Simple fusion systems and the Solomon 2-local groups

We introduce a notion of simple fusion systems which imitates the corresponding notion for finite groups and show that the fusion system on the Sylow-2-subgroup of a 7-dimensional spinor group over a field of characteristic 3 considered by Ron Solomon [18] and by Ran Levi and Bob Oliver [11] is simple in this sense

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 Automorphisms and Focal Subgroups of Blocks

Given a p-block B of a finite group with defect group P and fusion system on P, we show that the rank of the group is invariant under stable equivalences of Morita type. The main ingredients are the construction, due to Broué and Puig, a theorem of Weiss on linear source modules, arguments of Hertweck and Kimmerle applying Weiss’ theorem to blocks, and connections with integrable derivations in the Hochschild cohomology of block algebras

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)

Fusion category algebras

The fusion system F on a defect group P of a block b of a finite group G over a suitable p-adic ring O does not in general determine the number l(b) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class α of the constant functor with value k× on the orbit category F¯c of F-centric subgroups Q of P of b which “glues together” the second cohomology classes α(Q) of AutF¯(Q) with values in k× in K¨ulshammer-Puig [13, 1.8]. We show that if α exists, there is a canonical quasi-hereditary k-algebra F¯(b) such that Alperin’s weight conjecture becomes equivalent to the equality l(b) = l(F¯(b)). By work of Broto, Levi, Oliver [3], the existence of a classifying space of the block b is equivalent to the existence of a certain extension category L of Fc by the center functor Z. If both invariants α, L exist we show that there is an O-algebra L(b) associated with b having F¯(b) as quotient such that Alperin’s weight conjecture becomes again equivalent to the equality l(b) = l(L(b)); furthermore, if b has an abelian defect group, L(b) is isomorphic to a source algebra of the Brauer correspondent of 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