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

$$\begin{aligned} F_n^A=\{\sigma \in I_n\mid \sigma \text{ has } \text{ exactly } a \text{ fixed } \text{ points } \text{ for } \text{ some } a \in A\}. \end{aligned}$$

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), recently proved by Can et al.