Promotion and evacuation on standard Young tableaux of rectangle and staircase shape

TL;DR

This paper explores promotion and evacuation bijections on standard Young tableaux of rectangular and staircase shapes, demonstrating a new embedding that preserves these operations and relates to the cyclic sieving phenomenon.

Contribution

It introduces a promotion- and evacuation-preserving embedding of staircase-shaped tableaux into rectangular-shaped tableaux, aiding the understanding of cyclic sieving.

Findings

Embedding of SYT(staircase shape) into SYT(rectangular shape) preserves promotion and evacuation.

Promotion on rectangular tableaux exhibits the cyclic sieving phenomenon.

The embedding helps analyze promotion actions on staircase-shaped tableaux.

Abstract

(Dual-)promotion and (dual-)evacuation are bijections on SYT(\lambda) for any partition \lambda. Let c^r denote the rectangular partition (c,...,c) of height r, and let sc_k (k > 2) denote the staircase partition (k,k-1,...,1). B. Rhoades showed representation-theoretically that promotion on SYT(c^r) exhibits the cyclic sieving phenomenon (CSP). In this paper, we demonstrate a promotion- and evacuation-preserving embedding of SYT(sc_k) into SYT(k^{k+1}). This arose from an attempt to demonstrate the CSP of promotion action on SYT(sc_k).