Statistics for random representations of Lie algebras

Abstract: How does a typical finite-dimensional representation of a complex Lie algebra look like? In this talk, we address this question for the infinite family $\mathfrak{sl}_{r+1}(\mathbb{C})$ with $r \geq 2$. Specifically, we derive the asymptotic statistical properties of a representation sampled uniformly from all representations with a given large dimension. This naturally extends similar studies on integer partitions with methods inspired from statistical mechanics. The multivariable generalization we consider contains some new features compared to integer partitions, which have been studied from a large range of points of view from combinatorics to modularity. In the talk, we will review those aspects familiar from integer partitions, describe the physics inspired approach to the problem, and finally detail our results for the general case. This is joint work with Walter Bridges and Kathrin Bringmann.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.