The Littlewood-Richardson rule for Schur $P$-, $Q$-multiple zeta functions

TL;DR

This paper extends the Littlewood-Richardson rule to Schur P- and Q-multiple zeta functions, providing new expansion formulas involving symmetric group summations and subgroup restrictions.

Contribution

It introduces a novel Littlewood-Richardson type expansion formula for Schur P- and Q-multiple zeta functions, generalizing classical product expansions.

Findings

Derived a linear expansion formula for Schur P-multiple zeta functions.

Established a skew Schur Q-multiple zeta function expansion via symmetric group summation.

Refined the expansion by restricting to specific subgroups of the symmetric group.

Abstract

The Schur $P$-, $Q$-multiple zeta functions were defined by Nakasuji and Takeda inspired by the tableau representation of Schur $P$-, $Q$-functions. While a product of two Schur $P$-functions expands as a linear combination of Schur $P$-functions, we obtain a similar expansion formula for the Schur $P$-multiple zeta functions by taking summation over the symmetric group permutating all the variables. We also introduce a expansion formula of skew Schur $Q$-multiple zeta functions by taking summation over the symmetric group. Furthermore, this skew type formula can be refined by restricting the symmetric group to its specific subgroup.