Proofs and generalizations of a homomesy conjecture of Propp and Roby

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.