91 research outputs found

    Orbits of Plane Partitions of Exceptional Lie Type

    Full text link
    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 We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case XX is an even-dimensional quadric hypersurface.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

    Resonance in orbits of plane partitions

    Get PDF
    International audienceWe introduce a new concept of resonance on discrete dynamical systems. Our main result is an equivariant bijection between plane partitions in a box under rowmotion and increasing tableaux under K-promotion, using a generalization of the equivariance of promotion and rowmotion [J. Striker–N. Williams '12] to higher dimensional lattices. This theorem implies new results for K-promotion and new proofs of previous results on plane partitions

    Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass

    Full text link
    Let PP be a graded poset of rank rr and let c\mathbf{c} be a cc-element chain. For an order ideal II of P×cP \times \mathbf{c}, its rowmotion ψ(I)\psi(I) is the smallest ideal containing the minimal elements of the complementary filter of II. The map ψ\psi defines invertible dynamics on the set of ideals. We say that that PP has NRP ("not relatively prime") rowmotion if no ψ\psi-orbit has cardinality relatively prime to r+c+1r+c+1. In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product P=a×bP = \mathbf{a} \times \mathbf{b} of two chains, the poset whose order ideals correspond to the Schubert varieties of a Grassmann variety Gra(Ca+b)\mathrm{Gr}_a(\mathbb{C}^{a+b}) under containment. Here, we initiate the general study of posets with NRP rowmotion. Our first main result establishes NRP rowmotion for all minuscule posets PP, posets whose order ideals reflect the Schubert stratification of minuscule flag varieties. Our second main result is that NRP promotion depends only on the isomorphism class of the comparability graph of PP.Comment: 15 pages, 5 figure
    • …
    corecore