### 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} } _{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.

Original language | English |
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Pages (from-to) | 617-638 |

Number of pages | 22 |

Journal | Journal of Combinatorial Optimization |

Volume | 25 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2013 May 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Journal of Combinatorial Optimization*,

*25*(4), 617-638. https://doi.org/10.1007/s10878-012-9495-6