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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.

A mix of human-written and AI-generated textHuman understanding: some partsMacdonald polynomialPlancherel measureSchur positivityplethysmq-Foulkes conjecturestochastic order

Nov 02, 2026PDF ↗View work →

hexagon:2610.00718v1

math.PR — Probability

Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series

Contributed by Suvrit Sra

We prove a functional extension of Gaussian operator-norm small-ball inequalities. Let $X=\sum_{i=1}^d g_iB_i$ , where $g\sim N(0,I_d)$ , the matrices $B_i\in\mathbb{R}^{d\times d}$ are symmetric, and $\sum_iB_i^2\preceq I_d$ . For every even log-concave $f\colon\mathbb{R}\to[0,\infty)$ , we obtain $\mathbb{E}\det f(X)\ge(\mathbb{E} f(\kappa G))^d$ at dilation $\kappa=13/5$ , where $G$ is standard Gaussian. The proof reduces each function to one small-ball estimate, obtained from matrix-weighted Poincaré inequalities, and controls the rest by averages over the spectrum. We give an equivalent entropy formulation. An explicit counterexample shows that no universal dilation below $\sqrt{\pi/2}$ can work; we conjecture that this value is optimal.

A mix of human-written and AI-generated textHuman understanding: all partsdeterminant inequalitylog-concave functionsmatrix Gaussian seriesrandom matricessmall-ball probabilities

Nov 02, 2026PDF ↗View work →

Matching works

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

Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series

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Matching works

The qqq-Foulkes problem II: Schur positivity with high probability

Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series

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