The Broué invariant of a p-permutation equivalence.

Robert Boltje

University of California Santa Cruz

**Abstract:** Let G and H be finite groups and let B and C be p-block algebras of G and H, respectively. In a landmark paper, Broué defined the notion of a perfect isometry I between B and C as a bijection between their irreducible characters with signs satisfying certain arithmetic conditions. He proved that the ratio of the codegrees of corresponding irreducible characters (including the sign) leads to a nonzero element β(I) in the field with p elements which is independent of the irreducible characters. We call β(I) the Broué invariant of I and show that if I comes from a p-permutation equivalence or a splendid Rickard equivalence between A and B then - up to a sign - it is independent of the equivalence and explicitly determined by local invariants of B and C.

Higher limits over the fusion orbit category

Ergün Yalçın

Bilkent University

**Abstract:** An homology decomposition of a discrete group is sharp if certain higher limits
vanish. For the subgroup decomposition these higher limits are either over the orbit category
or over the fusion orbit category of a discrete group. I will introduce these categories and discuss
how the higher limits over an orbit category can be calculated. At the end, I will state some results
for the vanishing of higher limits over the fusion orbit category of a discrete group.

Monomial posets and their Lefschetz invariants

Hatice Mutlu Akatürk

University of California Santa Cruz

**Abstract:** The Euler-Poincaré characteristic of a given poset X is defined as the alternating sum of
the orders of the sets of chains Sd_{n}(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
Sd_{n}(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
B_{A}(G). We will also introduce Lefschetz invariants of A-monomial G-posets, which are
elements of B_{A}(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.

Transfer in the Cohomology of Categories Using Biset Functors

Peter Webb

University of Minnesota

**Abstract:** The cohomology of categories includes as a special case the theory of group cohomology. Defining a transfer on the cohomology of categories is problematic: most attempts to do this in related contexts require induction and restriction functors to be adjoint on both sides, which typically does not happen with categories. We get round this by defining category cohomology as a functor on a biset category where the objects are small categories, extending the usual theory of biset functors when the objects are groups. After summarizing this theory, we show how to make the definition of category cohomology as a biset functor.

Fusion Systems and Brauer Indecomposability of Scott Modules

İpek Tuvay

Mimar Sinan Güzel Sanatlar Üniversitesi

**Abstract:** The Brauer indecomposability of Scott modules is important for obtaining splendid Morita equivalences between the principal blocks of two finite groups whose fusion systems on their Sylow p-subgroups are isomorphic. In this talk, first the connection between the Brauer indecomposability of a p-permutation module and the saturation of the corresponding fusion system will be discussed. Then, recent results along these lines for some special families of 2-groups, including semidihedral and wreathed 2-groups, will be presented. Part of this work is joint with S. Koshitani.

A Functorial Resolution of Units of Burnside Rings

Serge Bouc

Université de Picardie Jules Verne

**Abstract:** Most of the structural properties - prime spectrum, species, idempotents, ... - of the Burnside ring of a finite group have been precisely described a few years after its introduction in 1967. An important missing item in this list is its group of units. After a - non exhaustive - review of this subject, I will present some recent results on the functorial aspects of this group..