blocks of symmetric groups are iterated wreath products of cyclic groups of order p, see [JK, 6.2.39]. This gives a characterization of the representation type for blocks of symmetric groups which is analogous to our theorem; but the proof is quite different.

1.3 Acknowledgements: The second author would like to acknowledge the Mathematical Institute at Oxford University for their support and hospitality during his visit in June 1997.

2 Complexity

2.1 Complexity and representation type: Let A be a finite-dimensional algebra over a field k and M be a finite-dimensional A-module. Let Ω^{0}(M) = M and P_{0} be the projective cover of Ω^{0}(M). Now let Ω^{1}(M) be the kernel of the map P_{0} → Ω^{0}(M). Recursively, one can now let Ω^{j+1}(M) be the kernel of map P_{j} → Ω^{j}(M) where P_{j} is the projective cover of Ω^{j}(M). In this process we have constructed a minimal projective resolution of M in mod(A)

⋯ → P_{2} → P_{1} → P_{0} → M → 0.

For V = {V_{t} : t ∈ ℕ} a sequence of k-vector spaces, let r(V) be the smallest integer s ≥ 0 such that dim_{k} V_{t} ≤ C · t^{s−1} for some constant C ≥ 0 and all t ∈ ℕ. If no such integer exists we say that r(V) is infinite. The complexity of M in mod(A), c_{A}(M), is then equal to r({P_{t} : t ∈ ℕ}).

There are some elementary properties about the complexity of modules which are presented below.

(2.1.1) c_{A}(M) = r({Ext_{A}^{t}(M, S) : t ∈ ℕ}) where S is the direct sum of all simple A-modules.

(2.1.2) If B is a subalgebra of A such that A is a free (left) B-module then c_{B}(M) ≤ c_{A}(M) for M ∈ mod(A); and c_{A}(U ⊗_{B} A) ≤ c_{B}(U) for U ∈ mod(B).

(2.1.3) If N is a direct summand of M in mod(A) then c_{A}(N) ≤ c_{A}(M).

The following proposition demonstrates that the notion of complexity is is related to the representation type of the algebra provided that the algebra A is self-injective.

Proposition . Let A be a finite-dimensional self-injective algebra and M ∈ mod(A) be indecomposable.

(a) If c_{A}(M) ≥ 1, then A is not semisimple.

(b) If c_{A}(M) ≥ 2, then A has infinite representation type.

(c) If c_{A}(M) ≥ 3, then A has wild representation type.

Proof. (a) Observe that if A is self-injective then c_{A}(M) = 0 occurs precisely when M is a projective A-module. Therefore, A is semisimple if and only if c_{A}(M) = 0 for all M ∈ mod(A). (b) The modules Ω^{•}(M) are indecomposable and there are infinitely many non-isomorphic ones. Thus A has infinite representation type. (c) Let T be a composition factor of M in mod(A). Then c_{A}(T) ≥ c_{A}(M) ≥ 3. Therefore, by [Ric, Thm. 2] A has wild representation type. _{□}

2.2 Young modules: Let λ be a composition of r and let Σ_{λ} be the corresponding Young subgroup of Σ_{r}, so Σ_{λ} ≅ Σ_{λ_{1}} × Σ_{λ_{2}} × · · ·. Let

x_{λ} = ∑_{w∈Σ^{λ}} T_{w}.

For any partition λ, let H_{q}(λ) be the subalgebra generated by T_{w}, w ∈ Σ_{λ}; this is isomorphic to the tensor product ⊗_{i}H_{q}(λ_{i}) of algebras. It is important to note that H_{q}(r) is free as a module over H_{q}(λ), of rank equal to the index of Σ_{λ} in Σ_{d}. The element x_{λ} above lies in this algebra, and ⟨x_{λ}⟩ is a module for H_{q}(λ) which is isomorphic to the trivial module. Set M^{λ} := x_{λ}H_{q}(r) ≅ ⟨x_{λ}⟩ ⊗_{H_{q}(λ)} H_{q}(r). The module M^{λ} is often called the q-permutation module.

The module M^{λ} has a unique submodule isomorphic to the Specht module S^{λ}, and there is a unique indecomposable direct summand of M^{λ} containing S^{λ}; this is the Young module Y^{λ}. In particular, Y^{λ}, S^{λ} belong to the same block B_{λ}. Furthermore, Y^{λ} ≅ Y^{μ} if and only if λ = μ. Each M^{λ} is a direct sum of Young modules Y^{μ} (see [Mar]).

