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. They proved that the cardinality statistic on order ideals of the product of two chains poset under rowmotion exhibits homomesy. In this paper, 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 conclude with a generalization of recombination to any ranked poset and a homomesy result for the Type B minuscule poset cross a two element chain.
Highlights
Homomesy is a surprisingly ubiquitous phenomenon, isolated by J
Rowmotion on an order ideal is defined as the order ideal generated by the minimal poset elements that are not in the order ideal; this action has generated significant interest in recent the electronic journal of combinatorics 26(4) (2019), #P4.30 algebraic combinatorics, giving rise to many beautiful results [2, 4, 7, 10, 17]
In Propositions 38 and 39, we show that our homomesy result does not generalize to order ideals of [a]×[b]× [c] or order ideals of [2]×⋅ ⋅ ⋅×[2] under promotion with cardinality statistic
Summary
Homomesy is a surprisingly ubiquitous phenomenon, isolated by J. Theorem 21, says that the order ideals of [2] × [b] × [c] exhibit homomesy with average value bc under promotion when using the cardinality statistic To prove this theorem, we generalize the recombination result of Einstein and Propp from a product of chains [a]×[b] to a product of chains [a1]×⋅ ⋅ ⋅×[an] in our second main theorem, Theorem 25. In Theorem 56, we generalize the recombination result of Theorem 25 from a product of chains to any ranked poset We use this for Corollary 57, a homomesy result on order ideals of a type B minuscule poset cross a chain of size two under promotion with cardinality statistic.
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