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.

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.

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.

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.