COHOMOLOGY OF HECKE ALGEBRAS

DAVE BENSON, KARIN ERDMANN, AND ARAM MIKAELIAN

Abstract. We compute the cohomology H^{∗}(H, k) = Ext_{H}^{∗}(k, k) where H = H(n, q) is the Hecke algebra of the symmetric group 𝔖_{n} at a primitive ℓth root of unity q, and k is a field of characteristic zero. The answer is particularly interesting when ℓ = 2, which is the only case where it is not graded commutative. We also carry out the corresponding computation for Hecke algebras of type B_{n} and D_{n} when ℓ is odd.

1. Introduction

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. Write n = ℓm + a where 0 ≤ a < ℓ, and let B = H(λ, q) where λ is the partition

λ = (ℓ^{m}, 1^{a}).

That is, B is the subalgebra of H generated by all T_{i} where i < ℓm and ℓ does not divide i. Then B is isomorphic to the tensor product of m copies of H(ℓ, q), and is a maximal ℓ-parabolic subalgebra of H. It has been proved by Du [7] that every H-module is relatively B-projective. This suggests that B should play a role similar to that of the group algebra of a Sylow subgroup of a finite group.

The algebra H has a trivial module k, so it has cohomology H^{∗}(H, k) = Ext_{H}^{∗}(k, k). Similarly we define H^{∗}(B, k) = Ext_{B}^{∗}(k, k). Here we relate the cohomology of H to that of B. We prove an analogue of the result for group algebras which states that if G is a finite group and F is a field of characteristic p then H^{∗}(G, F) is isomorphic to the stable part of H^{∗}(P, F) where P is a Sylow p-subgroup of G (Cartan and Eilenberg [2], Theorem XII.10.1), and that if furthermore P is abelian then the stable elements are the invariants of the action of N_{G}(P)/P (Swan [13], corrected in [14]).

The symmetric group 𝔖_{m} acts naturally on B and on H^{∗}(B, k) = Ext_{B}^{∗}(k, k). Our main theorem is as follows.

Theorem 1.1. The restriction map in cohomology induces an isomorphism

H^{∗}(H, k) → H^{∗}(B, k)^{𝔖_{m}}.

Supported by EPSRC grant EP/D077656/1.

Case 1. If ℓ > 2 then H^{∗}(B, k) = Λ(y_{1}, …, y_{m}) ⊗_{k} k[x_{1}, …, x_{m}] with |y_{i}| = 2ℓ − 3 and |x_{i}| = 2ℓ − 2. Defining a derivation d on H^{∗}(B, k) via d(x_{i}) = y_{i}, d(y_{i}) = 0, we have

H^{∗}(H, k) ≅ H^{∗}(B, k)^{𝔖_{m}} = Λ(dσ_{1}, …, dσ_{m}) ⊗_{k} k[σ_{1}, …, σ_{m}]

where σ_{i} is the ith elementary symmetric polynomial in x_{1}, …, x_{m}.

Case 2. If ℓ = 2 then H^{∗}(B, k) = k⟨z_{1}, …, z_{m}⟩/(z_{i}z_{j} + z_{j}z_{i}, i ≠ j) with |z_{i}| = 1. This algebra is not graded commutative, because the degree one elements z_{i} do not anticommute or square to zero.

Let v_{i} be the ith elementary symmetric function in z_{1}^{2}, …, z_{m}^{2}, so that |v_{i}| = 2i. There are elements u_{i} ∈ H^{2i−1}(B, k)^{𝔖_{m}} (1 ≤ i ≤ m) satisfying

(1.2)  u_{i}^{2} = ∑_{l=0}^{i−1}(2l + 1)v_{i−l−1}v_{i+l}

for 1 ≤ i ≤ m (for i = 1 this relation says that u_{1}^{2} = v_{1}), and

(1.3)  u_{i}u_{j} + u_{j}u_{i} = 2 ∑_{l=0}^{j−1}(i − j + 2l + 1)v_{j−l−1}v_{i+l}

for 1 ≤ j < i ≤ m, where v_{i} is taken to be zero if i > m and v_{0} = 1. We have

H^{∗}(H, k) ≅ H^{∗}(B, k)^{𝔖_{m}} = k⟨u_{1}, …, u_{m}, v_{2}, …, v_{m}⟩/(R)

where (R) is the following set of relations:

(1) v_{i}v_{j} = v_{j}v_{i} (1 ≤ i, j ≤ m)

(2) u_{i}v_{j} = v_{j}u_{i} (1 ≤ i, j ≤ m)

(3) relation (1.2) (2 ≤ i ≤ m)

(4) relation (1.3) (1 ≤ j < i ≤ m) where, in the right hand side of relations (1.2) and (1.3), we take v_{1} to be u_{1}^{2}.

In both cases, ℓ > 2 and ℓ = 2, the following is the Poincaré series for the cohomology:

∑_{i≥0} t^{i} dim_{k} H^{i}(H, k) = \frac{(1 + t^{2(ℓ−1)−1})(1 + t^{4(ℓ−1)−1}) … (1 + t^{2m(ℓ−1)−1})}{(1 − t^{2(ℓ−1)})(1 − t^{4(ℓ−1)}) … (1 − t^{2m(ℓ−1)})}.

In the final section of the paper we prove the analogous theorem for the Hecke algebras of types B_{n} and D_{n} when ℓ is odd.

2. Background on Hecke algebras

The standard approach to working with the representation theory of Hecke algebras of type A was developed by Dipper and James [5]. Given a standard Young subgroup S_{λ} of 𝔖_{n}, they work with a set D_{λ} of right coset representatives of minimal length. Since we are working with left modules rather than right modules, we use the set D_{λ}^{−1} of inverses of these elements.

In detail, let λ be a partition of n. Let t^{λ} be the tableau of shape λ in which the numbers 1, 2, …, n appear in order along successive rows. Then we take for S_{λ} the standard Young subgroup, in which the rows of t^{λ} are the orbits of S_{λ}. Then the distinguished set D_{λ} of right coset representatives of S_{λ} in 𝔖_{n} consists precisely of elements g ∈ 𝔖_{n} such that the tableau t^{λ}g is row standard. We write D_{λ}^{−1} for the set of g^{−1} with g ∈ D_{λ}.

We recall from [5] the basic properties of the distinguished coset representatives.

Lemma 2.1. Let d ∈ D_{λ}^{−1}. Then (i) For each w ∈ S_{λ}, we have l(dw) = l(w) + l(d). (ii) Write S for the set of transpositions of the form s_{i} = (i, i + 1) in 𝔖_{n}. If v ∈ S then

either vd ∈ D_{λ}^{−1}, or d^{−1}vd ∈ S_{λ} ∩ S and l(vd) = l(d) + 1.

We will also need to work with double cosets of Young subgroups, and we recall what we need from Lemma 1.6 of Dipper and James [5].

Lemma 2.2. Let D_{λ,λ} = D_{λ} ∩ D_{λ}^{−1}. Note that D_{λ,λ} = D_{λ,λ}^{−1}.

(a) D_{λ,λ} is a system of (S_{λ}, S_{λ}) double coset representatives in 𝔖_{n}.

(b) Each d ∈ D_{λ,λ} is the unique element of minimal length in its double coset.

(c) If d ∈ D_{λ,λ} then dS_{λ}d^{−1} ∩ S_{λ} is a standard Young subgroup, which we denote by S_{λ(d)} where λ(d) is a composition of n.

(d) If v = dud^{−1} ∈ S_{λ(d)} for u ∈ S_{λ} then l(u) = l(v).

(e) Every element w ∈ 𝔖_{n} has a unique expression as w = udv with u ∈ D_{λ(d)}^{−1} ∩ S_{λ} and d ∈ D_{λ,λ} and v ∈ S_{λ}. Moreover, l(w) = l(u) + l(d) + l(v) and

D_{λ}^{−1} = ⋃_{d∈D_{λ,λ}}(D_{λ(d)}^{−1} ∩ S_{λ})d.

By part (e) of the lemma, any w ∈ D_{λ}^{−1} has a unique expression of the form w = td for d ∈ D_{λ,λ} and t ∈ D_{λ(d)}^{−1} ∩ S_{λ}. We set T_{d} = D_{λ(d)}^{−1} ∩ S_{λ} for d ∈ D_{λ,λ}, and then we have

(2.3)  D_{λ}^{−1} = ⋃_{d∈D_{λ,λ}} ⋃_{t∈T_{d}}{td}.

The q-analogue of Lagrange’s theorem. Suppose that S_{μ} ⊆ S_{λ} ⊆ 𝔖_{n} are standard parabolic subroups. Then S_{λ} can be written as a disjoint union of left cosets

S_{λ} = ⋃_{d∈D_{μ}^{−1}∩S_{λ}} dS_{μ}.

Definition 2.4. We define the q-index of S_{μ} in S_{λ} to be the number

(S_{γ} : S_{μ})_{q} = ∑_{d∈D_{μ}^{−1}∩S_{γ}} q^{l(d)}.

The following is an analogue for Hecke algebras of Lagrange’s Theorem for groups:

Lemma 2.5. We have

(S_{λ} : S_{μ})_{q}(S_{μ} : 1)_{q} = (S_{λ} : 1)_{q}.

Proof. If w ∈ S_{γ} then w has a unique factorisation as w = dv with v ∈ S_{μ} and d a distinguished coset representative, and l(w) = l(d) + l(v). □

Furthermore, we have a summation formula, corresponding to the double coset decomposition.

Lemma 2.6. We have

(𝔖_{n} : S_{μ})_{q} = ∑_{d∈D}(S_{μ} : S_{μ(d)})_{q}q^{l(d)}

Proof. This follows from Lemma 2.5, using (2.3), since

∑_{d∈D_{λ}^{−1}} q^{l(d)} = ∑_{d∈D_{λ,λ}}( ∑_{t∈T_{d}} q^{l(t)})q^{l(d)} = ∑_{d∈D_{λ,λ}}(S_{μ} : S_{μ(d)})_{q}.  □

A divisibility lemma. From now we take λ = (ℓ^{m}, a) where n = ℓm + a and 0 ≤ a < ℓ. We shall need to use the fact that the length of an element

g ∈ N(S_{λ}) ∩ D_{λ} = N(S_{λ}) ∩ D_{λ}^{−1} = N(S_{λ}) ∩ D_{λ,λ}

which fixes the fixed points of S_{λ}, is divisible by ℓ. This will play a role several times in simplifying identities.

Recall that the length of a permutation x ∈ 𝔖_{n} is the size of its inversion set, which is defined to be

inv(x) = {(i, j) : 1 ≤ i < j ≤ n and ix > jx}

(see Exercise 2 in Section 1.6 of Humphreys [10]). Take x ∈ D_{λ}, and assume x fixes each fixed point of S_{λ}. Then the tableau t^{λ}x is row standard, and hence if i < j and i, j are in the same row of t^{λ} then (i, j) is not in inv(x). Writing R_{i} for the ith row of t^{λ}, the inversion set is therefore the disjoint union of sets inv(x)_{b,c} = inv(x) ∩ (R_{b} × R_{c}) where 1 ≤ b < c ≤ m.

Lemma 2.7. Let x ∈ D_{λ}. Assume one of R_{b}x or R_{c}x is equal to some row of t^{λ} where b < c ≤ m. Then the size of inv(x)_{b,c} is divisible by ℓ.

Proof. Say R_{b}x is some row of t^{λ}, then R_{b}x consists of consecutive numbers. Fix some j ∈ R_{c}, then jx is different from the numbers in R_{b}x, so either ix > jx for all i ∈ R_{b}, or ix < jx for all i ∈ R_{b}. So if M is the set of all j ∈ R_{c} such that (i, j) ∈ inv(x) for some i ∈ R_{b} then inv(x)_{b,c} has size |R_{b}| · |M| = ℓ · |M|. □

Corollary 2.8. Suppose g ∈ N(S_{λ}) ∩ D_{λ} and g fixes each fixed point of S_{λ}. Then the length of g is divisible by ℓ.

Proof. In this case, for any 1 ≤ b ≤ m, the set R_{b}g is a row of t^{λ} and hence the size of inv(x)_{b,c} is divisible by ℓ. Furthermore, the fixed points of S_{λ} do not contribute to the inversion set of g. □

Some explicit q-indices. Recall that we are assuming that λ = (ℓ^{m}, 1^{a}) where n = ℓm + a and 0 ≤ a < ℓ, so that S_{λ} is maximal ℓ-parabolic.

Lemma 2.9. (1) We have (S_{n} : 1)_{q} = [n]_{q}!. Furthermore, if μ = (μ_{i}) is a composition

of n, then (S_{μ} : 1)_{q} = ∏_{i}(S_{μ_{i}} : 1)_{q}.

(2) We have (S_{n} : S_{λ})_{q} = m![a]_{q}!.

(3) If S_{λ(d)} is a proper subgroup of S_{λ} then (S_{λ} : S_{λ(d)})_{q} = 0.

Recall that the q-factorial is defined as [a]_{q}! = [1]_{q}[2]_{q} … [a]_{q} where [a]_{q} denotes 1 + q + ⋯ + q^{a−1}.

Proof. Part (1) is well known, and part (2) is proved in Corollary 2.5 of [4]. To prove part (3), note that λ(d) is a refinement of λ, so we can factorise the q-index, and it suffices to show that (𝔖_{ℓ} : S_{μ})_{q} = 0 if μ is a composition of ℓ and μ ≠ (ℓ). We have

(𝔖_{ℓ} : S_{μ})_{q}(S_{μ} : 1)_{q} = (𝔖_{ℓ} : 1)_{q} = [ℓ]_{q}! = 0

and (S_{μ} : 1)_{q} is non-zero since all parts of μ are strictly less than ℓ. So it follows that (𝔖_{ℓ} : S_{μ})_{q} = 0. □

The transfer map. Assume that M and N are H-modules, and y ∈ Hom_{B}(M, N). Define

tr_{B,H}(y)(x) = ∑_{d∈D_{λ}^{−1}} q^{−l(d)}T_{d}y(T_{d^{−1}}x) for x ∈ M.

Lemma 2.10. The map tr_{B,H}(y) is a H-module homomorphism.

Proof. The argument for this is given in [4], but since the context and notation are slightly different we give the proof here for the convenience of the reader.

Fix s ∈ S where S is the set of basic transpositions. We must show that tr_{B,H}(y) commutes with T_{s}. Let

D_{1} = {d ∈ D_{λ}^{−1} : d^{−1}sd ∈ S_{λ} ∩ S, l(sd) = l(d) + 1}
D_{2} = {d ∈ D_{λ}^{−1} : sd ∈ D_{λ}^{−1}, l(sd) = l(d) + 1}
D_{3} = {d ∈ D_{λ}^{−1} : sd ∈ D_{λ}^{−1}, l(sd) = l(d) − 1} = sD_{2}

By Lemma 2.1, these sets form a partition of D_{λ}^{−1}. So we have

(2.11)  tr_{B,H}(y) = ∑_{d∈D_{1}} q^{−l(d)}T_{d}yT_{d^{−1}} + ∑_{d∈D_{2}}(q^{−l(d)}T_{d}yT_{d^{−1}} + q^{−l(d)−1}T_{sd}yT_{(sd)^{−1}}).

If d ∈ D_{1}, write s′ = d^{−1}sd. Then

T_{d^{−1}}T_{s} = T_{d^{−1}s} = T_{s′d^{−1}} = T_{s′}T_{d^{−1}}
and  T_{d}T_{s′} = T_{ds′} = T_{sd} = T_{s}T_{d}.
So  T_{d}yT_{d^{−1}}T_{s} = T_{d}yT_{s′}T_{d^{−1}} = T_{d}T_{s′}yT_{d^{−1}} = T_{s}T_{d}yT_{d^{−1}}.

This proves that T_{s} commutes with the first sum in (2.11).

Now let d ∈ D_{2}, then

(q^{−l(d)}T_{d}yT_{d^{−1}}+q^{−l(d)−1}T_{(sd)}yT_{(sd)^{−1}})T_{s} = q^{−l(d)}T_{d}yT_{d^{−1}s} + q^{−l(d)−1}T_{sd}yT_{d^{−1}}(T_{s})^{2}
= q^{−l(d)}T_{d}yT_{d^{−1}s} + q^{−l(d)−1}(q − 1)T_{sd}yT_{d^{−1}s} + q^{−l(d)}T_{sd}yT_{d^{−1}}
= T_{s}(q^{−l(d)−1}T_{(sd)}yT_{(sd)^{−1}} + q^{−l(d)}T_{d}yT_{d^{−1}}).

This proves that T_{s} commutes with the second sum in (2.11). □

We will only use this when N = k, the trivial module. Then the formula becomes

tr_{B,H}(y)(x) = ∑_{d∈D_{λ}^{−1}} y(T_{d^{−1}}x)

Next, still assuming N = k, we write tr_{B,H}(y) = ∑_{d∈D_{λ,λ}} y_{d} where

(2.12)  y_{d}(x) = ∑_{t∈T_{d}} y(T_{d^{−1}}T_{t^{−1}}x) (x ∈ M)

and T_{d} = D_{λ(d)}^{−1} ∩ S_{λ}.

Definition 2.13. Suppose M is an H-module, then y ∈ Hom_{B}(M, k) is stable provided for all d ∈ D_{λ,λ} we have y[T_{d^{−1}}(−)] = q^{l(d)}y(−).

Remark 2.14. Any d in N(S_{λ}) ∩ D_{λ} can be written as d = d_{1}d_{2} = d_{2}d_{1} where d_{1} fixes the fixed points of S_{λ} and d_{2} fixes the remaining points. Furthermore, d_{1} and d_{2} are also in N(S_{λ}) ∩ D_{λ}. For such a d, the stability condition of Definition 2.13 reduces to the condition for d_{1}. This observation will be used in the proof of Proposition 3.3 (ii).

Lemma 2.15. If y ∈ Hom_{B}(M, k) is an H-module homomorphism then it is stable.

Proof. In this case, for x ∈ M we have

y[T_{d^{−1}}(x)] = T_{d^{−1}}y(x) = q^{l(d)}y(x)

since y maps into the trivial module. □

Now assume y ∈ Hom_{B}(M, k) is stable, and consider the map y_{d} defined in (2.12). Since y is a B-module homomorphism, this can then be written as

(2.16)  y_{d}(x) = q^{l(d)} ∑_{t∈T_{d}} T_{t^{−1}}y(x) = q^{l(d)}[∑_{t} q^{l(t)}]y(x) = q^{l(d)}(S_{λ} : S_{λ(d)})_{q}y(x).

3. Relating cohomology of H and B

In the following we write H^{∗}(H, k) for Ext_{H}^{∗}(k, k) and H^{∗}(B, k) for Ext_{B}^{∗}(k, k). The multiplicative structure on cohomology is given by Yoneda composition. Since H and B are not Hopf algebras, there is no a priori reason why these cohomology rings should be graded commutative, and indeed we shall see that for ℓ = 2 it is not.

To compute H^{∗}(H, k), we take a projective resolution of k as an H-module

(3.1)  ⋯ → P_{i} → P_{i−1} → ⋯ → P_{0} → 0

and then take the cohomology of the complex

0 → Hom_{H}(P_{0}, k) → ⋯ → Hom_{H}(P_{i−1}, k) → Hom_{H}(P_{i}, k) → ….

We take for P_{∗} a minimal resolution for H, so that we have H^{i}(H, k) ≅ Hom_{H}(Ω^{i}k, k) where Ω^{i}k is the kernel of P_{i−1} → P_{i−2} for i ≥ 2, Ωk is the kernel of P_{0} → k, and Ω^{0}k = k.

Since H is a free B-module (see for example Lemma 2.4 of [5]), the restriction of a projective H-module to B is a projective B-module. So we may compute H^{∗}(B, k) using the same resolution, but we need to note that for B it is not the minimal resolution. So H^{i}(B, k) is the quotient \underline{Hom}_{B}(Ω^{i}k, k) of Hom_{B}(Ω^{i}k, k) by the maps P Hom_{B}(Ω^{i}k, k) that factor through a projective module.

In particular, the composite of

res_{H,B}: Hom_{H}(Ω^{i}k, k) → Hom_{B}(Ω^{i}k, k)

with the surjection Hom_{B}(Ω^{i}i, k) → \underline{Hom}_{B}(Ω^{i}k, k) gives the restriction map

res_{H,B}: H^{i}(H, k) → H^{i}(B, k)

which is a ring homomorphism with respect to Yoneda composition.

We also have the transfer map

tr_{B,H}: Hom_{B}(P_{i}, k) → Hom_{H}(P_{i}, k)

which commutes with the differentials, and hence induces a transfer map in cohomology

tr_{B,H}: H^{i}(B, k) → H^{i}(H, k).

As an intermediary, the transfer map

tr_{B,H}: Hom_{B}(Ω^{i}k, k) → Hom_{H}(Ω^{i}k, k)

sends maps that factor through a projective module to zero, and this gives the map in cohomology.

Lemma 3.2. Let y ∈ Hom_{H}(Ω^{i}k, k). Then we have

tr_{B,H}res_{H,B}(y) = (𝔖_{n} : S_{λ})_{q}y.

In particular, (𝔖_{n} : S_{λ})_{q} ≠ 0, so that

res_{H,B}: H^{∗}(H, k) → H^{∗}(B, k)

is injective.

Proof. Since y is an H-module homomorphism we can write for x ∈ Ω^{i}k

tr_{B,H}res_{H,B}(y)(x) = ∑_{d∈D_{λ}^{−1}} y(T_{d^{−1}}x) = ∑_{d∈D_{λ}^{−1}} T_{d^{−1}}y(x) = (𝔖_{n} : S_{λ})_{q}y(x),

and by Lemma 2.9 the q-index is non-zero. This shows that

res_{H,B}: Hom_{H}(Ω^{i}k, k) → Hom_{B}(Ω^{i}k, k)

is injective. Since the transfer of an element of P Hom_{B}(Ω^{i}k, k) is always zero, we deduce that res_{H,B}: H^{∗}(H, k) → H^{∗}(B, k) is injective as required. □

This shows that H^{∗}(H, k) is isomorphic to the image of the restriction. We will now give a description of this in terms of stable elements.

Proposition 3.3. (i) The intersection of the stable elements of Hom_{B}(Ω^{i}k, k) with the projective homomorphisms P Hom_{B}(Ω^{i}k, k) is equal to {0}, so that it makes sense to talk of stable elements of H^{i}(B, k) ≅ \underline{Hom}_{B}(Ω^{i}k, k). The image of the restriction map

res_{H,B}: H^{∗}(H, k) → H^{∗}(B, k)

consists precisely of the stable elements. (ii) The inclusion H(ℓm, q) ⊆ H induces an isomorphism H^{∗}(H, k) → H^{∗}(H(ℓm, q), k).

Proof. (i) Suppose that y ∈ H^{i}(H, k). Then res_{H,B}(y) ∈ Hom_{B}(Ω^{i}k, k) is stable, see Lemma 2.15.

Conversely, suppose that y ∈ Hom_{B}(Ω^{i}k, k) and assume that y is stable. Using the Mackey formula, we have res_{H,B}tr_{B,H}(y) = ∑_{d∈D_{λ,λ}} y_{d} with y_{d} as in (2.12). Since y is assumed to be stable, we can apply (2.16) to get

y(x) = ∑_{d∈D_{λ,λ}} q^{l(d)}(S_{λ} : S_{λ(d)})_{q}y(x)

where x ∈ Ω^{i}(k) and S_{λ(d)} = dS_{λ}d^{−1} ∩ S_{λ}. By Lemma 2.9 if S_{λ(d)} is a proper subset of S_{λ} then the q-index is zero. So we get

(3.4)  y(x) = ∑_{d∈N(S_{λ})∩D_{λ}} q^{l(d)}y(x).

By Lemma 2.6 and parts (2) and (3) in 2.9 we have that ∑_{d∈N(S_{λ})∩D_{λ}} q^{l(d)} = m![a]_{q}! hence is non-zero. Therefore we have proved that y = res_{H,B}[tr_{B,H}(y)] · \frac{1}{m![a]_{q}!} so it is in the image of the restriction map.

Finally, the transfer of any projective map is zero, so that if a stable map is also projective then it is equal to zero.

(ii) It follows from Remark 2.14 and equation (3.4) that an element of Hom_{B}(Ω^{i}k, k) is stable with respect to H if and only if it is stable with respect to H(ℓm, q). □

4. Action of 𝔖_{m} on B and on H^{∗}(B, k)

The next aim is to characterise the image of the restriction map as fixed points of an action. According to Proposition 3.3 (ii), we may assume without loss of generality that n = ℓm, i.e., that a = 0. We write λ for the partition (ℓ^{m}).

The braid group. We begin by recalling that the braid group 𝔅_{n} on n strings has a presentation with generators T_{i} (1 ≤ i ≤ n − 1) and relations T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1} and T_{i}T_{j} = T_{j}T_{i} for |i − j| ≥ 2. For any w ∈ 𝔖_{n}, let w = s_{i_{1}} … s_{i_{l}} be a shortest word in the transpositions s_{i} = (i i + 1) giving w, and set T_{w} = T_{i_{1}} … T_{i_{l}}; this is independent of the choices.

There is a homomorphism from 𝔅_{n} to H(n, q) taking T_{i} to the element of the same name. Note that in H(n, q), T_{i} is invertible with inverse q^{−1}(T_{i} − q + 1).

For each i with 1 ≤ i ≤ m, let d_{i} ∈ N(S_{λ}) be the element swapping the ith block of ℓ elements with the i + 1st. Thus

jd_{i} = \begin{cases}j + ℓ & if ℓi < j ≤ ℓ(i + 1) \\ j − ℓ & if ℓ(i + 1) < j ≤ ℓ(i + 2) \\ j & otherwise. \end{cases}

Then d_{i} swaps the ith and i + 1st rows of the tableau t^{λ} and leaves the order of the elements within each row unchanged. Thus d_{i} is an involution and t^{λ}d_{i} is row standard, so d_{i} ∈ D_{λ,λ}.

It is not hard to verify directly that the elements T_{d_{i}} (1 ≤ i ≤ m) satisfy the braid relations. So inside 𝔅_{n} we have a wreath product 𝔅_{ℓ} ≀ 𝔅_{m} generated by the elements T_{i} (1 ≤ i < n, ℓ ∤ i) and the elements T_{d_{i}} (1 ≤ i ≤ m). This is the same as the group generated by the T_{w} for w ∈ N(S_{λ}).

Conjugation by d_{i} interchanges the generators in the ith factor of S_{λ} with the generators of the i + 1st factor and fixes all other generators. So for s_{j} ∈ S_{λ} ∩ S (i.e., ℓ ∤ j) we have

d_{i}s_{j} = \begin{cases}s_{j+ℓ}d_{i} & ℓi < j < ℓ(i + 1) \\ s_{j−ℓ}d_{i} & ℓ(i + 1) < j < ℓ(i + 2) \\ s_{j}d_{i} & otherwise \end{cases}

Lemma 4.1. We have

T_{d_{i}}T_{j} = \begin{cases}T_{j+ℓ}T_{d_{i}} & ℓi < j < ℓ(i + 1) \\ T_{j−ℓ}T_{d_{i}} & ℓ(i + 1) < j < ℓ(i + 2) \\ T_{j}T_{d_{i}} & otherwise \end{cases}

Proof. Since d_{i} is a distinguished coset representative, we have for any s_{j} in S_{λ} that l(d_{i}s_{j}) = l(d_{i}) + 1 = l(s_{j}d_{i}) and therefore if for example s_{j} is in the ith factor then

T_{d_{i}}T_{j} = T_{d_{i}s_{j}} = T_{s_{j+l}d_{i}} = T_{j+l}T_{d_{i}}.

Similarly one gets the other identities. □

The action of 𝔖_{m} on B. The algebra B is the tensor product of m copies of H(ℓ, q), say B = ⨂_{i=1}^{m} B_{i}, where B_{i} is supported on the numbers in the ith row of t^{λ}. Therefore the symmetric group 𝔖_{m} acts by permuting the factors. This will induce an action on the cohomology of B, and we want to show that the fixed points under this action are precisely the stable elements. We shall realise this action through conjugation by elements of 𝔅_{ℓ} ≀𝔅_{m}. Namely, if y ∈ Hom_{B}(Ω^{i}k, k) and w ∈ N(S_{λ}) then we define T_{w} · y via

(4.2)  (T_{w} · y)(x) = T_{w}(y(T_{w}^{−1}x)) = q^{l(w)}y(T_{w}^{−1}x)

for x ∈ Ω^{i}k.

This is the usual formula for group actions on homomorphisms, but it is not the same as the formula for the action of the Hecke algebra on a dual space. Recall that for an H-module M, the dual space Hom_{k}(M, k) is an H-module, using the anti-involution on H defined by

(T_{g})^{∗} = T_{g^{−1}}

and linear extension, see §4 in [5]. We write the action of T_{g} ∈ H on Hom_{k}(M, k) as

(4.3)  T_{g}f(m) = f(T_{g^{−1}}m).

We view Hom_{B}(M, k) and Hom_{H}(M, k) as subspaces of M^{∗}.

The advantage of (4.2) over (4.3) is that it gives a well defined action of the group 𝔅_{ℓ} ≀𝔅_{m} on Hom_{B}(Ω^{i}k, k). Our strategy is as follows. The next proposition shows that the normal subgroup 𝔅_{ℓ}^{×m} acts trivially, so that we are reduced to an action of 𝔅_{m}. Then Proposition 4.7 will show that the pure braid group acts trivially up to projective homomorphisms, so that we are reduced to an action of 𝔖_{m} on H^{∗}(B, k).

Proposition 4.4. Let M be an H-module, y ∈ Hom_{B}(M, k) and g ∈ S_{λ}. Then T_{g} · y = y.

Proof. Using T_{i}^{−1} = q^{−1}(T_{i} − q + 1) we see that T_{g}^{−1} belongs to B and then the statement is clear, from the definition. □

Lemma 4.5. If g ∈ N(S_{λ}) ∩ D_{λ} then the element T_{g}T_{g^{−1}} ∈ H(n, q) centralises B. In particular T_{d_{i}}^{2} centralises B.

Proof. The algebra B is generated by the T_{j} for s_{j} ∈ S_{λ} ∩ S. If s_{j} ∈ S_{λ} ∩ S then for some j′ we have s_{j}g = gs_{j′}. Thus

T_{g}T_{g^{−1}}T_{j} = T_{g}T_{g^{−1}s_{j}} = T_{g}T_{s_{j′}g^{−1}} = T_{g}T_{j′}T_{g^{−1}} = T_{gs_{j′}}T_{g^{−1}} = T_{s_{j}g}T_{g^{−1}} = T_{j}T_{g}T_{g^{−1}}. □

The following lemma is copied from a standard argument in group cohomology. We shall need to use it not only when Q = B, but also when Q is a proper parabolic subalgebra of B.

Lemma 4.6. Let Q be a parabolic subalgebra of H and γ ∈ H be an element centralising Q and acting as the identity on k. Then γ: Ω^{i}k → Ω^{i}k is a Q-module map, and γ − 1 factors through a projective Q-module.

Thus for y ∈ Hom_{Q}(Ω^{i}k, k), the element y ∘ γ: x ↦ y(γx) of Hom_{Q}(Ω^{i}k, k) differs from y by an element of P Hom_{Q}(Ω^{i}k, k), so that y and y∘γ represent the same element of H^{i}(Q, k).

Proof. Multiplication by γ commutes with the action of Q, and therefore induces a map of chain complexes on the resolution (3.1), lifting the identity on k. Thus there is a chain homotopy to the identity, consisting of maps h_{i}: P_{i} → P_{i−1}:

\xymatrix{… \ar[r] & P_{i} \ar[r]^{δ} \ar[d]^{γ} & P_{i−1} \ar[r]^{δ} \ar[d]^{γ} \ar[dl]^{h_{i−1}} & P_{i−2} \ar[r]^{δ} \ar[d]^{γ} \ar[dl]^{h_{i−2}} & … \ar[r]^{δ} & P_{0} \ar[d]^{γ} \\ … \ar[r] & P_{i} \ar[r]^{δ} & P_{i−1} \ar[r]^{δ} & P_{i−2} \ar[r]^{δ} & … \ar[r]^{δ} & P_{0}}

satisfying δh_{i−1} + h_{i−2}δ = γ − 1. On Ω^{i}k = ker(δ: P_{i−1} → P_{i−2}), we have δ = 0 and so

γ − 1: Ω^{i}k → Ω^{i}k factors as Ω^{i}k ⟶^{h_{i−1}} P_{i} ⟶^{δ} Ω^{i}k. □

Proposition 4.7. (i) If y ∈ Hom_{B}(Ω^{i}k, k) then T_{d_{i}} · y and T_{d_{i}}y differ by an element of

P Hom_{B}(Ω^{i}k, k), and hence represent the same element of H^{i}(B, k). (ii) The elements T_{d_{i}}^{2} ∈ 𝔅_{m} act trivially on H^{∗}(B, k).

Proof. The elements T_{d_{i}} act as multiplication by q^{l(d_{i})} on k, and by Lemma 2.8 we have q^{l(d_{i})} = 1. Thus T_{d_{i}}^{2} also acts trivially on k. By Lemma 4.5, T_{d_{i}}^{2} also centralises B. From (4.2) and (4.3) we have T_{d_{i}} · y = y ∘ T_{d_{i}}^{−1} while T_{d_{i}}y = y ∘ T_{d_{i}}. Thus T_{d_{i}}y = (T_{d_{i}} · y) ∘ T_{d_{i}}^{2}, and (i) now follows from Lemma 4.6. Similarly for part (ii) we have T_{d_{i}}^{2} · y = y ∘ T_{d_{i}}^{−2} and we can again apply Lemma 4.6. □

Theorem 4.8. The action of 𝔅_{ℓ} ≀𝔅_{m} on H^{∗}(B, k) factors through the surjection 𝔅_{ℓ} ≀𝔅_{m} → 𝔖_{m}. The formulas (4.2) and (4.3) both describe the resulting action of 𝔖_{m} on H^{∗}(B, k).

Proof. This follows from Propositions 4.4 and 4.7. □

Theorem 4.9. The image of the restriction map H^{∗}(H, k) → H^{∗}(B, k) consists precisely of the fixed points in H^{∗}(B, k) under the action of the symmetric group 𝔖_{m}.

Proof. By Proposition 3.3, H^{∗}(H, k) consists of the stable elements of H^{∗}(B, k). If y ∈ H^{i}(B, k) is stable then y is in particular a fixed point. For the converse, let y ∈ H^{i}(B, k) be a fixed point under 𝔖_{m}. We must show that then y is stable.

Take d ∈ D_{λ}^{−1}, and let Q be the parabolic subalgebra of H corresponding to S_{λ(d)}. We must show that on restriction to Q we have T_{d}y = q^{l(d)}y as an element in H^{i}(Q, k)

By the Lemma 4.11 below, there is w ∈ D_{λ}^{−1} such that dS_{λ}d^{−1} = wS_{λ}w^{−1} and w centralises S_{λ(d)} and such that moreover if g = w^{−1}d then q^{l(g)+l(w)} = q^{l(d)}.

Since w centralises Q, we know by applying Lemma 4.6 to q^{−l(w)}T_{w^{−1}} that T_{w}y = q^{l(w)}y, as elements in H^{i}(Q, k). We must relate T_{d}y and T_{w}y.

We have d = wg and therefore (T_{d})^{∗} = (T_{g})^{∗}(T_{w})^{∗} and

(4.10)  T_{d}y(x) = y((T_{d})^{∗}x) = y((T_{g})^{∗}(T_{w})^{∗}x) = (T_{g}y)((T_{w})^{∗}x).

Since g ∈ N(S_{λ}) and y is a fixed point, we know T_{g}y = q^{l(g)}y, so

T_{d}y(x) = q^{l(g)}(T_{w}y)(x) = q^{l(g)+l(w)}y(x).

By Lemma 4.11 this is precisely q^{l(d)}y(x) as required. □

Lemma 4.11. Let d ∈ D_{λ,λ} and Q = dS_{λ}d^{−1} ∩ S_{λ}. Then there is some w ∈ C(Q) ∩ D_{λ,λ} such that wS_{λ}w^{−1} = dS_{λ}d^{−1}; and if g = w^{−1}d then ℓ divides l(g), and

q^{l(d)} = q^{l(w)+l(g)} = q^{l(w)}.

Proof. Since the rows of t^{λ} are the support sets of S_{λ}, we see that the rows of t^{λ}d^{−1} are the support sets of dS_{λ}d^{−1}. Hence the rows which are common to t^{λ} and t^{λ}d^{−1} (not necessarily in the same places) are then precisely the support sets of Q. Note that the natural order on numbers induces a linear order on the rows of t^{λ}.

Let R_{i_{1}} < R_{i_{2}} < … < R_{i_{t}} be the rows of t^{λ} such that if R_{j_{m}} = R_{i_{m}}d^{−1} then the R_{j_{m}} are the support sets of the intersection Q. Then let A_{1} < A_{2} < … < A_{r−t} be the rows of t^{λ} other than the R_{i_{m}}, and B_{1} < B_{2} < … < B_{r−t} be the rows of t^{λ} other than the rows R_{j_{m}}.

We define now g ∈ 𝔖_{n} by

B_{i}g = A_{i}, R_{j_{r}}g = R_{i_{r}}

Then g induces a permutation of the rows of t^{λ}, keeping each row in order. Hence g ∈ N(S_{λ}) ∩ D_{λ}^{−1} (and therefore g ∈ D_{λ,λ}). Define w^{−1} = gd^{−1}. Then

R_{j_{r}}w^{−1} = R_{j_{r}}, B_{i}w^{−1} = A_{i}d^{−1}

Hence w^{−1} ∈ D_{λ} and it fixes the support of Q pointwise and therefore it centralises Q. We will show that this satisfies the condition on the length. By Corollary 2.7, we know that l(g) is divisible by ℓ.

So we must show that l(d^{−1}) ≡ l(w^{−1}) mod ℓ. Let b < c. If one of R_{b}d^{−1} or R_{c}d^{−1} is a row of t^{λ} then inv(d^{−1})_{b,c} is divisible by ℓ, by Corollary 2.7. So we only need to consider such sets where neither R_{b}d^{−1} nor R_{c}d^{−1} is a row of t^{λ}; and the same reduction holds for the inversion set of w^{−1}.

Suppose that b < c are such that neither R_{b}d^{−1} nor R_{c}d^{−1} are rows of t^{λ}. We show that there are u < v such that |inv(d^{−1})_{b,c}| = |inv(w^{−1})_{u,v}| where neither R_{u}w^{−1} nor R_{v}w^{−1} are rows of t^{λ} and where this produces a bijection between the relevant parts in the partitions of inv(d^{−1}) and inv(w^{−1}). Since the remaining parts of inv(d^{−1}) and inv(w^{−1}) have size divisible by ℓ, the Lemma will follow.

By assumption, R_{b} = A_{b′} and R_{c} = A_{c′} for b′ < c. Now, A_{b′}d^{−1} = B_{b′}w^{−1} and A_{c′}d^{−1} = B_{c′}w^{−1}. There are unique u, v such that R_{u} = B_{b′} and R_{v} = B_{c′}, and then u < v. Let (i, j) ∈ inv(d^{−1})_{b,c}. Then id^{−1} > jd^{−1}, so

id^{−1} = i′w^{−1}, jd^{−1} = j′w^{−1} (i′ ∈ B_{b′}, j′ ∈ B_{c′})

and then (i′, j′) ∈ inv(w^{−1})_{u,v}. The converse also holds, and this gives the required bijection between inv(d^{−1})_{b,c} and inv(w^{−1})_{u,v}. □

5. Invariant theory

Let k be a field of characteristic zero, and let H = H(n, q) be the Hecke algebra of degree n with parameter q a primitive ℓth root of unity, ℓ ≥ 2. Let H_{1} = H(ℓ, q), let n = ℓm + a with 0 ≤ a < ℓ, and let B = H_{1}^{⊗m} ⊆ H. From the stable element computation in the last section, we have

(5.1)  H^{∗}(H, k) = H^{∗}(B, k)^{𝔖_{m}}.

In this section we compute the right hand side using invariant theory.

