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.