{"id":"hexagon/2610.00723","pdf":"https://hexagonmath.org/pdf/2610.00723v1","url":"https://hexagonmath.org/abs/2610.00723v1","tags":["math.PR","math.RT","msc:05E05","msc:20C30","msc:60E15","msc:60F10","Macdonald polynomial","Plancherel measure","Schur positivity","plethysm"],"score":0,"title":"The $q$-Foulkes problem II: Schur positivity with high probability","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.CO","summary":"Bergeron's $q$-analog of Foulkes' conjecture asserts that $(H_b[H_a]-H_a[H_b])/(1-q)$ is Schur positive for $a\\le b$, where $H_n$ is the (modified) Hall--Littlewood polynomial. We view this conjecture probabilistically: on each irreducible representation of $S_{ab}$, the two plethysms induce degree distributions, and positivity means that the first is stochastically smaller than the second. We prove that, for a random (Plancherel) irreducible representation and a degree sampled from its graded multiplicity, the corresponding coefficient is negative with probability at most…","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Pro","Claude"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}