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

作者：Eu Sen Peng^*; Fu Tung Shan; Pan Yeh Jong

来源：Journal of Combinatorial Optimization, 2013, 25(4): 617-638.

DOI：10.1007/s10878-012-9495-6

• Cyclic sieving phenomenon
• Coxeterhedron
• Permutohedron
• Boxed ordered partition

For a Coxeter system (W,S), the subgroup W (J) generated by a subset JaS dagger S is called a parabolic subgroup of W. The Coxeterhedron PW associated to (W,S) is the finite poset of all cosets {wW (J) } (waW,JaS dagger 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 a"e (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.

• 出版日期2013-5