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

Sen Peng Eu*, Tung Shan Fu, Yeh Jong Pan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)617-638
Number of pages22
JournalJournal of Combinatorial Optimization
Issue number4
Publication statusPublished - 2013 May
Externally publishedYes


  • 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


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