Trivial source bimodule rings for blocks and p-permutation equivalences

We associate with any p-block of a finite group a Grothendieck ring of certain p-permutation bimodules. We extend the notion of p-permutation equivalences introduced by Boltje and Xu [4] to source algebras of p-blocks of finite groups. We show that a p-permutation equivalence between two source algebras A, B of blocks with a common defect group and same local structure induces an isotypy

Blocks of minimal dimension

Any block with defect group P of a finite group G with Sylow-p-subgroup S has dimension at least |S|2/|P|; we show that a block which attains this bound is nilpotent, answering a question of G. R. Robinson

Finite generation of Hochschild cohomology of Hecke algebras of finite classical type in characteristic zero

We show that the Hochschild cohomology HH*(ℋ) of a Hecke algebra ℋ of finite classical type over a field k of characteristic zero and a non-zero parameter q in k is finitely generated, unless possibly if q has even order in k× and ℋ is of type B or D

On graded centres and block cohomology

We extend the group theoretic notions of transfer and stable elements to graded centers of triangulated categories. When applied to the center H∗Db(B)) of the derived bounded category of a block algebra B we show that the block cohomology H∗(B) is isomorphic to a quotient of a certain subalgebra of stable elements of H∗(Db(B)) by some nilpotent ideal, and that a quotient of H∗(Db(B)) by some nilpotent ideal is Noetherian over H∗(B)

Hochschild and block cohomology varieties are isomorphic

We show that the varieties of the Hochschild cohomology of a block algebra and its block cohomology are isomorphic, implying positive answers to questions of Pakianathan and Witherspoon in [16] and [17]. We obtain as a consequence that the cohomology H*(G; k) of a finite group G with coefficients in a field k of characteristic p is a quotient of the Hochschild cohomology of the principal block of kG by a nilpotent ideal

On dimensions of block algebras

Following a question by B. K¨ulshammer, we show that an inequality, due to Brauer, involving the dimension of a block algebra, has an analogue for source algebras, and use this to show that a certain case where this inequality is an equality can be characterised in terms of the structure of the source algebra, generalising a similar result on blocks of minimal dimensions. Let p be a prime and k an algebraically closed field of characteristic p. Let G be a finite group and B a block algebra of kG; that is, B is an indecomposable direct factor of kG as k-algebra. By a result of Brauer in [2], the dimension of B satisfies the inequality dimk(B) ≥ p2a−d · ℓ(B) · u2 B where pa is the order of a Sylow-p-subgroup of G, pd is the order of a defect group of B, ℓ(B) is the number of isomorphism classes of simple B-modules and uB is the unique positive integer such that pa−d · uB is the greatest common divisor of the dimensions of the simple B-modules. It is well-known that uB is prime to p. K¨ulshammer raised the question whether an equality could be expressed in terms of the structure of a source algebra of B, generalising the result in [3] on blocks of minimal dimension. We show that this is the case. The first observation is an analogue for source algebras of Brauer’s inequality. We keep the notation above and refer to [5] for block theoretic background material

The orbit space of a fusion system is contractible

Given a fusion system F on a finite p-group P, where p is a prime, we show that the partially ordered set of isomorphism classes in F of chains of non-trivial subgroups of P, considered as topological space, is contractible, further generalising Symonds’ proof [19] of a conjecture of Webb [23, 24] and its generalisation to non-trivial Brauer pairs associated with a p-block by Barker [1]

Quasi-hereditary twisted category algebras

We give a sufficient criterion for when a twisted finite category algebra over a field is quasi-hereditary, in terms of the partially ordered set of L-classes in the morphism set of the category. We show that this is a common generalisation of a long list of results in the context of EI-categories, regular monoids, Brauer algebras, Temperley–Lieb algebras, partition algebras