Chains in Weak Order Posets Associated to Involutions

TL;DR

This paper provides a combinatorial description of W-sets in weak order posets related to involutions in the symmetric group, aiding the study of Borel orbits in classical symmetric spaces.

Contribution

It explicitly characterizes W-sets and maximal chains in weak order posets of involutions, fixed point free involutions, and signed fixed point involutions.

Findings

Complete combinatorial descriptions of W-sets for three involution classes

Characterization of maximal chains in lower order ideals

Applications to Borel orbits in symmetric spaces

Abstract

The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W-sets of the weak order posets of three different sets of involutions in the symmetric group, namely, the set of all involutions, the set of all fixed point free involutions, and the set of all involutions with signed fixed points (or "clans"). These distinguished sets of involutions parameterize Borel orbits in the classical symmetric spaces associated to the general linear group. In particular, we give a complete characterization of the maximal chains of an arbitrary lower order ideal in any of these three posets.