Asymptotic behaviour of normalized dimensions of standard and strict Young diagrams -- growth and oscillations

TL;DR

This paper investigates the asymptotic behavior of normalized dimensions of Young diagrams related to symmetric group representations, analyzing growth, oscillations, and providing refined estimates of limit constants through finite difference analysis.

Contribution

It introduces a novel finite difference approach to study growth and oscillations of normalized Young diagram dimensions, advancing understanding of their asymptotics.

Findings

Sequences with large Young diagram dimensions constructed

Evidence supporting the existence of a limit for normalized dimensions

More precise estimates of limit constants achieved

Abstract

In this paper, we present the results of a computer investigation of asymptotics for maximum dimensions of linear and projective representations of the symmetric group. This problem reduces to the investigation of standard and strict Young diagrams of maximum dimensions. We constructed some sequences for both standard and strict Young diagrams with extremely large dimensions. The conjecture that the limit of normalized dimensions exists was proposed 30 years ago [A.~M.~Vershik and S.~V.~Kerov, 1985] and has not been proved yet. We studied the growth and oscillations of the normalized dimension function in sequences of Young diagrams. Our approach is based on analyzing finite differences of their normalized dimensions. This analysis also allows us to give much more precise estimation of the limit constants.