Eigenvalue multiplicities of group elements in irreducible representations of simple linear algebraic groups

Let $k$ be an algebraicallly closed field of characteristic $p\geq 0$ and let $G$ be a linear algebraic group of rank $\ell\geq 1$ over $k$. Let $V$ be a rational finite-dimensional $kG$-module and let $V_g(\mu)$ denote the eigenspace corresponding to the eigenvalue $\mu\in k^*$ of $g \in G$ on $V$. We set $\nu_G(V)=\min\{\dim(V)-\dim(V_g(\mu))| g \in G \setminus Z(G), \mu \in k^*\}$. In this talk we will identify pairs $(G,V)$ of simple simply connected linear algebraic groups and of rational irreducible tensor-indecomposable $kG$-modules with the property that $\nu_G(V)\leq \sqrt{\dim(V)}$. This problem is an extension of the classification result obtained by Guralnick and Saxl for $\nu_G(V)\leq \max\{2,\frac{\sqrt{\dim(V)}}{2}\}$. One motivation for studying such problems is to identify subgroups of linear algebraic groups based on element behaviour. 

Date and Venue

Start Date
Online seminar
End Date


Ana Retegan

Speaker's Institution

Ecole Polytechnique Fédérale de Lausanne



Algebra, Combinatorics and Number Theory