Schur function averages for the real Ginibre ensemble

TL;DR

This paper derives a simple explicit formula for the expectations of all Schur functions in the real Ginibre ensemble, enabling the calculation of averages of analytic series of elementary symmetric functions.

Contribution

It provides a novel explicit formula for Schur function expectations in the real Ginibre ensemble, facilitating broader analytic computations.

Findings

Explicit formula for Schur function expectations

Positive integer values for even partition entries

Zero expectations for odd partition entries

Abstract

We derive an explicit simple formula for expectations of all Schur functions in the real Ginibre ensemble. It is a positive integer for all entries of the partition even and zero otherwise. The result can be used to determine the average of any analytic series of elementary symmetric functions by Schur function expansion.