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- Hultman, Axel, 1975-, et al.
(författare)
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Boolean Complexes of Involutions
- 2023
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Ingår i: Annals of Combinatorics. - : SPRINGER BASEL AG. - 0218-0006 .- 0219-3094. ; 27, s. 129-147
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Tidskriftsartikel (refereegranskat)abstract
- Let (W,S) be a Coxeter system. We introduce the boolean com-plex of involutions ofWwhich is an analogue of the boolean complex ofWstudied by Ragnarsson and Tenner. By applying discrete Morse theory,we determine the homotopy type of the boolean complex of involutionsfor a large class of (W,S), including all finite Coxeter groups, finding thatthe homotopy type is that of a wedge of spheres of dimension |S|-1. In addition, we find simple recurrence formulas for the number of spheres inthe wedge
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| 3. |
- Umutabazi, Vincent, 1982-
(författare)
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Boolean complexes of involutions and smooth intervals in Coxeter groups
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)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 (s, s′)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 S 2n. 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.
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| 4. |
- Umutabazi, Vincent, 1982-
(författare)
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Smooth Schubert varieties and boolean complexes of involutions
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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