Kronecker Comultiplication of Stable Characters and Restriction From $S_{mn}$ to $S_m \times S_n$

TL;DR

This paper explores the relationship between symmetric functions and representation theory, demonstrating how certain structure constants relate to restriction multiplicities of symmetric group representations and revealing stability properties.

Contribution

It establishes a connection between Kronecker comultiplication structure constants and restriction multiplicities, and shows two-row stability properties using symmetric functions.

Findings

Structure constants for $ ilde{s}_ u$ encode restriction multiplicities.

Restriction multiplicities stabilize for large $m,n$.

Demonstrates two-row stability in restriction multiplicities.

Abstract

A family of symmetric functions $\tilde{s}_\lambda$ was introduced in [OZ], and independently in [AS]. The $\tilde{s}_\lambda$ encode many stability properties of representations of symmetric groups (e.g. when multiplied, the structure constants are reduced Kronecker coefficients). We show that the structure constants for the Kronecker comultiplication $\Delta^*$ are multiplicities for the restriction of irreducible representations from $S_{mn}$ to $S_m \times S_n$ (provided $m$ and $n$ are sufficiently large), and use the structure of $\tilde{s}_\lambda$ to demonstrate two-row stability properties of these restriction multiplicities.