Cyclic Symmetry of the Scaled Simplex

2012/07/22 by Hugh Thomas, Nathan Williams, Thomas, Hugh +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics

paper · doi:10.48550/arxiv.1207.5240

openalex publication_date 2012/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Z_m^k consist of the m^k alcoves contained in the m-fold dilation of the fundamental alcove of the type A_k affine hyperplane arrangement. As the fundamental alcove has a cyclic symmetry of order (k+1), so does Z_m^k. By bijectively exchanging the natural poset structure of Z_m^k for a natural cyclic action on a set of words, we prove an instance of the cyclic sieving phenomenon.

Citations

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra

Related

• Cyclic sieving for longest reduced words in the hyperoctahedral group
• Cyclic Sieving for a Family of Semistandard Tableaux
• The cyclic sieving phenomenon: a survey
• Crystals, semistandard tableaux and cyclic sieving phenomenon
• A new cyclic sieving phenomenon for Catalan objects