We give a local method for calculating modulo \(p\) congruence classes of relative multiplicities between some of the local pointed groups associated with a maximal Brauer pair on a group algebra. We prove invariance under the associated fusion system and, in some cases, we deduce that a biset associated with the source algebra satisfies a divisibility condition. Some of this work is joint with Matthew Gelvin.

Trivial source modules of a finite group G over a field F of characteristic p > 0 are direct summands of finite-dimensional permutation FG-modules. Their isomorphism classes form a monoid under the direct sum and tensor product operations. The associated Grothendieck ring is called the trivial source ring, T(FG). We give a description of the trivial source ring as a subring of a product of character rings. This allows us to characterize the group of orthogonal units (or equivalently, torsion units) of T(FG) as a direct product of two subgroups. The first factor is the unit group of the Burnside ring of the p-fusion system of G, and the second factor consists of ‘coherent’ linear characters on normalizers of p-subgroups. The main motivation of studying the torsion unit group of T(FG) is its connection with the group of p-permutation self-equivalences of FG. This is joint work with Robert Carman.

By Yoshida's theorem, a Mackey functor is cohomological precisely if (when viewed as an additive functor on spans of G-sets) it factors through the category of permutation modules. Now let us categorify: Mackey 2-functors are an analogue of Mackey functors, taking values in additive categories rather than abelian groups; examples abound throughout equivariant mathematics. They can be viewed as being additive 2-functors defined on a certain bicategory of "Mackey 2-motives", built out of spans. In this talk, I will argue for what the correct notion of a "cohomological" Mackey 2-functor should be. Typical exemples arise in linear representation theory, such as abelian, derived or stable categories of group representations, but there are also examples from geometry. The analogue of Yoshida's theorem holds: a Mackey 2-functor is cohomological precisely if (when viewed as an additive 2-functor on Mackey 2-motives) it factors through the bicategory of right-free permutation bimodules. This involves "linearising" spans to bisets and bimodules, and yields a surprisingly concrete connection between the abstract Mackey 2-motives and the usual blocks of group algebras. This is joint work with Paul Balmer.

In 2010, Claire Amiot conjectured that algebraic 2-Calabi-Yau categories with cluster-tilting object must come from quivers with potential. This would extend a structure theorem obtained with Idun Reiten in the case where the endomorphism algebra of the cluster-tilting object is hereditary. Many other classes of examples are also known. We will report on recent progress in the general case.

Mackey functors encode in a categorical fashion the way that many constructions on groups arise, such as sending a finite group to its cohomology group. They have been studied in depth for finite groups, and the results obtained have been useful in group representation theory in particular. In this talk, we will discuss an attempt to a possible generalisation of Mackey functors to totally disconnected locally compact groups, with focus on the profinite groups. This is joint work in progress with Ilaria Castellano and Brita Nucinkis.

The Euler-Poincaré characteristic of a given poset X is defined as the alternating sum of the orders of the sets of chains Sdn (X) with cardinality n + 1 over the natural numbers n. Given a finite gorup G, Thévenaz extended this definition to G-posets and defined the Lefschetz invariant of a G-poset X as the alternating sum of the G-sets of chains Sdn (X) with cardinality n+1 over the natural numbers n which is an element of Burnside ring B(G). Let A be an abelian group. We will introduce the notions of A-monomial G-posets and A-monomial G-sets, and state some of their categorical properties. The category of A-monomial G-sets gives a new description of the A-monomial Burnside ring BA (G). We will also introduce Lefschetz invariants of A-monomial G-posets, which are elements of BA (G). An application of the Lefschetz invariants of A-monomial G-posets is the A-monomial tensor induction. Another application is a work in progress that aims to give a reformulation of the canonical induction formula for ordinary characters via A-monomial G-posets and their Lefschetz invariants. For this reformulation we will introduce A-monomial G-simplicial complexes and utilize the smooth G-manifolds and complex G-equivariant line bundles on them.

A quasi-hereditary algebra is a pair consisting of an algebra and a partial order on the isomorphism classes of its simple modules satisfying conditions coming from the representation theory of semisimple complex Lie algebras. By a famous result of Peter Webb, we know that the category of biset functors over a field of characteristic zero has a similar behavior and we will see how this can be used to study the quasi-hereditary property of the double Burnside algebra. We should remark that the definition of quasi-hereditary algebra involves the choice of a partial order, hence an algebra is in general not `canonically' quasi-hereditary. We will see in a simple example that in general there is a large number of such partial orders.

The cohomological Mackey functors for a finite group are modules over a given finitely generated algebra, called the cohomological Mackey algebra. This algebra shares many properties with the usual group algebra, and most questions about modules over the group algebra and methods used for them can be extended to Mackey functors: e.g. relative projectivity, vertex and source theory, Green correspondence, the central role played by the elementary abelian \(p\)-groups.

In this talk, based on my first joint work with Serge, I will give a presentation of the basic properties of cohomological Mackey functors, and explain the structure of the graded algebra of self extensions of some simple such functors.

An informal Talk

We consider a gluing problem for a destriction functor defined on subquotients of a finite group as a generalization of gluing problems that were considered by Bouc and Thévenaz for the Dade group functor. We develop an obstruction theory for this general gluing problem and apply it to some well-known p-biset functors, such as the Burnside ring functor, the rational representation ring functor, and the Dade group functor at odd primes. We recover the results due to Bouc and Thévenaz for the torsion part of the Dade group and improve them for the Dade group functor.

This is a joint work with Olçay Coskun.

Let \(k\) be an algebraically closed field of positive characteristic \(p\) and let \(\mathbb{F}\) be an algebraically closed field of characteristic \(0\). In this talk we introduce diagonal \(p\)-permutation functors: consider the \(\mathbb{F}\)-linear category \(\mathbb{F}pp_k^{\Delta}\) of finite groups, in which the set of morphisms from \(G\) to \(H\) is the \(\mathbb{F}\)-linear extension of the Grothendieck group of \(p\)-permutation \((kH, kG)\)-bimodules with (twisted) diagonal vertices. We call the \(\mathbb{F}\)-linear functors from \(\mathbb{F}pp_k^{\Delta}\) to \(\mathbb{F}\)-\(\mathrm{Mod}\) as diagonal \(p\)-permutation functors.

We first consider the diagonal \(p\)-permutation functor of the \(p\)-permutation ring. We then show that the category of diagonal \(p\)-permutation functors is semisimple and give a description of the evaluations of simple functors at finite groups. Finally, we introduce the diagonal \(p\)-permutation functors arising, in a natural way, from the blocks of finite groups and show that \(p\)-permutation equivalent blocks give rise to isomorphic functors.

This is joint work with Serge Bouc.

Biset functors may be defined on a biset category where the objects are all finite categories, not just all finite groups. Key examples of such biset functors are functors related to the Burnside ring of a category and the representation ring of a category. Many properties of these functors for categories are formally the same as they are for functors defined on groups, such as the construction of simple functors and their restriction to subcategories. Other properties, such as the general parametrization of simple functors in a convenient way, are more delicate. In this case the problem is brought about by the fact that non-isomorphic categories may become isomorphic in the biset category, and that we lack a useful factorization of bisets of the kind that exists for groups. I will survey the properties of these biset functors for categories, pointing out similarities and differences with the group situation.