Kronecker Coefficients and Simultaneous Conjugacy Classes

TL;DR

This work links Kronecker coefficients to simultaneous conjugacy classes in finite groups via the Kronecker Hecke algebra $\mathrm{Kr}_d(G)$, showing $|\mathrm{Conj}_d(G)|=\sum \kappa(V_1,\dots,V_{d+1})^2$ and $|\mathrm{RConj}_d(G)|=\sum_{(V_i)} \sigma(V_1)\cdots\sigma(V_{d+1})\kappa(V_1,\dots,V_{d+1})$, with $\sigma(V)$ the Frobenius–Schur indicator. A representation-theoretic characterization ties multiplicity-free $d$-fold tensor products to commutativity of $\mathrm{Kr}_d(G)$ and the $d$-real property, yielding that doubly real groups (2-real) have MFTP, while non-Abelian odd-order groups, non-Abelian simple groups, and most general linear groups do not. The paper develops a comprehensive framework for real simultaneous conjugacy classes, including Frame-type identities and growth via $r(g)$, and classifies several families (dihedral, extraspecial 2-groups, generalized quaternion) as doubly real with MF tensor products. It also provides extensive negative results showing which groups fail to have MF tensor products, including symmetric/alternating groups, nilpotent groups, odd-order groups, non-Abelian finite simple groups, and most $GL(n,q)$, thereby delineating the landscape of multiplicity-free phenomena in finite-group tensor products.

Abstract

A Kronecker coefficient is the multiplicity of an irreducible representation of a finite group $G$ in a tensor product of irreducible representations. We define Kronecker Hecke algebras and use them as a tool to study Kronecker coefficients in finite groups. We show that the number of simultaneous conjugacy classes in a finite group $G$ is equal to the sum of squares of Kronecker coefficients, and the number of simultaneous conjugacy classes that are closed under elementwise inversion is the sum of Kronecker coefficients weighted by Frobenius-Schur indicators. We use these tools to investigate which finite groups have multiplicity-free tensor products. We introduce the class of doubly real groups, and show that they are precisely the real groups which have multiplicity-free tensor products. We show that non-Abelian groups of odd order, non-Abelian finite simple groups, and most finite general linear groups do not have multiplicity-free tensor products.

Figures (1)

• Figure 1: A summary of our findings on groups with multiplicity-free tensor products and doubly real groups.

Theorems & Definitions (72)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof