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

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

*此作品的通信作者

研究成果: 雜誌貢獻期刊論文同行評審

摘要

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.

原文英語
頁(從 - 到)617-638
頁數22
期刊Journal of Combinatorial Optimization
25
發行號4
DOIs
出版狀態已發佈 - 2013 五月

ASJC Scopus subject areas

  • 電腦科學應用
  • 離散數學和組合
  • 控制和優化
  • 計算機理論與數學
  • 應用數學

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