A proof of the $q$-Foulkes conjecture for Gaussian coefficients when $a$ divides $c$

TL;DR

This paper proves a case of the $q$-Foulkes conjecture for Gaussian coefficients when the parameter $a$ divides $c$, extending known results from specific small cases to infinitely many values of $a$.

Contribution

It provides the first proof for infinitely many $a$, specifically when $a$ divides $c$, advancing the understanding of the $q$-Foulkes conjecture.

Findings

Proves the $q$-Foulkes conjecture for infinitely many $a$ when $a$ divides $c$

Includes all prime values of $a$ in the proof

Extends previous results from $a=2,3$ to a broader class of $a$

Abstract

Foulkes' conjecture has several generalisations due to Doran, Abdesselam--Chipalkatti, Bergeron, and Troyka. For the special linear Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, these assert that given $a \le c \le d \le b$ with $ab=cd$, the $\mathfrak{sl}_2(\mathbb{C})$-representation $\mathrm{Sym}^a\mathrm{Sym}^b\mathbb{C}^2$ is a subrepresentation of $\mathrm{Sym}^c\mathrm{Sym}^d\mathbb{C}^2$. We present a short proof in the case where $a$ divides $c$ or $d$, which includes all prime values of $a$. This is the first proof in this family of conjectures valid for infinitely many values of $a$; previously only the cases $a=2$ and $a=3$ were known.