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The Littlewood-Richardson rule for Schur multiple zeta functions

Hikari Hanaki

TL;DR

This work extends the Littlewood-Richardson framework to Schur multiple zeta functions by refining the product formula: instead of symmetrizing over all variables, the authors restrict to a subgroup determined by the body of the shape, and express the product as a sum over LR coefficients $c_{\mu\nu}^{\lambda}$ times transformed zeta-functions. Central to the approach are Knuth equivalence and jeu de taquin, which link permutations of variables to LR combinatorics via Rect maps and LR coefficients. The main results include a skew-shape LR-type expansion and a specialization that yields a product formula for Schur multiple zeta functions, with a further Winged type generalization that handles more complex diagram shapes. These contributions deepen the combinatorial toolkit for SMZFs and connect multivariable zeta structures with classical tableau theory, offering sharper expansions and potential applications in algebraic combinatorics and number theory.

Abstract

The Schur multiple zeta function was defined as a multivariable function by Nakasuji-Phuksuwan-Yamasaki. Inspired by the product formula of Schur functions, the products of Schur multiple zeta functions have been studied. While the product of two Schur functions expands as a linear combination of Schur functions, it is known that a similar expansion for the product of Schur multiple zeta functions can be obtained by symmetrizing, i.e., by taking the summation over all permutations of the variables. In this paper, we present a more refined formula by restricting the summation from the full symmetric group to its specific subgroup.

The Littlewood-Richardson rule for Schur multiple zeta functions

TL;DR

This work extends the Littlewood-Richardson framework to Schur multiple zeta functions by refining the product formula: instead of symmetrizing over all variables, the authors restrict to a subgroup determined by the body of the shape, and express the product as a sum over LR coefficients times transformed zeta-functions. Central to the approach are Knuth equivalence and jeu de taquin, which link permutations of variables to LR combinatorics via Rect maps and LR coefficients. The main results include a skew-shape LR-type expansion and a specialization that yields a product formula for Schur multiple zeta functions, with a further Winged type generalization that handles more complex diagram shapes. These contributions deepen the combinatorial toolkit for SMZFs and connect multivariable zeta structures with classical tableau theory, offering sharper expansions and potential applications in algebraic combinatorics and number theory.

Abstract

The Schur multiple zeta function was defined as a multivariable function by Nakasuji-Phuksuwan-Yamasaki. Inspired by the product formula of Schur functions, the products of Schur multiple zeta functions have been studied. While the product of two Schur functions expands as a linear combination of Schur functions, it is known that a similar expansion for the product of Schur multiple zeta functions can be obtained by symmetrizing, i.e., by taking the summation over all permutations of the variables. In this paper, we present a more refined formula by restricting the summation from the full symmetric group to its specific subgroup.
Paper Structure (9 sections, 13 theorems, 46 equations, 3 figures)

This paper contains 9 sections, 13 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Combinatorial background
  3. Knuth equivalence
  4. Jeu de Taquin
  5. Proof of main Theorems
  6. Preparation
  7. Proof of Theorem \ref{['skew']}
  8. Proof of Theorem \ref{['main']}
  9. Winged type

Key Result

Theorem 1.1

Let $\mu, \nu$ be partitions, and let $\mu', \nu'$ be the conjugates of $\mu, \nu$, respectively. Let $\bm{s}=(s_{ij})\in T(\mu, \mathbb{C}), \bm{t}=(t_{ij})\in T(\nu,\mathbb{C})$, and assume that the entries are variables. Assume that the real parts of the variables $s_{i1}$ with $\mu'_2+\nu'_1\le Here $\sum_{{\rm Sym}(B(\bm{s} \ast \bm{t}))}$ is the symmetric group permuting all the variables e

Figures (3)

  • Figure 1: Example of body $\bm{v}$
  • Figure 2: Elementary slides
  • Figure 3: Example of slide

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1: Knuth relation
  • Theorem 2.2: cf. F§2
  • Theorem 2.3: cf.F§2
  • Theorem 2.4: cf.F§5
  • Theorem 3.1: $\tilde{\mathbb{N}}$ analogue of Theorem \ref{['inv']}
  • Theorem 3.2: $\tilde{\mathbb{N}}$ analogue of Theorem \ref{['rect_word']}
  • ...and 23 more