HOCHSCHILD COHOMOLOGY OF HECKE ALGEBRAS

DAVID J. BENSON AND KARIN ERDMANN

Abstract. We compute the Hochschild cohomology HH^{∗}(B) of an arbitrary block B of the Hecke algebra H = H(n, q) of the symmetric group 𝔖_{n} at a primitive ℓth root of unity q, over a field of characteristic zero.

1. Introduction

In this paper we compute the Hochschild cohomology of the Hecke algebra of the symmetric group at a root of unity in characteristic zero. Most of the work is already in the literature, and this paper really just puts the pieces together. This paper complements our paper [1] where we compute the Ext ring of the trivial module for H.

Let H = H(n, q) be the Hecke algebra of the symmetric group 𝔖_{n} over a field k of characteristic zero and where q is a primitive ℓth root of unity. This has generators T_{1}, …, T_{n−1} satisfying braid relations together with the relations (T_{i} + 1)(T_{i} − q) = 0. We assume that ℓ ≥ 2.

In the smallest interesting case, namely n = ℓ, the principal block B_{0}(H(ℓ, q)) is a Brauer tree algebra for a tree which is a straight line with ℓ vertices with no exceptional multiplicity. We write A for this algebra.

Let w be a positive integer. Then there is a natural action of the symmetric group 𝔖_{w} on A^{⊗w}, permuting the tensor multiplicands.

Specht modules for H(n, q) are in the same block if and only if they have the same ℓ-core. The weight w of such a block is the number of rim ℓ-hooks that need to be removed to reach the ℓ-core.

Our main theorem is as follows.

Theorem 1.1. Let B be a block of H(n, q) of weight w. Then

HH^{∗}(B) ≅ HH^{∗}(A^{⊗w} ⋊ 𝔖_{w})

≅ ⨁_{g} HH^{∗}(A^{⊗w}, A^{⊗w}g)^{C_{𝔖w} (g)}

≅ ⨁_{λ⊢w} ⨂_{i≥1} HH^{∗}(A^{⊗p_{i}(λ)})^{𝔖_{p_{i}(λ)}}.

Here, the first sum is indexed by a set of representatives g of the conjugacy classes of elements of 𝔖_{w}. The second is indexed by partitions λ ⊢ w, and p_{i}(λ) denotes the number of parts of λ of length i.

Remark 1.2. The Hochschild cohomology of Brauer tree algebras is computed in Holm [8], Erdmann and Holm [5], so that HH^{∗}(A) may be explicitly computed.

This research was partly supported by EPSRC grant EP/D077656/1.

Remark 1.3. Elements are multiplied by a double coset formula described in Witherspoon [11], Theorem 3.16.

Remark 1.4. This theorem should give a new and more explicit proof of the finite generation theorem of Linckelmann [10] in the case of type A_{n}. Since Witherspoon’s formula for the multiplicative structure is hard to use in practice, we do not give the details.

2. Quoted results

We quote the following results from the literature.

Theorem 2.1 (Chuang–Rouquier [4], Theorem 7.12). Let B_{1} be a block of H(n_{1}, q) and B_{2} a block of H(n_{2}, q). Then B_{1} and B_{2} are derived equivalent if and only if they are Rickard equivalent, if and only if they have the same weight.

The following is a version for Hecke algebras of the theorem of Chuang and Kessar [2] for symmetric groups. Rouquier blocks of weight w are by definition the blocks with the property that the core, written in abacus notation, has at least w − 1 more beads on each runner that on the runner to the left of it. The effect of this is that when w rim hooks are added, the different possible locations are sufficiently far apart that they behave as though they are independent of each other.

Theorem 2.2 (Chuang–Miyachi [3], §5.3, Theorem 18). Let B be a Rouquier block of H(n, q) of weight w. Then B is Morita equivalent to A^{⊗w} ⋊ k𝔖_{w}.

Theorem 2.3. Suppose that G is a finite group whose order is invertible in k, and suppose that G acts by algebra automorphisms on a k-algebra S. Then the Hochschild cohomology of the algebra S ⋊ kG is additively given by

HH^{∗}(S ⋊ kG) = ⨁_{g} HH^{∗}(S, Sg)^{C_{G}(g)}.

Proof. See for example Witherspoon [11], Lemma 3.5. Note that the Hochschild–Serre spectral sequence which computes the term Ext_{Δ(C_{G}(g))}(S, Sg) appearing there collapses to HH^{∗}(S, Sg)^{C_{G}(g)} because |C_{G}(g)| is invertible in k. □

Note that Witherspoon [11], Theorem 3.16 gives the multiplicative structure of HH^{∗}(S ⋊ kG) in terms of this decomposition via a double coset formula. For our application, we want to take for S the tensor product A^{⊗w} and for G the symmetric group 𝔖_{w}.

In a paper of Etingof and Oblomkov [7], HH^{∗}(A^{⊗w}, A^{⊗w}g)^{C_{G}(g)} is identified explicitly in terms of the cycles of g. They first describe the action of a single cycle.

Proposition 2.4. If g is the n-cycle (1 2 … n) and S = A^{⊗n} then

HH^{∗}(S, Sg) ≅ HH^{∗}(A).

Furthermore, g acts trivially on HH^{∗}(S, Sg).

