A ring close to finite representation type
Noyan Fevzi Er
Dokuz Eylül University
Abstract: This is a rehash of a previous talk aiming to interest enthusiasts of mathematics in a long standing question situated at a junction of various subfields of algebra. With its origins dating back to the work of Koethe (1930s) and Cohen-Kaplansky (1950s), the question has tapped a rather large variety of techniques, thus engaging mathematicians wielding them. The talk begins with systems of linear equations.
Theories of Action
David Austin Pierce
Abstract: As an illustration of some model-theoretic concepts, we consider a complete (logical) theory of mathematical actions. It is the so-called model-completion of the theory of actions, and the model-companion of the theory of group actions, while not itself being a theory of group actions. It shares certain properties with the theory of algebraically closed fields with distinguished generic multiplicative subgroup. Meanwhile, we do not forget mathematics itself as an (ethical) action.
Two new concepts in finite group theory
Middle East Technical University
Abstract: Let G be a group acted on by a group A by automorphisms. The nature of this action is very restrictive and hence informative about the structure of G. We have been carrying on research in this area, especially on length type problems, in several collaborated works over the years.
The action is said to be coprime if G and A have coprime orders. The existence of nice conditions which are valid in this case made it almost traditional to assume that the action is coprime. After many attacks to a longstanding noncoprime conjecture we have recently introduced the concept of a good action of A on G in a joint work with Güloğlu and Jabara. We say the action is “good” if H=[H, B]CH(B) for every subgroup B of A and for every B-invariant subgroup H of G. It can be regarded as a generalization of the coprime action due to the fact that every coprime action is good. However there are noncoprime actions which are good. It is expected that this concept may help to understand the real difficulties in studying a noncoprime action.
On the other hand Khukhro, Makarenko and Shumyatsky studied the case where A is a Frobenius group and obtained very nice results showing the dependence of certain invariants of the group G on the corresponding invariants of CG(H) where H is a Frobenius complement of A. During some efforts to understand the real relation between this assumption on A and its conclusions, in a joint work with Güloğlu we have defined the concept of a Frobenius-like group. A finite group A is said to be Frobenius-like if it contains a nontrivial nilpotent normal subgroup F, which is called the kernel of A; and a nontrivial complement H to F in A, which is called the complement in A such that [F, h]=F for all nonidentity elements h of H. Every Frobenius group is a Frobenius-like group and Frobenius-like groups seem to be a very significant generalization, because they are much more probable to be encountered in practice.
With this talk I aim to present a survey of our results extending several coprime results to good action case, and studying the action of Frobenius-like groups.
Birational equivalences and Kac-Moody algebras
İstanbul Teknik Üniversitesi
Abstract: In this talk, I am going to talk about what would be a good notion of birational equivalence for noncommutative varieties. As an application, I will look into Kac-Moody algebras from the birational equivalence point of view.
Diagonal p-permutation functors
Université de Picardie Jules Verne
Abstract: Let p be an algebraically closed field of positive characteristic p and let 𝔽 be an algebraically closed field of characteristic 0. In this talk we introduce diagonal p-permutation functors: consider the 𝔽-linear category 𝔽ppkΔ of finite groups, in which the set of morphisms from G to H is the 𝔽-linear extension of the Grothendieck group of p-permutation (kH, kG)-bimodules with (twisted) diagonal vertices. We call the 𝔽-linear functors from 𝔽ppkΔ to 𝔽-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.
Semisimplicity of some deformations of the subgroup category and the biset category
Laurence John Barker
Abstract: Working over a field of characteristic zero, the biset category is a linear category whose objects are the finite groups and whose morphisms G ← H have basis elements corresponding to conjugacy classes of subgroups of G × H. Its composition formula resembles the composition formula for the subgroup category, whose morphisms have basis elements corresponding to the subgroups of G × H. The biset category is not contained in the subcategory. But we shall generalize, realizing both of those categories as particular cases of some deformations, and we shall find that each deformed biset category is contained in the corresponding deformed subgroup category. We shall show that, for almost all values of the deformation parameter, the two deformed categories have a semisimplicity property. One motive is with a view to a future harnessing of the semisimplicity, treating the biset category as if it were semisimple. This is joint work with İsmail Alperen Öğüt.
The Brou\'e invariant of a $p$-permutation equivalence.
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, Brou\'e 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 $\beta(I)$ in the field with $p$ elements which is independent of the irreducible characters. We call $\beta(I)$ the Brou\'e 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
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 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.
Transfer in the Cohomology of Categories Using Biset Functors
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
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
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..