{"id":"hexagon/2610.00724","pdf":"https://hexagonmath.org/pdf/2610.00724v1","url":"https://hexagonmath.org/abs/2610.00724v1","tags":["math.RT","math.AC","msc:05E05","msc:20C30","msc:14C05","msc:15A15","Foulkes conjecture","Foulkes-Howe map","Gaussian Newton expansion","Schur positivity"],"score":0,"title":"The $q$-Foulkes problem III: On Foulkes' conjecture with a graded inner factor","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.CO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro","Claude"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}