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Cyclic sieving for a class of rectangular domino tableaux

Laura Colmenarejo, Bridget Eileen Tenner, Camryn E. Thompson

Abstract

The cyclic sieving phenomenon (CSP) provides valuable data about symmetry classes of cyclic actions, and has applications to representation theory. In this paper, we enumerate domino tableaux of shape 2-by-n, and use this result to prove a new CSP on these objects. We then enumerate the rectangular domino tableaux of any dimensions, and conjecture a more general CSP on rectangular domino tableaux. As a consequence of the enumerative results, we obtain several identities involving Fibonacci and Catalan numbers.

Cyclic sieving for a class of rectangular domino tableaux

Abstract

The cyclic sieving phenomenon (CSP) provides valuable data about symmetry classes of cyclic actions, and has applications to representation theory. In this paper, we enumerate domino tableaux of shape 2-by-n, and use this result to prove a new CSP on these objects. We then enumerate the rectangular domino tableaux of any dimensions, and conjecture a more general CSP on rectangular domino tableaux. As a consequence of the enumerative results, we obtain several identities involving Fibonacci and Catalan numbers.
Paper Structure (12 sections, 22 theorems, 37 equations)

This paper contains 12 sections, 22 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Domino tableaux
  4. Examples of cyclic sieving
  5. Enumerating $\mathrm{DT}(n^2)$
  6. New identities involving Fibonacci and Catalan numbers
  7. Cyclic sieving on $\mathrm{DT}(n^2)$
  8. The cyclic sieving function
  9. The cyclic sieving action
  10. The CSP on $\mathrm{DT}(n^2)$
  11. Enumerating domino tableaux of shape $(n^m)$
  12. Open problems

Key Result

Theorem 2.8

The map $\Gamma$ is a bijection.

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4: DTSYT
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8: DTSYT
  • Definition 2.9
  • Example 2.10
  • ...and 43 more