This thesis is composed of two papers both in algebraic combinatorics and Coxeter groups. In Paper I, we concentrate on smoothness of Schubert varieties indexed by involutions from finite simply laced types. We show that if a Schubert variety indexed by an involution of a finite and simply laced Coxeter group is smooth, then that involution must be the longest element of a parabolic subgroup. Given a Coxeter system (W, S), we introduce in Paper II the boolean complex of involutions of W as an analogue of the boolean complex of W studied by Ragnarsson and Tenner. By using discrete Morse Theory, we compute the homotopy type for a large class of W, including all finite Coxeter groups. In all cases, the homotopy type is that of a wedge of spheres of dimension |S| − 1. In addition, we provide a recurrence formula for the number of spheres in the wedge.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:liu-179060 |
| Date | January 2021 |
| Creators | Umutabazi, Vincent |
| Publisher | Linköpings universitet, Algebra, geometri och diskret matematik, Linköpings universitet, Tekniska fakulteten, Linköping |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Licentiate thesis, comprehensive summary, info:eu-repo/semantics/masterThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Linköping Studies in Science and Technology. Licentiate Thesis, 0280-7971 ; 1916 |
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