Stability Properties Of The Plethysm: A Combinatorial Approach

TL;DR

This paper introduces a purely combinatorial approach to prove the stability of plethysm coefficients in symmetric functions, using polytope counting and bijections to establish eventual constancy.

Contribution

The paper presents a new combinatorial method to prove stability properties of plethysm coefficients, avoiding geometric and vertex operator techniques.

Findings

Plethysm coefficients can be decomposed into counts of integer points in polytopes.

Stability of these coefficients is established through bijections between integer point sets.

The approach provides a purely combinatorial proof of known stability results.

Abstract

An important family of structural constants in the theory of symmetric functions and in the representation theory of symmetric groups and general linear groups are the plethysm coefficients. In 1950, Foulkes observed that they have some stability properties: certain sequences of plethysm coefficients are eventually constant. Such stability properties were proven by Brion with geometric techniques, and by Thibon and Carr\'e by means of vertex operators. In this paper we present a new approach to prove such stability properties.Our proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients related to the complete homogeneous basis of symmetric functions. We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope.