Proofs and generalizations of a homomesy conjecture of Propp and Roby

TL;DR

This paper proves a conjecture that certain statistics on semistandard Young tableaux are homomesic under promotion actions, and extends the results to more general posets and related tableau classes.

Contribution

It proves the homomesy conjecture for semistandard Young tableaux and generalizes it to cominuscule posets, also exploring related tableau promotion scenarios.

Findings

Proved the homomesy conjecture for SSYT under promotion.

Extended homomesy results to cominuscule posets.

Discussed partial results for K-promotion on tableaux with increasing rows and columns.

Abstract

Let $G$ be a group acting on a set $X$ of combinatorial objects, with finite orbits, and consider a statistic $\xi : X \to \mathbb{C}$. Propp and Roby defined the triple $(X, G, \xi)$ to be \emph{homomesic} if for any orbits $\mathcal{O}_1, \mathcal{O}_2$, the average value of the statistic $\xi$ is the same, that is \[\frac{1}{{|\mathcal{O}_1|}}\sum_{x \in \mathcal{O}_1} \xi(x) = \frac{1}{|\mathcal{O}_2|}\sum_{y \in \mathcal{O}_2} \xi(y).\] In 2013 Propp and Roby conjectured the following instance of homomesy. Let $\mathrm{SSYT}_k(m \times n)$ denote the set of semistandard Young tableaux of shape $m \times n$ with entries bounded by $k$. Let $S$ be any set of boxes in the $m \times n$ rectangle fixed under $180^\circ$ rotation. For $T \in \mathrm{SSYT}_k(m \times n)$, define $\sigma_S(T)$ to be the sum of the entries of $T$ in the boxes of $S$. Let $\langle \mathcal{P} \rangle$ be a cyclic group of order $k$ where $\mathcal{P}$ acts on $\mathrm{SSYT}_k(m \times n)$ by promotion. Then $(\mathrm{SSYT}_k(m \times n), \langle \mathcal{P} \rangle, \sigma_S)$ is homomesic. We prove this conjecture, as well as a generalization to cominuscule posets. We also discuss analogous questions for tableaux with strictly increasing rows and columns under the K-promotion of Thomas and Yong, and prove limited results in that direction.