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

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