Representation stability and finite linear groups

Representation stability and finite linear groups

We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes– Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with t...

Representation theory of finite groups II: Linear algebra

Representation Theory for Finite Groups

We cover some of the foundational results of representation theory including Maschke’s Theorem, Schur’s Lemma, and the Schur Orthogonality Relations. We consider character theory, constructions of representations, and conjugacy classes. Finally, we touch upon Pontryagin Duality for abelian groups and Young Tableaux for the Symmetric Group.

Representation Theory of Finite Groups

1.1. Let G be a finite group. We write the multiplication in G using either concatenation or · (so gg′ = g · g′) and we will denote the unit in G by 1. Throughout these notes, C denotes any algebraically closed field of characteristic 0. Of course, it’s fine to have the complex numbers in mind, but there’s no problem with using Q or anything else. In particular, the use of analytic properties o...

Representation Theory of Finite Groups