Putting the pieces together, they obtain

Theorem 2.5 (Etingof and Oblomkov). The Hochschild cohomology of A^{⊗w} ⋊ 𝔖_{w} is given as follows. For each λ ⊢ w and i > 0 we let p_{i}(λ) be the number of parts of λ of length i. Then

HH^{∗}(A^{⊗w} ⋊ 𝔖_{w}) = ⨁_{λ⊢w} ⨂_{i≥1}(HH^{∗}(A)^{⊗p_{i}(λ)})^{𝔖_{p_{i}(λ)}}.

Their discussion is in Hochschild homology and k = ℂ, but Hochschild cohomology over fields of characteristic zero works essentially the same way.

3. Proof of the theorem

Let B be a block of H(n, q) of weight w. Then there is a Rouquier block B′ of some H(m, q) with the same weight w, but for m possibly different from n. By Theorem 2.1, B and B′ are derived equivalent. A theorem of Keller [9] states that derived equivalent algebras have isomorphic Hochschild cohomology rings. Therefore, we may assume without loss of generality that B is a Rouquier block of H(n, q). By Theorem 2.2, it follows that the Hochschild cohomology of B is isomorphic to that of A^{⊗w} ⋊𝔖_{w}. The Hochschild cohomology of this ring is computed using Theorems 2.3 and 2.5. This gives the main theorem.

4. Examples

To illustrate the main theorem, we compute the case n = 4 and ℓ = 2. That is, we compute HH^{∗}(B) where B is the block of H(4, −1) of weight two (in fact H(4, −1) only has one block, so B = H(4, −1), but this is irrelevant to the argument). To do this, we must first compute HH^{∗}(A), where A is the block of H(2, −1) of weight one. We have A ≅ k[t]/(t^{2}), and so

HH^{∗}(A) = k[t, u, v]/(t^{2}, tu, tv, u^{2})

with |t| = 0, |u| = 1, |v| = 2. Thus

HH^{∗}(A ⊗ A) = HH^{∗}(A) ⊗ HH^{∗}(A)

has generators t_{1}, t_{2}, u_{1}, u_{2}, v_{1}, v_{2} with the obvious set of relations. Taking invariants, we get generators a = t_{1} + t_{2}, b = u_{1} − u_{2}, c = v_{1} + v_{2}, d = u_{1}v_{1} + u_{2}v_{2}, e = v_{1}v_{2}, and relations (a^{3}, b^{2}, a^{2}b, a^{2}c, abc, a^{2}d, ae). The Poincaré series for HH^{∗}(A^{⊗2})^{𝔖_{2}} is

3 + 2t + 2t^{2} + 3t^{3} + 4t^{4} + 4t^{5} + 4t^{6} + 5t^{7} + 6t^{8} + 6t^{9} + 6t^{10} + 7t^{11} + ⋯

For the other piece, with g = (1 2), using Proposition 2.4 we get the Poincaré series of HH^{∗}(A^{⊗2}, A^{⊗2}g)^{𝔖_{2}} equal to that of HH^{∗}(A):

2 + t + t^{2} + t^{3} + ⋯

so that the Poincaré series of HH^{∗}(B) is

5 + 3t + 3t^{2} + 4t^{3} + 5t^{4} + 5t^{5} + 5t^{6} + 6t^{7} + 7t^{8} + 7t^{9} + 7t^{10} + 8t^{11} + ⋯

This gives the same answer as Erdmann and Schroll [6].

More generally, let n = 2ℓ. We can compute HH^{∗}(A) where A is the block of H(ℓ, q) of weight one, by [5] the dimension of HH^{r}(A) is 1 for r ≥ 1 and = ℓ for r = 0. Let B be the block of H(2ℓ, q) of weight two, then

dim HH^{n}(B) = \begin{cases}2ℓ + \binom{n}{2} & n = 0 \\ ℓ + 2k + 2 & n = 4k + 3 \\ ℓ + 2k + 1 & n = 4k + i ≥ 1, i = 0, 1, 2 \end{cases}

References

[1] D. J. Benson, K. Erdmann, and A. Mikaelian, Cohomology of Hecke algebras, Preprint, 2010.

[2] J. Chuang and R. Kessar, Symmetric groups, wreath products, Morita equivalences, and Broué’s abelian defect group conjecture, Bull. London Math. Soc. 34 (2002), 174–184.

[3] J. Chuang and H. Miyachi, Runner removal Morita equivalences, Preprint, 2007.

[4] J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and 𝔰𝔩_{2}-categorification, Ann. of Math. 167 (2008), 245–298.

[5] K. Erdmann and T. Holm, Twisted bimodules and Hochschild cohomology for self-injective algebras of class A_{n}, Forum Math. 11 (1999), 177–201.

[6] K. Erdmann and S. Schroll, On the Hochschild cohomology of tame Hecke algebras, Arch. Math. (Basel) 94 (2010), 117–127.

[7] P. Etingof and A. Oblomkov, Quantization, orbifold cohomology, and Cherednik algebras, Jack, Hall-Littlewood and Macdonald polynomials, Contemp. Math., vol. 417, American Math. Society, 2006, pp. 171–182.

[8] T. Holm, Hochschild cohomology of Brauer tree algebras, Commun. in Algebra 26 (1998), 3625–3646.

[9] B. Keller, Hochschild cohomology and derived Picard groups, J. Pure & Applied Algebra 190 (2004), 177–196.

[10] M. Linckelmann, Finite generation of Hochschild cohomology of Hecke algebras of finite classical type in characteristic zero, Preprint, 2010.

[11] S. J. Witherspoon, Products in Hochschild cohomology and Grothendieck rings of group crossed products, Adv. in Math. 185 (2004), 136–158.

Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK

Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK