Hochschild cohomology of symmetric groups and generating functions, II

We relate the generating functions of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups with those of blocks of the symmetric groups. We show that the first Hochschild cohomology of a positive defect block of a symmetric group is nonzero, answering in the affirmative a question of the third author. To do this, we prove a formula expressing the dimension of degree one Hochschild cohomology as a sum of dimensions of centres of blocks of smaller symmetric groups. This in turn is a consequence of a general formula that makes more precise a theorem of our previous paper describing the generating functions for the dimensions of Hochschild cohomology of symmetric groups.


Introduction
Let p be a prime number and k a field of characteristic p.As a consequence of results in [5], using the classification of finite simple groups, if G is a finite group of order divisible by p, then HH 1 (kG) is non-zero.It is an open question [6,Question 7.4] whether for G a finite group and B a block of kG, if the defect groups of B are non-trivial, then HH 1 (B) is non-zero.This question has been shown to have a positive answer in some cases in [8] and [7], for instance.We prove that this question has an affirmative answer if G is a symmetric group S n on n letters.Theorem 1.1.Let B a block of kS n with non trivial defect groups.Then HH 1 (B) = 0.
In fact, we give a precise formula for the dimension of the first Hochschild cohomology of a block of a symmetric group as a sum of dimensions of the centres of blocks of smaller symmetric groups (Theorem 1.2), and this easily implies Theorem 1.1.
In order to state the formula, let us recall that to each block B of kS n is associated a nonzero integer w ⌊n/p⌋ called the weight of B, with the property that the Sylow p-subgroups of S pw are defect groups of B under the natural inclusion S pw S n .In particular, B has non-trivial defect groups if and only if w > 0.Moreover, by Theorem 7.2 of Chuang and Rouquier [3] if B and B ′ are blocks of possibly different symmetric groups, with the same weight, then B and B ′ are derived equivalent algebras, and consequently, for any r 0, we have dim k HH r (B) = dim k HH r (B ′ ).
For w 0, denote by B pw the principal block of kS pw .Then B pw has weight w and by the above, dim k HH r (B) = dim k HH r (B pw ) for any weight w block B of a symmetric group algebra.Thus Theorem 1.1 is a consequence of the following result.For w 0, let ρ(pw, ∅) equal the number of partitions of pw with empty p-core.Theorem 1.2.Let B be a weight w-block of a symmetric group algebra over k.If p = 2, then The proof of the above formula goes through the following theorem relating generating functions of dimensions of Hochschild cohomology of blocks of symmetric groups with those of the entire group algebra and then invoking the results of our previous paper [1].Denote by p(n) the number of partitions of n, and by P (t) the generating function 1, there exists a rational function φ(t) (depending on p and r) with φ(0) non-zero, such that and Remark 1.4.In [1], we proved that where R p,r (t) is a rational function of t.Ken Ono asked us whether R p,r (t) is a rational function of t p , and Theorem 1.3 proves that this is the case, with R p,r (t) = t p φ(t p ).
Remark 1.5.The constant coefficient of φ(t) in Theorem 1.3 is equal to ).An easy calculation, using the centraliser decomposition, shows that for r 1 we have y 1 = 2 if r ≡ 0 or −1 modulo 2(p − 1) (which is in particular the case if p = 2) and y 1 = 1 otherwise.
Remark 1.6.We note that the above results do not depend on the choice of the field k.If k ′ is an extension field of k and B a block of kS n for some positive integer n, then B ′ = k ′ ⊗ k B is a block of k ′ S n having the same defect groups as B, and for any finite-dimensional k-algebra A we have a graded k-algebra isomorphism

Proofs.
We begin with an elementary lemma.
Lemma 2.1.Let R be an integral domain and m a positive integer.If ) is a rational function, then h(t) is also a rational function.

t) .
If h(t) = 0, then there is nothing to prove.So, we assume that h(t) = 0. Write Comparing coefficients of powers of t, the equality for each s, 0 s m − 1. Choose s such that a s (t) = 0. Then h(t) = a s (t)/b s (t) is a rational function of t.
Proof of Theorem 1.3.Let r 1.For n 0, let c(n) denote the number of p-core partitions of n and for each s, 0 ) and y pw = dim k HH r (B pw ).We use the notation from the statement of Theorem 1.3 and we set Note that by [3,Theorem 7.2], y pw is the dimension of the degree r Hochschild cohomology of any weight w block of a symmetric group algebra.Further, note that by the Nakayama conjecture, proved by Brauer [2] and Robinson [4], the number of weight w blocks of kS n equals c(n−pw).Since the Hochschild cohomology of a finite dimensional algebra is the direct sum of the Hochschild cohomologies of its blocks, we have that for any s with 0 s p − 1, and any n 0, In other words, for any s, 0 s p − 1, On the other hand, we have y 0 = dim k HH r (B 0 ) = 0 and y 1 = dim k HH r (B p ) = 0 (cf.Remark 1.5).In particular, Y (t) is divisible by t but not by t 2 in Z [[t]].So we may define a power series in t, with non-zero constant term, and then we have (2.4) where we have used (2.3) for the second equality and (2.4) for the third equality.On the other hand, by (2.2), Now by Theorem 1.3 of [1] we have that t p φ(t p ) is a rational function of t.It then follows from Lemma 2.1 that φ(t) is a rational function of t, and by the above, φ(0) is non-zero.
Proof of Theorem 1.2.By Theorem 1.2 of [1], t p 1 − t p P (t) p 3. Note that in [1] the base field is taken to be F p , but the above result clearly holds as well for any field of characteristic p (cf. Remark 1.6).By Theorem 1.3 and its notation, applied with r = 1, of Theorem 1.2.The remaining equations follow from the Nakayama Conjecture, proved in [2,4], which states that characters of a symmetric group belong to the same p-block if and only if the corresponding partitions have the same p-core, implying in particular that dim k Z(B pj ) = ρ(pj, ∅).
Theorem 1.1 is an immediate consequence of Theorem 1.2.
and dim k HH r (kS pn+s ) = n w=0 y pw c(p(n − w) + s).