The Bruhat order on conjugation-invariant sets of involutions in the symmetric group

2015/02/12 by Mikael Hansson, Hansson, Mikael
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1502.03598

openalex publication_date 2015/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let I_n be the set of involutions in the symmetric group S_n, and for A ⊆ \0,1,…,n\, let F_n^A=\σ∈ I_n | σ has a fixed points for some a ∈ A\. We give a complete characterisation of the sets A for which F_n^A, with the order induced by the Bruhat order on S_n, is a graded poset. In particular, we prove that F_n^\1\ (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When F_n^A is graded, we give its rank function. We also give a short new proof of the EL-shellability of F_n^\0\ (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck. Keywords: Bruhat order, symmetric group, involution, conjugacy class, graded poset, EL-shellability

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