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

Sen-Peng Eu (), Tung-Shan Fu () and Yeh-Jong Pan ()
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Sen-Peng Eu: National University of Kaohsiung
Tung-Shan Fu: National Pingtung Institute of Commerce
Yeh-Jong Pan: Tajen University

Journal of Combinatorial Optimization, 2013, vol. 25, issue 4, No 11, 617-638

Abstract: Abstract 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 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.

Keywords: Cyclic sieving phenomenon; Coxeterhedron; Permutohedron; Boxed ordered partition (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-012-9495-6

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