Symmetric Schur multiple zeta functions

TL;DR

This paper introduces symmetric-structure multiple zeta functions related to representation theory, explores their fundamental properties, and derives explicit formulas including pfaffian, sum, and determinant expressions, extending to quasi-symmetric functions.

Contribution

It presents the first systematic study of symmetric Schur multiple zeta functions, deriving explicit formulas and generalizations to quasi-symmetric functions.

Findings

Pfaffian expression for Schur Q-multiple zeta functions

Sum formula for Schur P- and Q-multiple zeta functions

Determinant expressions for symplectic and orthogonal Schur multiple zeta functions

Abstract

We introduce the multiple zeta functions with structures similar to those of symmetric functions such as Schur $P$-, Schur $Q$-, symplectic and orthogonal functions in the representation theory. We first consider their basic properties such as a domain of absolute convergence. And then by restricting to the truncated multiple zeta functions, we obtain the pfaffian expression of the Schur $Q$-multiple zeta functions, the sum formula for Schur $P$- and Schur $Q$-multiple zeta functions, the determinant expressions of symplectic and orthogonal Schur multiple zeta functions under an assumption on variables. Finally, we generalize those to the quasi-symmetric functions.