On orbits of order ideals of minuscule posets

  • Authors:

  • David B. Rush;Xiaolin Shi

  • Affiliations:

  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2013

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Abstract

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脳[2] also exhibits the cyclic sieving phenomenon, only the proof is by appeal to the classification of minuscule posets and is not uniform.