Symmetrizing Tableaux and the 5th case of the Foulkes Conjecture

TL;DR

This paper verifies the 5th case of the Foulkes conjecture using computational and combinatorial methods, and provides new insights into the algebraic structure of Chow varieties.

Contribution

It confirms the Foulkes conjecture for a=5 and offers a detailed representation-theoretic decomposition of Chow varieties' ideals.

Findings

Foulkes conjecture verified for a=5

Complete decomposition of the 5th Chow variety's ideal

No degree 5 equations for the 6th Chow variety

Abstract

The Foulkes conjecture states that the multiplicities in the plethysm Sym^a(Sym^b V) are at most as large as the multiplicities in the plethysm Sym^b(Sym^a V) for all a <= b. This conjecture has been known to be true for a <= 4. The main result of this paper is its verification for a = 5. This is achieved by performing a combinatorial calculation on a computer and using a propagation theorem of Tom McKay from 2008. Moreover, we obtain a complete representation theoretic decomposition of the vanishing ideal of the 5th Chow variety in degree 5, we show that there are no degree 5 equations for the 6th Chow variety, and we also find some representation theoretic degree 6 equations for the 6th Chow variety.