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Super Major Index Cyclic Sieving

Stephan Pfannerer

Abstract

Recently, Armon and Swanson introduced signed standard tableaux and a corresponding super major index that refines the classical major index. In this paper, we prove that signed standard tableaux of rectangular shape exhibit a cyclic sieving phenomenon (CSP) under the combined action of Schützenberger promotion and cyclic shift of the signs, with the sieving polynomial given by the super major index generating function. This extends Rhoades's celebrated CSP for standard Young tableaux. Furthermore, by considering Cartesian products of tableaux, we generalize this result to arbitrary non-rectangular shapes.

Super Major Index Cyclic Sieving

Abstract

Recently, Armon and Swanson introduced signed standard tableaux and a corresponding super major index that refines the classical major index. In this paper, we prove that signed standard tableaux of rectangular shape exhibit a cyclic sieving phenomenon (CSP) under the combined action of Schützenberger promotion and cyclic shift of the signs, with the sieving polynomial given by the super major index generating function. This extends Rhoades's celebrated CSP for standard Young tableaux. Furthermore, by considering Cartesian products of tableaux, we generalize this result to arbitrary non-rectangular shapes.
Paper Structure (9 sections, 16 theorems, 48 equations)

This paper contains 9 sections, 16 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Standard Young tableaux
  4. Promotion
  5. $q$-analogues
  6. The Cyclic Sieving Phenomenon
  7. Border Strip Tableaux
  8. Signed Standard Tableaux and Super Major Index
  9. Proof of the Main Theorems

Key Result

Theorem 1.1

Let $\lambda \vdash n$ be a rectangular partition. For any $0 \le k \le n$, the triple exhibits the cyclic sieving phenomenon, where the action is Schützenberger promotion on the tableau and cyclic shift on the negative set. The sieving polynomial $f^\lambda_{\pm k}(q)$ is the generating function for the super major index over signed tableaux with $k$ negative entries, and the twist $

Theorems & Definitions (31)

  • Theorem 1.1: Main Theorem A
  • Theorem 1.2: Main Theorem B
  • Example 2.1
  • Example 2.2
  • Theorem 2.3: Hook-Length Formula
  • Theorem 2.4: $q$-Hook-Length Formula
  • Definition 2.5: RSW2004
  • Theorem 2.6: RSW2004
  • Theorem 2.7: Rhoades2010
  • Theorem 2.8: APRU2021
  • ...and 21 more