Entropy of Schur-Weyl Measures

TL;DR

This paper proves a conjecture that the normalized dimensions of isotypic components in tensor representations of symmetric groups converge to a constant as the ratio of N to sqrt(n) approaches a limit.

Contribution

It establishes the convergence of normalized isotypic component dimensions in tensor representations, confirming Olshanski's conjecture.

Findings

Normalized dimensions converge to a constant in the specified limit.

The result extends understanding of asymptotic behavior of symmetric group representations.

Provides a rigorous proof of a previously conjectured measure behavior.

Abstract

Relative dimensions of isotypic components of N-th order tensor representations of the symmetric group on n letters give a Plancherel-type measure on the space of Young diagrams with n cells and at most N rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel-type measures in the limit when N/sqrt{n} converges to a constant. The main result of the paper is the proof of this conjecture.