Two stability theorems on plethysms of Schur functions

TL;DR

This paper proves two stability theorems for plethysm coefficients of Schur functions, showing their behavior under certain partition operations, using combinatorial methods involving plethystic tableaux.

Contribution

It introduces two new stability theorems for plethysm coefficients, generalizing existing results and employing combinatorial proofs with plethystic tableaux.

Findings

Stability of plethysm coefficients under partition addition and joining.

Generalization of all known plethysm stability results.

Combinatorial proofs using plethystic semistandard tableaux.

Abstract

The plethysm product of Schur functions corresponds to composing polynomial representations of infinite general linear groups. Finding the plethysm coefficients $\langle s_\nu \circ s_\mu, s_\lambda\rangle$ that express an arbitrary plethysm $s_\nu \circ s_\mu$ as a sum $\sum_\lambda \langle s_\nu \circ s_\mu, s_\lambda \rangle s_\lambda$ of Schur functions is a fundamental open problem in algebraic combinatorics. We prove two stability theorems for plethysm coefficients under the operations of adding and/or joining an arbitrary partition to either $\mu$ or $\nu$. In both theorems $\mu$ may be replaced with an arbitrary skew partition. As special cases we obtain all stability results on the plethysm product of two Schur functions in the literature to date. The proofs are entirely combinatorial using plethystic semistandard tableaux with positive and negative entries.