In [DDu] it is proved that indecomposable H_{q}(r)-modules have “vertices” and “sources”, where vertices are certain parabolic subalgebras of the form H_{q}(μ). Moreover it is proved that the vertices of Young modules are (l − p)- parabolics given as follows. Fix a partition λ and take the (l − p)-adic decomposition of λ as

λ = λ(0) + ∑_{i=0}^{m} μ(i)lp^{i}

where λ(0) is an l-restricted partition and the μ(i) are p-restricted partitions. This decomposition is unique. Define the partition

ρ = (1^{a}, l^{b_{0}}, (lp)^{b_{1}}, ⋯ (lp^{m})^{b_{m}})

where a = |λ(0)| and b_{i} = |μ(i)|. Then Y^{λ} has Young vertex H_{q}(ρ), and its source is the trivial module for H_{q}(ρ). So we have that Y^{λ} is a direct summand of M^{ρ} and the restriction of Y^{λ} to H_{q}(ρ) has k as a direct summand. The following result describes the complexity of Young modules.

Theorem . Let H_{q}(ρ) be a Young vertex of Y^{λ}. Then c_{H_{q}(r)}(Y^{λ}) = c_{H(ρ)}(k).

Proof. Recall that H := H_{q}(r) is free as a module over L := H(ρ). Therefore, induction and restriction are exact and projective resolutions of L-modules induce to projective resolutions. By (2.1.2), it follows that c_{L}(k) ≥ c_{H}(k ⊗_{L} H). Since Y^{λ} is a direct summand of k ⊗_{L} H, we also have that c_{H}(k ⊗_{L} H) ≥ c_{H}(Y^{λ}) by (2.1.3), thus c_{L}(k) ≥ c_{H}(Y^{λ}).

On the other hand, c_{H}(Y^{λ}) ≥ c_{L}(Y^{λ}) by (2.1.2). By [DDu], k is a direct summand of Y^{λ} as L-module, thus c_{L}(Y^{λ}) ≥ c_{L}(k) and c_{H}(Y^{λ}) ≥ c_{L}(k). _{□}

2.3 Let A and B be algebras and let F be a functor from mod(A) to mod(B). We say that the functor F is bounded if for all d ≥ 1 there is a constant c = c(d) such that for all M ∈ mod(A) of length ≤ d, the module F (M) has length ≤ c(d).

Proposition . Let A and B be algebras and F : mod(A) → mod(B) be a bounded functor. Moreover, let G : mod(B) → mod(B) be a functor such that M is a direct summand of GF (M) for all M ∈ mod(A). If A is wild then so is B.

Proof. Assume that B is not wild; then by Drozd’s theorem B is tame, and we will show that then A is also tame. By [dlP] it suffices to show that A is weakly tame, which is to say that for every d ≥ 1 there are finitely many k[T] − A-bimodules M_{i} which are finitely generated free as k[T]-modules such that every indecomposable A-module of dimension ≤ d is a direct summand of some S ⊗_{k[T]} M_{i} where S is a simple k[T]-module. (That is, the weakening consists of allowing direct summands, instead of taking the whole module).

Let d ≥ 1, consider A-modules M of dimension ≤ d. Since B is tame, there are finitely many k[T]−B-bimodules M_{i} such that every B-module X of dimension ≤ c(d) is of the form X ≅ S ⊗ M_{i}. Now take X = F (M), and let X = F (M) ≅ S ⊗ M_{i}. Then G(X) = GF (M) which is S ⊗ M_{i} restricted to A has M as a direct summand by our hypothesis. Hence, A is weakly tame. _{□} 3 Blocks of the Hecke algebra

3.1 If q is not a root of unity then the Hecke algebra is semisimple and w(λ) = 0 for all partitions λ. So assume that q is a primitive l-root of unity, and let B_{λ} be a block of H_{q}(r) of weight w(λ). It is known that the block B_{λ} is semisimple if and only if w(λ) = 0, and the block is of finite type if w(λ) = 1 [U] (see also §3.2). Therefore, Theorem 1.2 (main result) will be established by proving the the following theorem.

Theorem . Let B_{λ} be a block of H_{q}(r).

(a) If either w(λ) ≥ 2 and l ≥ 3, or if l = 2 and w(λ) ≥ 3 then B_{λ} is wild.

(b) If w(λ) = 2 and l = 2 then B_{λ} is tame.

