Promotion and cyclic sieving on families of SSYT

TL;DR

This paper investigates the cyclic sieving phenomenon in families of semistandard Young tableaux under promotion, including stretched hook shapes, skew rectangles, and ribbons, introducing new methods for charge computation.

Contribution

It extends cyclic sieving results to new families of tableaux and introduces a novel charge computation method for skew tableaux with uniform element frequency.

Findings

Cyclic sieving observed in stretched hook shapes using cocharge polynomial.

Cyclic sieving confirmed for skew shapes of rectangles with charge polynomial.

Promotion on skew ribbons exhibits a bicyclic sieving phenomenon.

Abstract

We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family we consider consists of skew shapes, consisting of rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier result by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occur with the same frequency.