math.CO — Combinatorics

The $q$-Foulkes problem II: Schur positivity with high probability

Contributed by Suvrit Sra

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 $C\exp(-c(ab)^{1/3})$, uniformly in $a\le b$. Consequently, the $q$-Foulkes conjecture holds after deleting a fraction at most $C\exp(-c(ab)^{1/3})$ of each of the two graded modules. Additionally, we show that exact conditional means and variances yield complete positivity for explicit width and height families.