Multidimensional Toggle Dynamics

Abstract

J. Propp and T. Roby isolated a phenomenon in which a statistic on a set has the same average value over any orbit as its global average, naming it homomesy. One set they investigated was order ideals of partially ordered sets (posets). They proved that the cardinality statistic on order ideals of the product of two chains poset under rowmotion or promotion exhibits homomesy. We prove an analogous result in the case of the product of three chains where one chain has two elements. In order to prove this result, we generalize from two to n dimensions the recombination technique that D. Einstein and Propp developed to study homomesy. We see that our main homomesy result does not fully generalize to an arbitrary product of three chains, nor to larger products of chains; however, we have a partial generalization to an arbitrary product of three chains. Additional corollaries include refined homomesy results in the product of three chains and a new result on increasing tableaux. We also generalize recombination to any ranked poset and from this, obtain a homomesy result for a type B minuscule poset cross a two-element chain. We conclude by extending the definition of promotion to infinite posets, exploring homomesy, recombination, and a connection to monomial ideals.