Fixed elements of pircon automorphisms

TL;DR

This paper demonstrates that fixed elements of automorphisms in pircons form subposets that are also pircons, and applies this to symmetric groups, showing certain fixed point free involutions form EL-shellable posets.

Contribution

It establishes that fixed elements of automorphisms in pircons form subpircons and proves EL-shellability for fixed point free involutions in type B.

Findings

Fixed elements of automorphisms induce subpircons.

Order complexes of open intervals are PL balls or spheres.

Fixed point free involutions form EL-shellable posets.

Abstract

We prove that the subposet induced by the fixed elements of any automorphism of a pircon is also a pircon. By a result of Abdallah, Hansson, and Hultman, the order complex of any open interval in a pircon is a PL ball or a PL sphere. We apply our main results to symmetric groups of the form $S_{2n}$. A consequence is that the fixed point free signed involutions form a pircon under the dual of the Bruhat order on the hyperoctahedral group. Finally, we prove that this poset is, in fact, EL-shellable, which is a type $B$ analogue of a result of Can, Cherniavsky, and Twelbeck.