Quadratic relations for ninth variations of Schur functions and application to Schur multiple zeta functions

TL;DR

This paper develops algebraic relations for Macdonald's ninth variation of Schur functions, extending classical identities, and applies these results to analyze Schur multiple zeta functions and their special values.

Contribution

It introduces new algebraic relations for the ninth variation of Schur functions using determinant formulas, generalizing classical Schur function identities.

Findings

Derived algebraic relations for ninth variation Schur functions

Connected these relations to properties of Schur multiple zeta functions

Analyzed special values for rectangular shapes in Schur multiple zeta functions

Abstract

Macdonald's ninth variation of Schur functions is a broad generalization of the classical Schur function and its variants, defined via the Jacobi-Trudi determinant formula. In this paper, we establish various algebraic relations for $S^{(r)}_{\lambda/\mu}(X)$, a class of the ninth variation introduced by Nakagawa, Noumi, Shirakawa, and Yamada, by combining the Jacobi-Trudi formula with determinant formulas such as the Desnanot-Jacobi adjoint matrix theorem and the Pl\"ucker relations, which generalize the corresponding relations for Schur functions. As an application, we investigate algebraic relations for "diagonally constant" Schur multiple zeta functions and examine their specific special values when the shape is rectangular.