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.
Categories: math.CO