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Conjectures on the reduced Kronecker coefficients

Tao Gui

TL;DR

The paper develops a program to generalize log-concavity phenomena from Littlewood--Richardson coefficients to the symmetric group setting via reduced Kronecker coefficients. It defines and studies the stable (reduced) tensor product, formulates a central conjecture that $\bar{g}_{(\lambda+\mu)/2\,(\lambda+\mu)/2}^{\nu} \ge \bar{g}_{\lambda\mu}^{\nu}$ when $(\lambda+\mu)/2$ is a partition, and recasts this into a concavity statement in the Grothendieck ring of $\operatorname{Rep}(S_{\infty})$. The authors prove the conjecture in special cases (two-row and one-column shapes) and show dimension-level consequences (a Schur-concavity-type inequality) while connecting to broader frameworks such as Schur positivity, Deligne categories, and the partition algebra. They also provide a suite of related conjectures (Gconj-2, Gconj-3, Gconj-4) and discuss computational evidence and potential geometric/combinatorial implications, including an interpretation via intersection cohomology and stability phenomena.

Abstract

We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of Okounkov's conjecture on the log-concavity of the Littlewood--Richardson coefficients and the Schur log-concavity theorem of Lam--Postnikov--Pylyavskyy. We prove our conjectures in some special cases and discuss some implications of these conjectures.

Conjectures on the reduced Kronecker coefficients

TL;DR

The paper develops a program to generalize log-concavity phenomena from Littlewood--Richardson coefficients to the symmetric group setting via reduced Kronecker coefficients. It defines and studies the stable (reduced) tensor product, formulates a central conjecture that when is a partition, and recasts this into a concavity statement in the Grothendieck ring of . The authors prove the conjecture in special cases (two-row and one-column shapes) and show dimension-level consequences (a Schur-concavity-type inequality) while connecting to broader frameworks such as Schur positivity, Deligne categories, and the partition algebra. They also provide a suite of related conjectures (Gconj-2, Gconj-3, Gconj-4) and discuss computational evidence and potential geometric/combinatorial implications, including an interpretation via intersection cohomology and stability phenomena.

Abstract

We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of Okounkov's conjecture on the log-concavity of the Littlewood--Richardson coefficients and the Schur log-concavity theorem of Lam--Postnikov--Pylyavskyy. We prove our conjectures in some special cases and discuss some implications of these conjectures.
Paper Structure (5 sections, 8 theorems, 39 equations)

This paper contains 5 sections, 8 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Okounkov conjecture on the Littlewood--Richardson coefficients
  3. Kronecker Coefficients and Their Convexity Property
  4. Reduced Kronecker Coefficients
  5. Log-concavity Conjectures of Stable Tensor Product of Irreducible Representations of Symmetric Groups

Key Result

Theorem 2

(chindris2007counterexamples) Let $n \geqslant 1$ be an integer and let $\lambda(n), \mu(n)$ be two partitions defined by Then Consequently, when $n \geqslant 21$, Conjecture Oconj fails for $\lambda_1=2\lambda(n)$, $\mu_1=\nu_1=2\mu(n)$, $\lambda_2=\lambda(n)$, $\mu_2=\nu_2=\mu(n)$, $\lambda_3=\mu_3=\nu_3=0$.

Theorems & Definitions (17)

  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Conjecture 6
  • Theorem 7
  • Conjecture 8
  • Conjecture 9: Restatement of Conjecture \ref{['Gconj-1']}
  • Theorem 10
  • ...and 7 more