Kronecker comultiplication of stable characters and restriction from Smn to Sm × Sn

Abstract

A family of symmetric functions s˜λ was introduced in [11], and independently in [1]. The s˜λ 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 Δ⁎ are multiplicities for the restriction of irreducible representations from Smn to Sm×Sn (provided m and n are sufficiently large), and use the structure of s˜λ to demonstrate two-row stability properties of these restriction multiplicities.