The remainder of this section will be devoted to the proof of this theorem. We will proceed by proving the following steps.

(3.1.1) If w(λ) ≥ 3 then B_{λ} is wild. Also for l ≥ 3 the principal block of H_{q}(2l) is wild.

(3.1.2) If w(λ) = 2 and l ≥ 3 then B_{λ} is wild.

(3.1.3) If w(λ) = 2 and l = 2 then B_{λ} is tame.

The statement (3.1.2) is verified by investigating [2 : t]-pairs of blocks, and for (3.1.3) we determine the basic algebra of B_{λ} by quiver and relations.

3.2 Quiver and relations for blocks of weight one: These were determined in [U]. Also by [Jo], any such block is Morita equivalent to the principal block A say, of H_{q}(l), which can described as follows.

There are l − 1 simple modules in the block, X_{j} := D^{λ} where λ = (l − j, 1^{j}) for j = 0, 1, 2, …, l−2. The projective covers for these simple modules have the following structures.

P^{(l)}  P^{(j,1^{l−j})} : j = l − 1, …, 3  P^{(2,1^{l−2})}
D^{(l)}  D^{(j,1^{l−j})}  D^{(2,1^{l−2})}
D^{(l−1,1)}  D^{(j+1,1^{l−j−1})} D^{(j−1,1^{l−j+1})}  D^{(3,1^{l−3})}
D^{(l)}  D^{(j,1^{l−j})}  D^{(2,1^{l−2})}

The quiver for the algebra is Figure 1 below with relations α_{j}α_{j+1} = 0 = β_{j+1}β_{j}, and β_{j}α_{j} = α_{j+1}β_{j+1} for j = 1, 2, …, l − 3. The only indecomposable modules of radical length three are the projective modules. Therefore, A has the same representation type as the algebra A/J^{2} (J = Rad A). But, the quiver for A/J^{2} which is the same as for A separates into a union of two quivers of type A_{l−1}. Hence, the principal block of H_{q}(l) is of finite representation type, by Gabriel’s Theorem.

[figure: •; X_{0}; •; X_{1}; •; X_{2}; •; X_{l−3}; •; X_{l−2}; α_{1}; β_{1}; α_{2}; β_{2}; ⋯; α_{l−2}; β_{l−2}] Figure 1

3.3 From the previous section the complexity of k as an H_{q}(l)-module is 1. Namely, H_{q}(l) is a self-injective Brauer tree algebra, and k is non-projective periodic module. This implies that the complexity of k as a module for L := H_{q}(l)⊗H_{q}(l)⊗⋯⊗H_{q}(l) (m factors) is m, by the Künneth formula. Now suppose that A = H_{q}(ρ) contains L as a parabolic subalgebra and that A is a free L-module. Then by (2.1.2), c_{H_{q}(ρ)}(k) ≥ m. We can now verify (3.1.1).

Proposition (A). Let B_{λ} be a block of H_{q}(r) of weight w(λ) ≥ 3. Then B_{λ} is wild.

Proof. Let b be the l-core of B_{λ}, and we may assume that λ is the largest partition in this block, that is

λ = b + (lw) = (b_{1} + lw, b_{2}, ⋯)

Any l-core is l-restricted and hence λ has l − p adic expansion

λ = b + ∑_{i}(w_{i})lp^{i}

where w = ∑ w_{i}p^{i} is the p-adic expansion of the integer w. Hence, Y^{λ} has Young vertex A := H_{q}(ρ) where ρ has |b| parts equal to 1, w_{0} parts equal to l, w_{1} parts equal to lp and so on. We see that A contains a subalgebra L = H_{q}(l) ⊗ H_{q}(l) ⊗ H_{q}(l) so that A is free over L. Consequently, c_{A}(k) ≥ 3, and by Theorem 2.2, c_{H_{q}(r)}(Y^{λ}) ≥ 3. Hence, B_{λ} is wild, by Proposition 2.1. _{□}

Proposition (B). The principal block of H_{q}(2l) is wild when l ≥ 3.

Proof. It suffices to show that H_{q}(2l) is wild, since all non-principal blocks have finite type; these blocks have weight less than or equal to one. For this, it suffices by Proposition 2.3 to show that L := H_{q}(l) ⊗ H_{q}(l) is wild because the induction functor is bounded and it is well-known that for any L-module M, M is a direct summand of M ⊗_{L} H_{q}(2l) restricted to L.

The principal block of H_{q}(l) has quiver as Figure 1 with l − 1 vertices. Hence the quiver of L has “lattice form”, with (l − 1)^{2} vertices. If l > 3 then this has a proper subquiver which is a circuit, so by [EN, Prop. 2.3(B)] it must be wild. This leaves us with the case l = 3. Here we need to look at the projectives in the principal block of L. We see easily that the module category of the connected quiver in [Ri, p.168] with three vertices and three arrows (and no relations) can be embedded into the module category of L. This is wild, see [Ri] and hence the algebra L is wild as well. In order to obtain this embedding, note that the algebra L has four simple modules, S_{i} ⊗ S_{j} where S_{1}, S_{2} are the two simple modules for the principal block H_{q}(l). The projective cover of the H_{q}(l)-module S_{i} is uniserial of length three, with factors S_{i}, S_{j}, S_{i}. It follows that the third Loewy layer of P (S_{1} ⊗ S_{1}) is the direct sum of two copies of S_{1} ⊗ S_{1} and one copy of S_{2} ⊗ S_{2}. So label the vertices of the above wild quiver with S_{2} ⊗ S_{2} and S_{1} ⊗ S_{1} twice (we take the separated quiver of the block). _{□}

3.4 For the study of blocks of weight w = 2, we will work with the set-up as in [Sc1] and [Jo]. For general w and t they define [w : t]-pairs of blocks B̄, B where B̄ is a block of H_{q}(r − t) and B is a block of H_{q}(r), both of weight w. Every block of weight w of some Hecke algebra (over the same field) is related to the principal block of H_{q}(lw) by a finite sequence of [w : t]-pairs, where t varies. In [Jo] it was proved that for t ≥ w the blocks B and B̄ are Morita equivalent.

For the setup in [Sc1], [Jo] one needs to use the abacus display of partitions. Let B be a block of H_{q}(r) with l-core b = (b_{1}, ⋯, b_{s}), b_{s} ≠ 0 and l-weight w. The partition b is a partition of r − lw. Display the (r + lw)-element β-set of b on an abacus with l runners. If there exists an i, 2 ≤ i ≤ l, such that the i-th runner contains t more beads than the (i − 1)-th runner, we can swap these two runners to obtain the abacus display of a partition b̄ of r − t − lw. Let B̄ be the block of H_{q}(r − t) which has l-core b̄ (and therefore l−weight w). We then say that B and B̄ form a [w : t]-pair.

We assume that w = 2 and t = 1. Then the functors used in [Sc1] and [Jo] are easy to describe. Take the pair of adjoint functors (G, F) where G : mod(B) → mod(B̄) and F : mod(B̄) → mod(B), where G is restriction to H_{q}(r − 1) followed by taking the block component, and F is (co)induction to H_{q}(r) followed by taking the block component. This can be described by taking the B−B̄-bimodule W = eH_{q}(r)ē where e is the block idempotent for B and ē ∈ H_{q}(r − 1) is the block idempotent for B̄. Then G ≅ (−) ⊗_{B} W and F ≅ HomB̄(W, −).

The results in [Sc2] for symmetric groups generalize to Hecke algebras: the proofs use only the abacus combinatorics and the functor set-up, and these work the same way for Hecke algebras (see [Jo]). The only other ingredient is column removal, but this is also known for Hecke algebras in [Don, 4.2(9), (15)].

From [Sc2] there are three partitions each in B̄, B respectively which are “exceptional” denoted by ᾱ, β̄, γ̄ and α, β, γ respectively. If λ ≠ α and λ in the block B is l-regular then G(D^{λ}) is simple, and if λ̄ in the block B̄ with λ̄ ≠ ᾱ and is l-regular then F (Dλ̄) is simple. The partitions α and ᾱ are always l-regular. Moreover G(D^{α}) is not simple but has a simple socle and top isomorphic to Dᾱ and precisely two composition factors isomorphic to Dᾱ; and F (Dᾱ) is not simple but has a simple socle and top isomorphic to D^{α} and precisely two composition factors isomorphic to D^{α}. The projectives P (α) and P (ᾱ) have Specht filtrations with S^{α}, S^{β}, S^{γ}; and with Sᾱ, Sβ̄ and Sγ̄ respectively. So P (α) has precisely three composition factors isomorphic to D^{α} and similarly P (ᾱ) has three composition factors isomorphic to Dᾱ, by reciprocity.

