We show that the reduction mod p of an orthogonal linear representation is orthogonal.
Tag - Representations of finite groups
This talk will be about the representations of a finite group (or a finite group scheme) G defined over a field k of positive characteristic. My plan is to explain the statement and proof of a recent result (obtained in collaboration with Dave Benson, Henning Krause, and Julia Pevtsova) to the effect that the stable module category of finite-dimensional representations of G has local Serre duality.
Modular representation theory of finite groups seeks to understand, and possibly classify, the algebras - called block algebras of finite groups - which arise as indecomposable direct factors of finite group algebras over a complete local principal ideal domain with residue field of prime characteristic p. The expectation is that 'few' algebras should arise in this way, and that this should in turn lead to significant structural connections between finite groups and their block algebras.
The key feature of block algebras of finite groups is the dichotomy of invariants attached to these algebras.
On the one hand, they have all the typical algebra-theoretic invariants - module categories, their derived categories and stable categories, as well as numerical invariants such as the numbers of isomorphism classes of simple modules, and cohomologivcal invariants such as their Hochschild cohomology.
On the other hand, they have p-local invariants, due to their provenance from group algebras - reminiscent of the local structure of a finite group which includes its Sylow p-subgroups and its associated fusion systems.
Essentially all prominent conjectures which drive modular representation theory revolve around the interplay between these two types of invariants. We describe this interplay with a focus on Hochschild cohomology and analogous cohomology rings which are defined p-locally. This involves a variety of angles - Hochschild cohomology is graded commutative, hence methods and notions from commutative algebra will play a role. Hochschild cohomology in positive degree is also a Lie algebra. We will investigate connections between the algebra structure of block algebras and the Lie algebra structure of its first Hochschild cohomology space.
We'll study the global structure of the stable module category StMod G or, equivalently, the category of singularities of representations of a finite group scheme G over a field of positive characteristic p. The goal of the lectures will be to classify the tensor ideal localizing subcategories in StMod G. The techniques involved in the classification include the theories of support and cosupport in modular representation theory, detection of projectivty for modules, Benson-Iyengar-Krause theory of local cohomology functors, and new methods inspired by commutative algebra which allow to relate local cohomology at closed and arbitrary points. This is based on joint work with Eric Friedlander and Dave Benson, Srikanth Iyengar and Henning Krause.
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