Equating Schur Functions

TL;DR

This paper explores the possibility of equating all Schur functions across representations by averaging over a Gaussian measure in independent time variables, revealing a complex generating function structure.

Contribution

It introduces a novel approach to relate Schur functions through Gaussian averaging, providing new insights into their collective behavior and generating functions.

Findings

Schur functions can be made equal on average using Gaussian measures.

The generating function for Young diagrams is reproduced in a non-trivial manner.

Averages of Schur functions relate to independent time variables.

Abstract

We wonder if there is a way to make all Schur functions in all representations equal. This is impossible for fixed value of time variables, but can be achieved for averages. It appears that the corresponding measure is just Gaussian in times, which are all independent. The generating function for the number of Young diagrams does not straightforwardly appear as a product, but is reproduced in a non-trivial way.