# Functional Analysis, Linear Structures and Applications Seminar

### The asymptotic behaviour of the Super-Plancherel measure

Let $\mathbb{F}_q$ be a finite field and denote $U_n(\mathbb{F}_q)$ the group of $n \times n$ uppertriangular matrices over $\mathbb{F}_q$ with only ones in the diagonal. In recent years the representation theory of $U_n(\mathbb{F}_q)$ has been approached via certain Supercharacter-Theories, not only due to their (non-commutative) combinatorial-analogues to the representation theory of the symmetric group $S_n$, but also as a useful tool to address Harmonic-analysis problems.

We consider a particular Supercharacter-Theory for $U_n(\mathbb{F}_q)$ which yields a natural measure on the set-partitions of $\{1,...,n \}$: the Super-plancherel measure $\textbf{SPl}_n$. The aim of this talk is to understand the asymptotic behaviour of $\textbf{SPl}_n$ as $n \rightarrow \infty$; in particular limit objects are interpreted in a representation-theoretical setting.

Current organizers: Helena Mascarenhas, Ângela Mestre.