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

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