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

Foulkes’ conjecture has several generalisations due to Doran, Abdesselam–Chipalkatti, Bergeron, and Troyka. For the special linear Lie algebra ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$, these assert that given $a\le c\le d\le b$ with $ab=cd$, the ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$-representation ${\mathrm{S}\mathrm{y}\mathrm{m}}^a{\mathrm{S}\mathrm{y}\mathrm{m}}^b{\mathbb{C}}^2$ is a subrepresentation of ${\mathrm{S}\mathrm{y}\mathrm{m}}^c{\mathrm{S}\mathrm{y}\mathrm{m}}^d{\mathbb{C}}^2$. 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.