Quasi-hereditary structure of twisted split category algebras revisited

Let $k$ be a field of characteristic $0$, let $\mathsf{C}$ be a finite split category, let $\alpha$ be a 2-cocycle of $\mathsf{C}$ with values in the multiplicative group of $k$, and consider the resulting twisted category algebra $A:=k_\alpha\mathsf{C}$. Several interesting algebras arise that way, for instance, the Brauer algebra. Moreover, the category of biset functors over $k$ is equivalent to a module category over a condensed algebra $\varepsilon A\varepsilon$, for an idempotent $\varepsilon$ of $A$. In [2] the authors proved that $A$ is quasi-hereditary (with respect to an explicit partial order $\le$ on the set of irreducible modules), and standard modules were given explicitly. Here, we improve the partial order $\le$ by introducing a coarser order $\unlhd$ leading to the same results on $A$, but which allows to pass the quasi-heredity result to the condensed algebra $\varepsilon A\varepsilon$ describing biset functors, thereby giving a different proof of a quasi-heredity result of Webb, see [26]. The new partial order $\unlhd$ has not been considered before, even in the special cases, and we evaluate it explicitly for the case of biset functors and the Brauer algebra. It also puts further restrictions on the possible composition factors of standard modules.Comment: 39 page

A ghost algebra of the double Burnside algebra in characteristic zero

For a finite group $G$, we introduce a multiplication on the \QQ-vector space with basis \scrS_{G\times G}, the set of subgroups of $G\times G$. The resulting \QQ-algebra \Atilde can be considered as a ghost algebra for the double Burnside ring $B(G,G)$ in the sense that the mark homomorphism from $B(G,G)$ to \Atilde is a ring homomorphism. Our approach interprets \QQ B(G,G) as an algebra $eAe$, where $A$ is a twisted monoid algebra and $e$ is an idempotent in $A$. The monoid underlying the algebra $A$ is again equal to \scrS_{G\times G} with multiplication given by composition of relations (when a subgroup of $G\times G$ is interpreted as a relation between $G$ and $G$). The algebras $A$ and \Atilde are isomorphic via M\"obius inversion in the poset \scrS_{G\times G}. As an application we improve results by Bouc on the parametrization of simple modules of \QQ B(G,G) and also of simple biset functors, by using results by Linckelmann and Stolorz on the parametrization of simple modules of finite category algebras. Finally, in the case where $G$ is a cyclic group of order $n$, we give an explicit isomorphism between \QQ B(G,G) and a direct product of matrix rings over group algebras of the automorphism groups of cyclic groups of order $k$, where $k$ divides $n$.Comment: 41 pages. Changed title from "Ghost algebras of double Burnside algebras via Schur functors" and other minor changes. Final versio

Alperin's weight conjecture in terms of equivariant Bredon cohomology

Alperin’s weight conjecture [1] admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system [7] of a fusion system of a block, which in turn can be interpreted as the equivariant Bredon cohomology of a certain functor on the G-poset of centric Brauer pairs. The underlying general constructions of categories and functors needed for this reformulation are described in §1 and §2, respectively, and §3 provides a tool for computing the cohomology of the functors arising in §2. Taking as starting point the alternating sum formulation of Alperin’s weight conjecture by Knörr-Robinson [10], the material of the previous sections is applied in §4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors, using further work of Robinson [16, 17, 18]

On stable equivalences and blocks with one simple module

Using a stable equivalence due to Rouquier, we show that Alperin’s weight conjecture holds for any p-block of a finite group with defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules