A combinatorial proof of the cyclic sieving phenomenon for faces of Coxeterhedra

For a Coxeter system (W,S), the subgroup W _J generated by a subset J⊆S is called a parabolic subgroup of W. The Coxeterhedron PW associated to (W,S) is the finite poset of all cosets {wW _J } _wεW,J⊆S of all parabolic subgroups of W, ordered by inclusion. This poset can be realized by the face lattice of a simple polytope, constructed as the convex hull of the orbit of a generic point in ℝ; ⁿ under an action of the reflection group W. In this paper, for the groups W=A _n-1, B _n and D _n in a case-by-case manner, we present an elementary proof of the cyclic sieving phenomenon for faces of various dimensions of PW under the action of a cyclic group generated by a Coxeter element. This result provides a geometric, enumerative and combinatorial approach to re-prove a theorem in Reiner et al. (J. Comb. Theory, Ser. A 108:17-50, 2004). The original proof is proved by an algebraic method that involves representation theory and Springer's theorem on regular elements.