Boolean complexes of involutions and smooth intervals in coxeter groups

Abstract

This dissertation is composed of four papers in algebraic combinatorics related to Coxeter groups. By a Coxeter group, we mean a group W generated by a subset S ⊂ W such that for all s ∈ S, we have s2 = e, and (ss )m(s,s ) = (s s)m(s,s ) = e, where m(s,s ) = m(s s) ≥ 2 for all s s ∈ S. The condition m(s,s ) = ∞ is allowed and means that there is no relation between s and s . There are some partial orders that are associated with every Coxeter group. Among them, the most notable one is the Bruhat order. Coxeter groups and their Bruhat orders have important properties that can be utilised to study Schubert varieties. In Paper I, we consider Schubert varieties that are indexed by involutions of a finite simply laced Coxeter group. We prove that the Schubert varieties which are indexed by involutions that are not longest elements of some standard parabolic subgroups are not smooth. Paper II is based on the Boolean complexes of involutions of a Coxeter group. These complexes are analogues of the Boolean complexes invented by Ragnarsson and Tenner. We use discrete Morse theory to compute the homotopy type of the Boolean complexes of involutions of some infinite Coxeter groups together with all finite Coxeter groups. In Paper III, we prove that the subposet induced by the fixed elements of any automorphism of a pircon is also a pircon. In addition, our main results are applied to the symmetric groups S2n. As a consequence, we prove that the signed fixed point free involutions form a pircon under the dual of the Bruhat order on the hyperoctahedral group. Let W be a Weyl group and I denote a Bruhat interval in W. In Paper IV, we prove that if the dual of I is a zircon, then I is rationally smooth. After examining when the converse holds, and being influenced from conjectures by Delanoy, we are led to pose two conjectures. Those conjectures imply that for Bruhat intervals in type A, duals of smooth intervals, zircons, and being isomorphic to lower intervals are all equivalent. We have verified our conjectures in types An, n ≤ 8, by using SageMath.