Some numerical and algorithmical probelms in the asymptotic representation theory

TL;DR

This paper explores computational and algorithmic challenges in asymptotic representation theory, focusing on Young diagrams, statistical estimates, and limit shapes of diagrams under specific distributions.

Contribution

It provides experimental results on statistical values and limit shapes of Young diagrams, advancing computational methods in asymptotic representation theory.

Findings

Estimated maximum and average dimensions of irreducible representations.

Presented computed limit shapes of 2D and 3D Young diagrams.

Analyzed statistical properties under Plancherel and Richardson distributions.

Abstract

The article presents the results of experiments in computation of statistical values related to Young diagrams, including the estimates on maximum and average (by Plancherel distribution) dimension of irreducible representation of symmetric group $S_n$. The computed limit shapes of two-dimensional and three-dimensional diagrams distributed by Richardson statistics are presented as well.