The $q$-Foulkes problem I: A fusion inequality for Macdonald plethysms
We prove that the plethysm $H_m[\widetilde H_\mu]$ of a Macdonald polynomial by the Hall–Littlewood polynomial $H_m$ lies above the Macdonald polynomial $\widetilde H_{m\mu}$ obtained by fusing $m$ copies of $\widetilde H_\mu$. More specifically, the quotient $(H_m[\widetilde H_\mu]-\widetilde H_{m\mu})/(1-q^m)$ is Schur positive with coefficients in $\mathbb N[q,t]$, and it is the character of an explicit module that we call the fusion defect. For one-row shapes the inequality places both $q$-Foulkes plethysms $H_b[H_a]$ and $H_a[H_b]$ above the same polynomial $H_{ab}$, and comparing their…
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