On Orbits of Order Ideals of Minuscule Posets

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

arXiv comment20 pages, to appear in J. Algebraic Combin

References in corpus (2)

• On Orbits of Order Ideals of Minuscule Posets
• Some Arithmetic Properties of the q-Euler Numbers and q-Salié Numbers

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• Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion
• The CDE property for skew vexillary permutations
• The birational Lalanne-Kreweras involution
• Orbits of Plane Partitions of Exceptional Lie Type
• The canonical join complex
• Promotion and cyclic sieving on families of SSYT
• Dynamics of plane partitions: Proof of the Cameron-Fon-Der-Flaass conjecture
• Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass
• Homomesy via Toggleability Statistics
• The Coxeter transformation on Cominuscule Posets
• P-flag spaces and incidence stratifications
• Orbits of antichains in certain root posets
• Order polynomial product formulas and poset dynamics
• Minuscule representations and Panyushev conjectures
• $P$-strict promotion and $Q$-partition rowmotion: the graded case