Cyclic sieving, promotion, and representation theory

TL;DR

This paper proves several conjectures related to the cyclic sieving phenomenon in the context of promotion on rectangular tableaux, using advanced algebraic and combinatorial tools, and explores dihedral group actions and applications to hyperoctohedral groups.

Contribution

It establishes new proofs of conjectures on cyclic sieving for tableaux, extends the theory to dihedral actions, and connects to hyperoctohedral groups and noncrossing partitions.

Findings

Proved conjectures on cyclic sieving with promotion on rectangular tableaux.

Extended cyclic sieving results to dihedral group actions.

Applied theory to hyperoctohedral groups and noncrossing partitions.

Abstract

We prove a collection of conjectures of D. White \cite{WComm}, as well as some related conjectures of Abuzzahab-Korson-Li-Meyer \cite{AKLM} and of Reiner and White \cite{ReinerComm}, \cite{WComm}, regarding the cyclic sieving phenomenon of Reiner, Stanton, and White \cite{RSWCSP} as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, ..., x_{nn}]$ due to Skandera \cite{SkanNNDCB}. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups and noncrossing partitions.