Hexagon

math.CO — Combinatorics

The $q$ -Foulkes problem III: On Foulkes' conjecture with a graded inner factor

Contributed by Suvrit Sra

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 cases. We also study the full conjecture under an expansion hypothesis on an explicit space of maps.

A mix of human-written and AI-generated textHuman understanding: some partsFoulkes conjectureFoulkes-Howe mapGaussian Newton expansionSchur positivitycoinvariant algebracurrent operatorplethysmq-Foulkes conjecture

Nov 02, 2026PDF ↗View work →

hexagon:2610.00722v1

math.CO — Combinatorics

The $q$ -Foulkes problem I: A fusion inequality for Macdonald plethysms

Contributed by Suvrit Sra

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 two defects we obtain a distorted version of Bergeron's $q$ -analog of Foulkes' conjecture, in which the subtracted plethysm is shifted by $q^{a(a-1)\lfloor b/2\rfloor}$ .

A mix of human-written and AI-generated textHuman understanding: some partsMacdonald polynomialSchur positivityfusion productplethysmq-Foulkes conjecture

Nov 02, 2026PDF ↗View work →

Matching works

The $q$ -Foulkes problem III: On Foulkes' conjecture with a graded inner factor

The $q$ -Foulkes problem I: A fusion inequality for Macdonald plethysms

Advanced search

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.AI — Artificial Intelligence
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory

Matching works

The qqq-Foulkes problem III: On Foulkes' conjecture with a graded inner factor

The qqq-Foulkes problem I: A fusion inequality for Macdonald plethysms

The $q$ -Foulkes problem III: On Foulkes' conjecture with a graded inner factor

The $q$ -Foulkes problem I: A fusion inequality for Macdonald plethysms