### Local representation theory: a topological perspective

Algebra and Geometry Seminar

20th February 2019, 2:30 pm – 3:30 pm

Howard House, 4th Floor Seminar Room

Let $p$ be a prime number. A saturated fusion system on a finite $p$-group $S$ is a category whose objects are the subgroups of $S$ where morphisms are injective homomorphisms satisfying certain axioms. A finite group $G$ containing $S$ as a Sylow $p$-subgroup provides an example, denoted $\mathcal{F}_S(G)$. A local-global conjecture in modular representation theory typically expresses an equality between some global invariant of $G$ (such as the number of representations with a particular property) and some local integer invariant associated to $\mathcal{F}_S(G)$. A modern attack tends to involve a reduction to the case $G$ is simple, and then a case by case check using the classification of finite simple groups. In an attempt to give a different perspective we forget the group $G$, and consider the behaviour of these integer invariants when $\mathcal{F}$ is an *arbitrary* saturated fusion system. Do they behave like the local-global conjectures predict they should when $\mathcal{F}$ is induced by a group? We will state some conjectures, discuss recent progress and look at some examples coming from algebraic topology.

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