Note that the B̄-module W is a projective generator. Namely it is projective since it is a block component of eH_{q}(r). To see that it is a generator, observe that HomB̄(W, Dλ̄) ≠ 0 for all l-regular partitions λ̄ in B̄ (which is just F (Dλ̄)).

Lemma . The module GF (Dᾱ) is isomorphic to Dᾱ ⊕ P (ᾱ).

Proof. Let X = GF (Dᾱ). By the adjointness we get from the above that socB̄(X) and topB̄(X) are both the direct sum of two copies of Dᾱ. Also, in total X has four composition factors isomorphic to Dᾱ.

It suffices to show that socB̄(X) ⊈ radB̄(X). If so then the simple is a direct summand. The remaining part has a simple socle and top and has three composition factors Dᾱ, and the only module with this property is the indecomposable projective.

So suppose that socB̄(X) ⊆ radB̄(X). The projective P (ᾱ) has precisely three composition factors Dᾱ. Therefore there is precisely one module (up to isomorphism) which is not simple or projective which has a simple socle and top isomorphic to Dᾱ, and hence this is G(D^{α}).

We have an epimorphism π : X → G(D^{α}). Let x ∈ X such that π(x) generates the image of π. Then ⟨x⟩ is a submodule of X which has simple top Dᾱ and its socle has only composition factors Dᾱ. It cannot have socle of length two because the projective P (ᾱ) does not have such a quotient. Therefore, the socle is simple and it follows by the uniqueness that ⟨x⟩ is isomorphic to G(D^{α}) ≅ Im(π). It follows that π is split. Moreover we have an embedding of G(D^{α}) into the kernel of π and it follows that X is the direct sum of two copies of G(D^{α}).

We will now get a contradiction by considering stable homomorphisms. For modules Y, Z let \underline{Hom}(Y, Z) be the quotient of Hom(Y, Z) factored out by maps which factor through a projective module. First we have \underline{Hom}(X, Dᾱ) is isomorphic to Hom(X, Dᾱ) since X has no projective summand and Dᾱ is simple, and this is 2-dimensional.

On the other hand, the adjointness is compatible with stable homomorphisms, see [DJ1]. So we have

\underline{Hom}(X, Dᾱ) ≅ \underline{Hom}(F (Dᾱ), F (Dᾱ))

Clearly Hom(F (Dᾱ), F (Dᾱ)) is 2-dimensional. So we will get a contradiction by showing that there are non-zero maps in this space which factor through a projective modules.

Let Z = F (Dᾱ), this has a simple socle and top isomorphic to D^{α}, so we have an inclusion j : Z → P (α) and an epimorphism π : P (α) → Z. Since D^{α} occurs precisely once in the heart of P (α) it follows that π ∘ j cannot be zero and must be the socle of Z. So π ∘ j is a non-zero endomorphism of Z which factors through a projective. _{□}

The following result allows us to relate blocks with weight two to blocks with weight two for smaller Hecke algebras.

Theorem . Let (B̄, B) be a [2 : 1]-pair. Then there are (bounded) functors G : mod(B) → mod(B̄) and F : mod(B̄) → mod(B) such that for any B̄-module M we have

GF (M) ≅ M ⊕ P,

where P is a projective B̄-module.

Proof. We proceed now by induction on the composition length. Note that by the adjointness we have a natural transformation ϵ : GF (−) → id which is “evaluation”. Since the B̄-module W is a projective generator, ϵ is onto.

If M is simple then the claim follows by Lemma 3.4. So assume now that M is not simple, then there is a commutative diagram with exact rows, where E is simple in B̄:

0 ⟶ GF (N) ⟶ GF (M) ⟶ GF (E) ⟶ 0
↓^{ϵ_{N}}  ↓^{ϵ_{M}}  ↓^{ϵ_{E}}
0 ⟶ N ⟶ M ⟶ E ⟶ 0

By the above remark, ϵ_{N} and ϵ_{E} are onto and then by inductive hypothesis, they are split epimorphisms. Now use the Snake Lemma and the fact that projectives are injective, to complete the proof. _{□}

We know that for l ≥ 3 the principal block of H_{q}(2l) is wild by Proposition 3.3(B). From Theorem 3.4 and Proposition 2.3, if B̄ is wild then so is B. Since Morita equivalent blocks have the same representation type, we can conclude by that all blocks of weight w = 2 for l ≥ 3 are wild which completes (3.1.2).

