We determine the Morita equivalence classes of 2-blocks with quaternion defect groups of arbitrary 2-power order, thereby completing the proof of Donovan's conjecture for blocks of tame representation type.

We show that the p-part of the conductor of a generalised character of a finite group is equal to the conductor of its generalised decomposition numbers. We use this to show that p-parts of conductors of irreducible characters are preserved under isotypies and perfect isometries that arise in the context of stable equivalences of Morita type with endopermutation source. We apply this to blocks with abelian defect and Frobenius inertial quotient.

In a previous paper, we stated and motivated counting conjectures for fusion systems that are purely local analogues of several local-to-global conjectures in the modular representation theory of finite groups. Here we verify some of these conjectures for fusion systems on an extraspecial group of order p^3, which contain among them the Ruiz-Viruel exotic fusion systems at the prime 7. As a byproduct we verify Robinson's ordinary weight conjecture for principal p-blocks of almost simple groups G realizing such (nonconstrained) fusion systems.

We give some criteria for the Lie algebra HH^1(B) to be solvable, where B is a p-block of a finite group algebra, in terms of the action of an inertial quotient of B on a defect group of B.

We show that dualising transfer maps in Hochschild cohomology of symmetric algebras over complete discrete valuations rings commutes with Tate duality. This is analogous to a similar result for Tate cohomology of symmetric algebras over fields. We interpret both results in the broader context of Calabi-Yau triangulated categories.

In this paper, we show that a conjecture of Giannelli on the degrees of height zero characters holds for blocks with a cyclic or Klein four defect group.

For G a finite group, k a field of prime characteristic p, and S a Sylow p-subgroup of G, the Sylow permutation module Ind^G_S(k) plays a role in diverse facets of representation theory and group theory, ranging from Alperin's weight conjecture to statistical considerations of S-S-double cosets in G. The Sylow permutation module breaks up along the block decomposition of the group algebra kG, but the resulting block components are not invariant under splendid Morita equivalences. We introduce a summand of the block component, which we call source permutation module, which is shown to be invariant under such equivalences. We investigate general structural properties of the source permutation module and we show that well-known results relating the self-injectivity of the endomorphism algebra of the Sylow permutation to Alperin's weight conjecture carry over to the source permutation module. We calculate this module in various cases, such as certain blocks with cyclic defect group and blocks with a Klein four defect group, and for some blocks of symmetric groups, prompted by a question in a recent paper by Diaconis-Giannelli-Guralnick-Law-Navarro-Sambale-Spink on the self-injectivity of the endomorphism algebra of the Sylow permutation module for symmetric groups.

We give some criteria for the Lie algebra to be solvable, where is a ‐block of a finite group algebra, in terms of the action of an inertial quotient of on a defect group of .

We extend a result due to Kawai on block varieties for blocks with abelian defect groups to blocks with arbitrary defect groups. This partially answers a question by J. Rickard.

The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial t -structures, and in particular, full abelian subcategories need not arise as hearts of a t -structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.

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.

We compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the Hochschild cohomology of symmetric groups.

We show that Thompson’s $$A\times B$$ A × B -lemma can be obtained as a consequence of Brauer’s third main theorem.

We show that the Green correspondence induces an injective group homomorphism from the linear source Picard group $\mathcal{L}(B)$ of a block $B$ of a finite group algebra to the linear source Picard group $\mathcal{L}(C)$, where $C$ is the Brauer correspondent of $B$. This homomorphism maps the trivial source Picard group $\mathcal{T}(B)$ to the trivial source Picard group $\mathcal{T}(C)$. We show further that the endopermutation source Picard group $\mathcal{E}(B)$ is bounded in terms of the defect groups of $B$ and that when $B$ has a normal defect group $\mathcal{E}(B)=\mathcal{L}(B)$. Finally we prove that the rank of any invertible $B$-bimodule is bounded by that of $B$.

We rule out a certain nine-dimensional algebra over an algebraically closed field to be the basic algebra of a block of a finite group, thereby completing the classification of basic algebras of dimension at most 12 of blocks of finite group algebras.

We give a simple algebraic recipe for calculating the components of the BV operator $\Delta$ on the Hochschild cohomology of a finite group algebra with respect to the centraliser decomposition. We use this to investigate the properties of $\Delta$ and to make some computations for some particular finite groups.

G. Navarro raised the question under what circumstancs two vertices of two indecomposable modules over a finite group algebra generate a Sylow $p$-subgroup. The present note provides a sufficient criterion for when this is the case. This generalises a result by Navarro for simple modules over finite $p$-solvable groups, which is the main motivation for this note.

The purpose of this note is to provide a reference for the fact that the strong Frobenius number, in the sense of Eaton and Livesey, of a block of a finite group with a cyclic defect group is equal to one. This answers a question of Farrell and Kessar.

We show that the subgroup of the Picard group of a p-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over a complete discrete valuation ring O of characteristic zero with an algebraic closure k of Fp as residue field is a colimit of finite Picard groups of blocks over p-adic subrings of O. We apply the results to blocks with an abelian defect group and Frobenius inertial quotient, and specialise this further to blocks with cyclic or Klein four defect groups.

Let A A be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext \operatorname {Ext} -quiver of A A is a simple directed graph, then H H 1 ( A ) H\!H^1(A) is a solvable Lie algebra. The second main result shows that if the Ext \operatorname {Ext} -quiver of A A has no loops and at most two parallel arrows in any direction, and if H H 1 ( A ) H\!H^1(A) is a simple Lie algebra, then char ⁡ ( k ) ≠ \operatorname {char}(k)\neq 2 2 and H H 1 ( A ) ≅ H\!H^1(A)\cong s l 2 ⁡ ( k ) \operatorname {\mathfrak {sl}}_2(k) . The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.