Refined Cyclic Sieving

Connor Ahlbach ; Joshua P. Swanson

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

Reiner-Stanton-White (2004) defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content.

There is a natural notion of refinement for many CSP's. We formulate and prove a refinement of the aforementioned CSP arising from tracking the cyclic descent type of a word in addition to its content. The argument presented is completely different from Reiner-Stanton-White's representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective.

A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner-Stanton-White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon-Wilf (1994).