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

TL;DR

This paper characterizes when certain sets of involutions in the symmetric group form graded posets under the Bruhat order, proving a conjecture and providing rank functions and shellability results.

Contribution

It provides a complete characterization of subsets of involutions forming graded Bruhat posets, including a proof of a conjecture and new insights into their structure.

Findings

F_n^{ ext{{1}}}} is graded, confirming Hultman's conjecture.

F_n^{ ext{{0}}}} is EL-shellable, with a new proof provided.

Rank functions are given for graded cases.

Abstract

Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for some $a \in A$}\}. \] 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