On orbits of order ideals of minuscule posets. (English) Zbl 1284.06008

Rush, David B.; Shi, Xiaolin

Proceedings of the 25th international conference on formal power series and algebraic combinatorics, FPSAC 2013, Paris, France, June 24–28, 2013. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 193-204, electronic only (2013).

Summary: 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.
See also the journal version of this paper in [J. Algebr. Comb. 37, No. 3, 545–569 (2013; Zbl 1283.06007)].
For the entire collection see [Zbl 1281.05001].

MSC:

06A07	Combinatorics of partially ordered sets
05E15	Combinatorial aspects of groups and algebras (MSC2010)

Keywords:

order ideals; antichains; minuscule posets; minuscule representations; fully commutative elements; cyclic sieving phenomenon