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 } w?W,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 ? 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 108:1750, 2004). The original proof is proved by an algebraic method that involves representation theory and Springers theorem on regular elements.
cyclic sieving phenomenon coxeterhedron permutohedron boxed ordered partition