{"id":"hexagon/2610.00722","pdf":"https://hexagonmath.org/pdf/2610.00722v1","url":"https://hexagonmath.org/abs/2610.00722v1","tags":["math.RT","msc:05E05","msc:20C30","msc:17B10","msc:14C05","Macdonald polynomial","Schur positivity","fusion product","plethysm","q-Foulkes conjecture"],"score":0,"title":"The $q$-Foulkes problem I: A fusion inequality for Macdonald plethysms","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.CO","summary":"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…","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Pro","Claude"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}