The structure of the algebra H^{∗}(H_{1}, k) depends on whether ℓ = 2 or ℓ > 2. If ℓ > 2 then

(5.2)  H^{∗}(H_{1}, k) = Λ(y) ⊗_{k} k[x],  |y| = 2ℓ − 3, |x| = 2ℓ − 2.

while if ℓ = 2 then

(5.3)  H^{∗}(H_{1}, k) = k[z],  |z| = 1

Here, Λ(y) denotes an exterior algebra in one variable. Thus we have y^{2} = 0.

To obtain H^{∗}(B, k) we apply the version of the Künneth theorem given in Yoneda [15], which describes the Ext algebra of a tensor product of algebras as the graded tensor product of the Ext algebras, with the usual sign conventions.

The case ℓ > 2. We deal with the easy case ℓ > 2 first. In this case, the Künneth theorem and (5.2) give

(5.4)  H^{∗}(B, k) = Λ(y_{1}, …, y_{m}) ⊗_{k} k[x_{1}, …, x_{m}],  |y_{i}| = 2ℓ − 3, |x_{i}| = 2ℓ − 2.

The action of the symmetric group 𝔖_{m} is by permutation of the variables y_{1}, …, y_{m} and simultaneously the variables x_{1}, …, x_{m}. The invariants are given by a theorem of Solomon [12] as follows. Define a derivation d on H^{∗}(B, k) via d(x_{i}) = y_{i}, d(y_{i}) = 0. This differential commutes with the action of 𝔖_{m}, so it sends invariants to invariants. Let σ_{1}, …, σ_{m} be the elementary symmetric polynomials in x_{1}, …, x_{m}, so that

k[x_{1}, …, x_{m}]^{𝔖_{m}} = k[σ_{1}, …, σ_{m}].

The main theorem of [12] shows in this case that

(5.5)  (Λ(y_{1}, …, y_{m}) ⊗_{k} k[x_{1}, …, x_{m}])^{𝔖_{m}} = Λ(dσ_{1}, …, dσ_{m}) ⊗_{k} k[σ_{1}, …, σ_{m}].

Combining (5.1), (5.4) and (5.5), we obtain

H^{∗}(H, k) = Λ(dσ_{1}, …, dσ_{m}) ⊗_{k} k[σ_{1}, …, σ_{m}].

A standard computation now shows that the Poincaré series for H^{∗}(H, k) is as given in Theorem 1.1.

The case ℓ = 2. What is different in the case ℓ = 2 is that the algebra (5.3) is not graded commutative, since there is an element of degree one that does not square to zero. So taking into account the signs given by the Künneth theorem, we get

H^{∗}(B, k) = k⟨z_{1}, …, z_{m}⟩/(z_{i}z_{j} + z_{j}z_{i}) (i ≠ j).

Here, the relations say that the variables anticommute, but do not square to zero.

In this case, we define a finite filtration on H^{∗}(B, k) and pass to the associated graded. The filtration is given by setting F_{i} equal to the linear span of the monomials in the generating variables in which at most i of the variables appear with odd exponent. We have

k[z_{1}^{2}, …, z_{m}^{2}] = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{m} = H^{∗}(B, k).

This is a multiplicative filtration, in the sense that F_{i}F_{j} ⊆ F_{i+j}, so the associated graded

Gr H^{∗}(B, k) = ⨁_{i=0}^{m} F_{i}/F_{i−1}

has a ring structure, where F_{−1} is interpreted as zero. In this ring, we write y_{i} for the image of z_{i} ∈ F_{0}, so that y_{i}^{2} = 0, and we write x_{i} for the image of z_{i}^{2} ∈ F_{2}. It is not hard to check that

Gr H^{∗}(B, k) = Λ(y_{1}, …, y_{m}) ⊗_{k} k[x_{1}, …, x_{m}].

Since the filtration is invariant under the action of 𝔖_{m}, we have

F_{i}^{𝔖_{m}} = F_{i} ∩ H^{∗}(B, k)^{𝔖_{m}}

and by Maschke’s theorem we have

(F_{i}/F_{i−1})^{𝔖_{m}} = F_{i}^{𝔖_{m}}/F_{i−1}^{𝔖_{m}}.

Setting

Gr H^{∗}(B, k)^{𝔖_{m}} = ⨁_{i=0}^{m} F_{i}^{𝔖_{m}}/F_{i−1}^{𝔖_{m}},

we again use Solomon’s formula (5.5) to obtain

(5.6)  Gr H^{∗}(B, k)^{𝔖_{m}} = Λ(dσ_{1}, …, dσ_{m}) ⊗_{k} k[σ_{1}, …, σ_{m}].

At this stage, it follows that the Poincaré series for H^{∗}(H, k) with ℓ = 2 is as given in Theorem 1.1, and it remains to ungrade the relations (dσ_{i})^{2} = 0 in the presentation for Gr H^{∗}(B, k)^{𝔖_{m}} to give a presentation for H^{∗}(H, k).

We define elements v_{i} ∈ H^{∗}(B, k) (1 ≤ i ≤ m) to be the elementary symmetric functions in z_{1}^{2}, …, z_{r}^{2}, with image σ_{i} in Gr H^{∗}(B, k), so that |v_{i}| = 2i, and we define elements u_{i} (1 ≤ i ≤ m) by lifting dσ_{i} in the obvious way to H^{∗}(B, k). Namely,

u_{i} = ∑ z_{α}z_{β_{1}}^{2} … z_{β_{i−1}}^{2},

where the sum is over indices that are all different and satisfy β_{1} < ⋯ < β_{i−1}. Then u_{1}^{2} = v_{1}, so v_{1} is a redundant generator and may be removed from the list. More generally, for i ≥ 1, u_{i}^{2} and u_{i}u_{j} + u_{j}u_{i} are polynomials of degree 2i − 1, respectively i + j − 1, in v_{1}, …, v_{m}. Our next task is to determine these polynomials explicitly.

Lemma 5.7. Let ν ≥ μ ≥ 0. Then

(ν − 2μ)\binom{ν}{μ} + (ν − 2μ + 2)\binom{ν}{μ − 1} + ⋯ + (ν − 2)\binom{ν}{1} + ν\binom{ν}{0} = (ν − μ)\binom{ν}{μ}.

Proof. This is an easy induction on μ using the fact that (ν − μ)\binom{ν}{μ} = (μ + 1)\binom{ν}{μ+1}. □

Proposition 5.8. Set v_{0} = 1. Then for 1 ≤ i ≤ m we have

u_{i}^{2} = ∑_{l=0}^{i−1}(2l + 1)v_{i−l−1}v_{i+l}
= v_{i−1}v_{i} + 3v_{i−2}v_{i+1} + 5v_{i−3}v_{i+2} + ⋯ + (2i − 1)v_{2i−1},

and for 1 ≤ j < i ≤ m we have

u_{i}u_{j} + u_{j}u_{i} = 2 ∑_{l=0}^{j−1}(i − j + 2l + 1)v_{j−l−1}v_{i+l}
= 2(i − j + 1)v_{j−1}v_{i} + 2(i − j + 3)v_{j−2}v_{i+1} + ⋯ + 2(i + j − 1)v_{i+j−1}.