3.5 Blocks of weight two for l = 2: The goal of this section is to verify statement (3.1.3) that any block for the Hecke algebra of weight two when l = 2 is tame. By [Jo] there are only two Morita equivalence classes of such blocks, and they are represented by the principal blocks of H_{q}(4) and H_{q}(5). Note that here q = −1 and hence the trivial module is isomorphic to the alternating module.

Proposition (A). The Hecke algebra H_{q}(4) for l = 2 has quiver given in Figure 2 with relations

βϵ = 0 = ϵα, (αβ)^{2} = ϵ^{2}.

This algebra is tame.

[figure: •; L(4); •; L(3, 1); α; β; ϵ] Figure 2

Proof. All partitions of 4 have empty 2-core, so there is only one block. The decomposition matrix for the Hecke algebra H_{q}(4) is

(\begin{matrix}[S^{(4)} : D^{μ}] \\ [S^{(3,1)} : D^{μ}] \\ [S^{(2,2)} : D^{μ}] \\ [S^{(2,1^{2})} : D^{μ}] \\ [S^{(1^{4})} : D^{μ}] \end{matrix}) = (\begin{matrix}1 & 0 \\ 1 & 1 \\ 0 & 1 \\ 1 & 1 \\ 1 & 0 \end{matrix})

We consider Young modules; clearly k = Y^{(4)} and from information from the decomposition matrix, Y^{(3,1)} is uniserial of length three, with heart the other simple module which we call E. Moreover, we see that Y^{(2,2)} has Specht quotients S^{(3,1)} and S^{(2,2)}, and hence it is uniserial with socle and top E and heart k. Clearly S^{(1^{4})} ≅ k.

So we have all decomposition numbers and the Cartan matrix. The projective P (E) has a quotient Y^{(2,2)} with kernel (necessarily ) uniserial of length two, factors E and k. It follows by self-duality that P (E) is uniserial of length five, with “alternating” composition factors.

This shows that Ext_{H_{q}(4)}^{1}(E, k) is 1-dimensional. Next, observe that P (k) has a quotient Y^{(3,1)} with kernel of length three, and also Y^{(3,1)} as a submodule. Using the fact that Ext_{H_{q}(4)}^{1}(E, k) is 1-dimensional, it follows that the heart of P (k) is the direct sum of k with a uniserial of length three, actually isomorphic to Y^{(2,2)}. Then it is straightforward to determine relations.

The algebra is special biserial and hence is tame. Furthermore, the algebra is of infinite type because the endomorphism ring of P (k) is a Kronecker algebra of infinite type. _{□}

Proposition (B). The principal block of H_{q}(5) for l = 2 has quiver given in Figure 8 with relations

ϵα = 0 = βϵ, αγ = 0 = γβ,
ϵ^{2} = αβ, γ^{2} = βα.

This algebra is tame.

[figure: •; L(5); •; L(3, 2); α; β; ϵ; γ] Figure 3

Proof. The partitions of five in the principal block are (5), (3, 2), (3, 1^{2}) and their conjugates. The decomposition matrix is given by

(\begin{matrix}[S^{(5)} : D^{μ}] \\ [S^{(3,2)} : D^{μ}] \\ [S^{(3,1^{2})} : D^{μ}] \\ [S^{(2^{2},1)} : D^{μ}] \\ [S^{(1^{5})} : D^{μ}] \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & 1 \\ 1 & 1 \\ 0 & 1 \\ 1 & 0 \end{matrix})

From this information we have Y^{(3,2)} = D^{(3,2)} =: E say; and the Young module Y^{(3,1^{2})} is indecomposable with Loewy length two, and both socle and top are k ⊕ E. The Cartan matrix has off-diagonal entries 1 and diagonal entries 3 and this together with self-injectivity and self-duality already implies the structure completely. This algebra is again special biserial of infinite type. Hence, the principal block of H_{q}(5) is tame for l = 2. _{□}

We remark that both H_{q}(4) and H_{q}(5) when l = 2 are not quite of “dihedral type” as defined in [Erd], since they have tubes of rank two. These algebras are of domestic representation type.

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Mathematical Institute, Oxford University, 24-29 St. Giles, Oxford, OX1 3LB

   E-mail address: erdmann@maths.ox.ac.uk

Department of Mathematics and Statistics, Utah St. University, Logan, UT. 84322, U.S.A.

   E-mail address: nakano@sunfs.math.usu.edu