Random partitions and asymptotic theory of symmetric groups, Hecke algebras and finite Chevalley groups

TL;DR

This thesis studies the asymptotic behavior of random partitions linked to the representation theory of symmetric groups and finite Chevalley groups, establishing laws of large numbers and central limit theorems.

Contribution

It introduces new techniques for analyzing random partitions using polynomial functions and cumulants, and generalizes the algebraic structures involved.

Findings

Proves laws of large numbers for q-Plancherel measures

Establishes central limit theorems for various measures

Connects polynomial functions on Young diagrams to algebraic structures

Abstract

In this thesis, we investigate the asymptotics of random partitions chosen according to probability measures coming from the representation theory of the symmetric groups $S_n$ and of the finite Chevalley groups $GL(n,F_q)$ and $Sp(2n,F_q)$. More precisely, we prove laws of large numbers and central limit theorems for the $q$-Plancherel measures of type A and B, the Schur-Weyl measures and the Gelfand measures. Using the RSK algorithm, it also gives results on longest increasing subsequences in random words. We develop a technique of moments (and cumulants) for random partitions, thereby using the polynomial functions on Young diagrams in the sense of Kerov and Olshanski. The algebra of polynomial functions, or observables of Young diagrams is isomorphic to the algebra of partial permutations; in the last part of the thesis, we try to generalize this beautiful construction.