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 \[\frac{1}{{|\mathcal{O}_1|}}\sum_{x \in \mathcal{O}_1} ξ(x) = \frac{1}{|\mathcal{O}_2|}\sum_{y \in \mathcal{O}_2} ξ(y).\] 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 $⟨\mathcal{P}⟩$ be a cyclic group of order $k$ where $\mathcal{P}$ acts on ${\mathrm{SSYT}}_k(m\times n)$ by promotion. Then $({\mathrm{SSYT}}_k(m\times n),⟨\mathcal{P}⟩,{\sigma }_S)$ 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.

19 Pages

References in corpus (1)

• Cyclic Sieving of Increasing Tableaux and small Schröder Paths

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