Partial sum of matrix entries of representations of the symmetric group and its asymptotics

TL;DR

This paper investigates the asymptotic behavior of partial sums of matrix entries in symmetric group representations, extending Kerov's results on characters through new limit theorems and symmetric function expansions.

Contribution

It introduces a novel analysis of partial traces and sums of representation matrices, providing central limit theorems and law of large numbers for these quantities.

Findings

Proves a central limit theorem for partial trace objects.

Establishes a law of large numbers for partial sums.

Uses symmetric function expansions on Jucys-Murphy elements.

Abstract

Many aspects of the asymptotics of Plancherel distributed partitions have been studied in the past fifty years, in particular the limit shape, the distribution of the longest rows, connections with random matrix theory and characters of the representation matrices of the symmetric group. Regarding the latter, we expand a celebrated result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose these objects into a main term and a reminder, proving a central limit theorem for both main terms and a law of large numbers for the partial sum itself. Our main tool is the expansion of symmetric functions evaluated on Jucys-Murphy elements.