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 ℝn 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.
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Dedicated to Gerard J. Chang on the occasion of his 60th birthday.
Research partially supported by NSC grants 98-2115-M-390-002 (S.-P. Eu), 99-2115-M-251-001 (T.-S. Fu), and 99-2115-M-127-001 (Y.-J. Pan).
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Eu, SP., Fu, TS. & Pan, YJ. A combinatorial proof of the cyclic sieving phenomenon for faces of Coxeterhedra. J Comb Optim 25, 617–638 (2013). https://doi.org/10.1007/s10878-012-9495-6
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DOI: https://doi.org/10.1007/s10878-012-9495-6