On orbits of order ideals of minuscule posets

By David B. Rush, X. Shi. Journal of Algebraic Combinatorics, 2011

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An action on order ideals of posets considered by Fon-Der-Flaass is analyzed in the case of posets arising from minuscule representations of complex simple Lie algebras. For these minuscule posets, it is shown that the Fon-Der-Flaass action exhibits the cyclic sieving phenomenon, as defined by Reiner, Stanton, and White. A uniform proof is given by investigation of a bijection due to Stembridge between order ideals of minuscule posets and fully commutative Weyl group elements. This bijection is proven to be equivariant with respect to a conjugate of the Fon-Der-Flaass action and an arbitrary Coxeter element. If P is a minuscule poset, it is shown that the Fon-Der-Flaass action on order ideals of the Cartesian product P×[2] also exhibits the cyclic sieving phenomenon, only the proof is by appeal to the classification of minuscule posets and is not uniform.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/aef91ff323173c5b0e39a89f8c1de075212722fc

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A uniform bijection between nonnesting and noncrossing partitions (2011, background): “Another recent paper related to the Panyushev complement is [24].”