Combinatorial proof for a stability property of plethysm coefficients

Abstract

Plethysm coefficients are important structural constants in the representation the- ory of the symmetric groups and general linear groups. Remarkably, some sequences of plethysm coefficients stabilize (they are ultimately constants). In this paper we give a new proof of such a stability property, proved by Brion with geometric representation theory techniques. Our new proof is purely combinatorial: we decompose plethysm coefficients as a alternating sum of terms counting integer points in poly- topes, and exhibit bijections between these sets of integer points.