Permutation modules and endotrivial complexes

Sam Miller (UCSC)
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PDL C-38

Title pre-seminar: Tensor-triangular geometry in modular representation theory

Abstract pre-seminar: To quote Everett C. Dade, "There are just too many modules over $p$-groups!" Indeed, the question of classifying indecomposable modular representations of finite groups is a seemingly impossible task. This suggests that instead, we should attempt to classify modules up to some notion of equivalence. Tensor-triangular geometry provides a framework to do this, by instead proposing that we classify all the thick tensor ideals of the stable module category stmod(kG). In this talk, I will give an overview of these topics, introducing tensor-triangular geometry with a focus on its application to modular representation theory. 

 

Title seminar:Permutation modules and endotrivial complexes

Abstract seminar: Let $G$ be a finite group and $k$ a field of characteristic $p > 0$. The recent work of Balmer and Gallauer has illuminated much about the bounded homotopy category of $p$-permutation modules, $K^b(p\operatorname{-perm}(kG))$, a tensor-triangulated category that is fundamentally linked with Broue's abelian defect group conjecture, in addition to structures in algebraic geometry and algebraic topology. The spectrum of this category is controlled by the Balmer spectrum of derived module categories of "$p$-local subgroups" via modular fixed-points functors - an example of "local-to-global" behavior in modular representation theory. In this talk, I will motivate this category, discuss its geometry, and describe the classification of invertible objects, which I call endotrivial complexes. If time permits, I will discuss ongoing work joint with Balmer and Gallauer which seeks to give a geometric description of the Picard group.

 

 
Event Type
Seminars
Event Subcalendar
Algebra and Algebraic Geometry Seminar