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

Hikari Hanaki

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.

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

Abstract

The Schur -, -multiple zeta functions were defined by Nakasuji and Takeda inspired by the tableau representation of Schur -, -functions. While a product of two Schur -functions expands as a linear combination of Schur -functions, we obtain a similar expansion formula for the Schur -multiple zeta functions by taking summation over the symmetric group permutating all the variables. We also introduce a expansion formula of skew Schur -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.
Paper Structure (1 section, 2 theorems, 11 equations, 1 figure)

This paper contains 1 section, 2 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction

Key Result

Theorem 1.1

Let $\mu, \nu$ be strict partitions. Let $\bm{s}=(s_{ij})\in ST(\mu, \mathbb{C}), \bm{t}=(t_{ij})\in ST(\nu,\mathbb{C})$ be variables. Assume that the real parts of all variables $\{s_{ij}\}, \{t_{ij}\}$ are greater than 1. For $\lambda \in \mathcal{G}^P(\mu,\nu)$, let $\bm{u}_{\lambda}(\bm{s}, \bm{ where the $\sum_{{\rm Sym}(\bm{s}, \bm{t})}$ is the summation over the symmetric group permuting al

Figures (1)

  • Figure 1: Example of diagram of skew strict partition: $\lambda/\mu = (6, 3, 1)/(3, 1)$

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3