The $q$-Foulkes problem III: On Foulkes' conjecture with a graded inner factor
We show that surjectivity of the canonical map $\mathrm{Sym}^b(\mathrm{Sym}^aV)\to\mathrm{Sym}^a(\mathrm{Sym}^bV)$ survives grading of the inner factor. Consequently $h_b[H_a]-H_a[h_b]$ is Schur positive whenever the canonical map is surjective, where $H_a$ is the (modified) Hall--Littlewood polynomial; the same holds for (modified) Macdonald polynomials. We prove this surjectivity for $b\ge(a-1)^2$, improving the known bound $a(a-1)$. For Bergeron's $q$-analog of Foulkes' conjecture that lets both factors be graded, we prove nonnegativity of every Schur coefficient for certain special…
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