74 research outputs found

    The toggle group, homomesy, and the Razumov-Stroganov correspondence

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    The Razumov-Stroganov correspondence, an important link between statistical physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello, relates the ground state eigenvector of the O(1) dense loop model on a semi-infinite cylinder to a refined enumeration of fully-packed loops, which are in bijection with alternating sign matrices. This paper reformulates a key component of this proof in terms of posets, the toggle group, and homomesy, and proves two new homomesy results on general posets which we hope will have broader implications.Comment: 14 pages, 10 figures, final versio

    Combinatorial, piecewise-linear, and birational homomesy for products of two chains

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    This article illustrates the dynamical concept of homomesyhomomesy in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and shows the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley, and then lifted to birational operations on the positive orthant in R∣P∣\mathbb{R}^{|P|} and indeed to a dense subset of C∣P∣\mathbb{C}^{|P|}. When the poset PP is a product of a chain of length aa and a chain of length bb, these lifted operations have order a+ba+b, and exhibit the homomesy phenomenon: the time-averages of various quantities are the same in all orbits. One important tool is a concrete realization of the conjugacy between rowmotion and promotion found by Striker and Williams; this recombinationrecombination mapmap allows us to use homomesy for promotion to deduce homomesy for rowmotion. NOTE: An earlier draft showed that Stanley's transfer map between the order polytope and the chain polytope arises as the tropicalization of an analogous map in the bilinear realm; in 2020 we removed this material for the sake of brevity, especially after Joseph and Roby generalized our proof to the noncommutative realm (see arXiv:1909.09658v3). Readers who nonetheless wish to see our proof can find the September 2018 draft of this preprint through the arXiv

    Promotion and Rowmotion

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    We present an equivariant bijection between two actions--promotion and rowmotion--on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and recent work of D. Armstrong, C. Stump, and H. Thomas on root posets and noncrossing partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Finally, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.Comment: 25 pages, 22 figures; final versio

    Orbits of Plane Partitions of Exceptional Lie Type

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    For each minuscule flag variety XX, there is a corresponding minuscule poset, describing its Schubert decomposition. We study an action on plane partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass (1995). For plane partitions of height at most 22, D. Rush and X. Shi (2013) proved an instance of the cyclic sieving phenomenon, completely describing the orbit structure of this action. They noted their result does not extend to greater heights in general; however, when XX is one of the two minuscule flag varieties of exceptional Lie type EE, they conjectured explicit instances of cyclic sieving for all heights. We prove their conjecture in the case that XX is the Cayley-Moufang plane of type E6E_6. For the other exceptional minuscule flag variety, the Freudenthal variety of type E7E_7, we establish their conjecture for heights at most 44, but show that it fails generally. We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case XX is an even-dimensional quadric hypersurface. Our argument uses ideas of K. Dilks, O. Pechenik, and J. Striker (2017) to relate the action on plane partitions to combinatorics derived from KK-theoretic Schubert calculus.Comment: 25 pages, 7 figures, 3 tables. Section 5 rewritten and simplifie

    Paths to Understanding Birational Rowmotion on Products of Two Chains

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    Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset PP, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [AST11, BW74, CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion to YY-systems of type Am×AnA_m \times A_n described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths rr and ss is r+s+2r+s+2 (first proved by D.~Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.Comment: 31 pages, to appear in Algebraic Combinatoric
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