Proof. The formula for u_{i}^{2} is really just the case i = j of the second formula, after dividing both sides by 2, so we shall concentrate on the second formula assuming 1 ≤ j ≤ i ≤ m.

Recall that u_{i} = ∑ z_{α}z_{β_{1}}^{2} … z_{β_{i−1}}^{2}, where the sum is over indices that are all different and satisfy β_{1} < ⋯ < β_{i−1}. Since z_{1}, …, z_{m} anticommute, we only get non-zero contributions to u_{i}u_{j} + u_{j}u_{i} if the z_{α} terms have the same index, and then we get a linear combination of monomials of the form z_{α_{1}}^{2} … z_{α_{s}}^{2}z_{β_{1}}^{4} … z_{β_{t}}^{4} where s = i + j − 2t − 1. The coefficient of such a monomial in u_{i}u_{j} + u_{j}u_{i} is

2\frac{s!}{(i − t − 1)!(j − t − 1)!} = 2(i − t)\binom{i + j − 2t − 1}{j − t − 1}

while the coefficient in v_{j−1−l}v_{i+l} is

\frac{s!}{(i + l − t)!(j − l − t − 1)!} = \binom{i + j − 2t − 1}{j − l − t − 1}.

The proposition then follows from the identity

∑_{l≥0}(i − j + 2l + 1)\binom{i + j − 2t − 1}{j − l − t − 1} = (i − t)\binom{i + j − 2t − 1}{j − t − 1}.

This identity is obtained from Lemma 5.7 by setting ν = i + j − 2t − 1 and μ = j − t − 1. □

It follows from Proposition 5.8 that after ungrading the invariants given in (5.6), we obtain

H^{∗}(H, k) ≅ H^{∗}(B, k)^{𝔖_{m}} ≅ k⟨u_{1}, …, u_{m}, v_{2}, …, v_{m}⟩/(R)

where (R) is the set of relations given in Theorem 1.1.

6. Some Hecke algebras for other types

We consider the other infinite families of finite Coxeter groups, of type B_{n} (n ≥ 2) and D_{n} (n ≥ 4). These groups are given by W (B_{n}) = 𝔖_{2} ≀ 𝔖_{n} and W (D_{n}) = W (B_{n}) ∩ A_{2n}, where A_{2n} is the alternating group of degree 2n. There is a Hecke algebra defined for any finite Coxeter system, see for example §68 of Curtis and Reiner [3] or §8.5 of Geck and Pfeiffer [8]. The Hecke algebra of type A_{n−1} is the algebra H(n, q) considered in previous sections. The Hecke algebra of a decomposable Coxeter system is isomorphic to the tensor product of the Hecke algebras for the indecomposable factors.

For type B_{n} the Hecke algebra involves two parameters q and Q, and we write H(B_{n}, Q, q). An explicit presentation can be found in §3 of Dipper and James [5]. There is a natural inclusions of H(n, q) in H(B_{n}, Q, q) which take the elements T_{i} (1 ≤ i ≤ n − 1) to the elements T_{s_{i}} of [5], ignoring the element T_{t}.

For type D_{n}, there is just one parameter and we write H(D_{n}, q). An explicit presentation in this case can be found in the introduction to Hu [9]. Again there is a natural inclusion of H(n, q) in H(D_{n}, q) which takes the elements T_{i} (1 ≤ i ≤ n − 1) to the elements of the same name in [9], ignoring T_{0}.

First we treat type B_{n} for n ≥ 2. We set f_{n}(Q, q) = ∏_{i=1−n}^{n−1}(Q + q^{i}).

Theorem 6.1. (Dipper-James, [5], Theorem 4.17)^{1} If f_{n}(Q, q) is invertible in k then the Hecke algebra H(B_{n}, Q, q) is Morita equivalent to the algebra

∏_{j=0}^{n} H(j, q) ⊗_{k} H(n − j, q).

It may be deduced from the character theory of the Hecke algebra (see for example §5.5 and §10.3 of Geck and Pfeiffer [8], and especially the remark at the bottom of page 165) that the trivial module corresponds to the pair of partitions ([n], ∅), and is therefore a representation of the factor Morita equivalent to H(n, q) corresponding to the term j = n in the above decomposition. Therefore we have the following.

Theorem 6.2. If f_{n}(Q, q) is invertible in k then the natural inclusion H(n, q) → H(B_{n}, Q, q) induces an isomorphism

H^{∗}(H(B_{n}, Q, q), k) ≅ H^{∗}(H(n, q), k).

Remark 6.3. If Q = q is an ℓth root of unity and k is a field of characteristic zero then the invertibility condition for f_{n}(Q, q) is equivalent to the statement that ℓ is odd, so that Case 1 of Theorem 1.1 gives the structure of H^{∗}(H(B_{n}, Q, q), k) in this case.

Next, we treat type D_{n} for n ≥ 4. We set f_{n}(q) = 2 ∏_{i=1}^{n−1}(1 + q^{i}). The next theorem is implicit in Theorems 3.6 and 3.7 of Pallikaros [11] and is made explicit in Hu [9].

Theorem 6.4. If f_{n}(q) is invertible in k and n is odd, then H(D_{n}, q) is Morita equivalent to the algebra

∏_{j=(n+1)/2}^{n} H(j, q) ⊗_{k} H(n − j, q).

The corresponding theorem for n even was proved by Hu [9]:

1There is a misprint in the statement of this theorem in [5]: they write (n, n−a) but they mean (a, n−a), where their a is our j.

Theorem 6.5. If f_{n}(q) is invertible in k and n is even, then H(D_{n}, q) is Morita equivalent to the algebra

A(n/2) × ∏_{j=(n+1)/2}^{n} H(j, q) ⊗_{k} H(n − j, q),

where A(n/2) is an explicitly described algebra.

In both cases, the trivial module again corresponds to the pair of partitions ([n], ∅) (see §5.6 and §10.4 of [8]). It is therefore a representation of the factor Morita equivalent to H(n, q) corresponding to the term j = n in the decomposition. Therefore we have the following.

Theorem 6.6. If f_{n}(q) is invertible in k then the natural inclusion H(n, q) → H(D_{n}, q) induces an isomorphism

H^{∗}(H(D_{n}, q), k) ≅ H^{∗}(H(n, q), k).

Remark 6.7. Again if q is an ℓth root of unity and k is a field of characteristic zero then the invertibility condition for f_{n}(q) is equivalent to the statement that ℓ is odd, so that Case 1 of Theorem 1.1 gives the structure of H^{∗}(H(D_{n}, q), k) in this case. We do not know the answers for type B_{n} and D_{n} for ℓ even, or for the Hecke algebras of exceptional type.

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