| 1. |
- Abdallah, Nancy, et al.
(författare)
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Topology of posets with special partial matchings
- 2019
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Ingår i: Advances in Mathematics. - : Academic Press. - 0001-8708 .- 1090-2082. ; 348, s. 255-276
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Tidskriftsartikel (refereegranskat)abstract
- Special partial matchings (SPMs) are a generalisation of Brentis special matchings. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Mariettis zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic W-sets constitute pircons. Together, these results extend a result of Can, Cherniaysky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.
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| 2. |
- Hansson, Mikael, 1986-
(författare)
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Combinatorics and topology related to involutions in Coxeter groups
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation consists of three papers in combinatorial Coxeter group theory.A Coxeter group is a group W generated by a set S, where all relations can be derived from the relations s2 = e for all s ? S, and (ss′)m(s,s′) = e for some pairs of generators s ≠ s′ in S, where e ? W is the identity element and m(s, s′) is an integer satisfying that m(s, s′) = m(s′, s) ≥ 2. Two prominent examples of Coxeter groups are provided by the symmetric group Sn (i.e., the set of permutations of {1, 2, . . . , n}) and finite reflection groups (i.e., finite groups generated by reflections in some real euclidean space). There are also important infinite Coxeter groups, e.g., affine reflection groups.Every Coxeter group can be equipped with various natural partial orders, the most important of which is the Bruhat order. Any subset of a Coxeter group can then be viewed as an induced subposet.In Paper A, we study certain posets of this kind, namely, unions of conjugacy classes of involutions in the symmetric group. We obtain a complete classification of the posets that are pure (i.e., all maximal chains have the same length). In particular, we prove that the set of involutions with exactly one fixed point is pure, which settles a conjecture of Hultman in the affirmative. When the posets are pure, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, established by Can, Cherniavsky, and Twelbeck.Paper B also deals with involutions in Coxeter groups. Given an involutive automorphism θ of a Coxeter system (W, S), letℑ(θ) = {w ? W | θ(w) = w−1}be the set of twisted involutions. In particular, ℑ(id) is the set of ordinary involutions in W. It is known that twisted involutions can be represented by words in the alphabet = { | s ? S}, called -expressions. If ss′ has finite order m(s, s′), let a braid move be the replacement of ′ ⋯ by ′ ′ ⋯, both consisting of m(s, s′) letters. We prove a word property for ℑ(θ), for any Coxeter system (W, S) with any θ. More precisely, we provide a minimal set of moves, easily determined from the Coxeter graph of (W, S), that can be added to the braid moves in order to connect all reduced -expressions for any given w ? ℑ(θ). This improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg.In Paper C, we investigate the topology of (the order complexes of) certain posets, called pircons. A special partial matching (SPM) on a poset is a matching of the Hasse diagram satisfying certain extra conditions. An SPM without fixed points is precisely a special matching as defined by Brenti. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti’s zircons. Our main result is that every open interval in a pircon is a PL ball or a PL sphere.An important subset of ℑ(θ) is the set ?(θ) = {θ(w−1)w | w ? W} of twisted identities. We prove that if θ does not flip any edges with odd labels in the Coxeter graph, then ?(θ), with the order induced by the Bruhat order on W, is a pircon. Hence, its open intervals are PL balls or spheres, which confirms a conjecture of Hultman. It is also demonstrated that Bruhat orders on Rains and Vazirani’s quasiparabolic W-sets (under a boundedness assumption) form pircons. In particular, this applies to all parabolic quotients of Coxeter groups.
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| 3. |
- 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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| 4. |
- Hultman, Axel, 1975-
(författare)
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Combinatorial complexes, Bruhat intervals and reflection distances
- 2003
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The various results presented in this thesis are naturallysubdivided into three different topics, namely combinatorialcomplexes, Bruhat intervals and expected reflection distances.Each topic is made up of one or several of the altogether sixpapers that constitute the thesis. The following are some of ourresults, listed by topic: Combinatorial complexes: Using a shellability argument, we compute the cohomologygroups of the complements of polygraph arrangements. These arethe subspace arrangements that were exploited by Mark Haiman inhis proof of the n! theorem. We also extend these results toDowling generalizations of polygraph arrangements. We consider certainB- andD-analogues of the quotient complex Δ(Πn)=Sn, i.e. the order complex of the partition latticemodulo the symmetric group, and some related complexes.Applying discrete Morse theory and an improved version of knownlexicographic shellability techniques, we determine theirhomotopy types. Given a directed graphG, we study the complex of acyclic subgraphs ofGas well as the complex of not strongly connectedsubgraphs ofG. Known results in the case ofGbeing the complete graph are generalized. We list the (isomorphism classes of) posets that appear asintervals of length 4 in the Bruhat order on some Weyl group. Inthe special case of symmetric groups, we list all occuringintervals of lengths 4 and 5. Expected reflection distances:Consider a random walk in the Cayley graph of the complexreflection groupG(r, 1,n) with respect to the generating set of reflections. Wedetermine the expected distance from the starting point aftertsteps. The symmetric group case (r= 1) has bearing on the biologist’s problem ofcomputing evolutionary distances between different genomes. Moreprecisely, it is a good approximation of the expected reversaldistance between a genome and the genome with t random reversalsapplied to it.
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| 5. |
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| 6. |
- Hultman, Axel, 1975-, et al.
(författare)
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The topology of the space of matrices of Barvinok rank two
- 2010
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Ingår i: Beiträge zur Algebra und Geometrie. - Lemgo, Germany : Heldermann Verlag. - 0138-4821. ; 51:2, s. 373-390
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Tidskriftsartikel (refereegranskat)abstract
- The Barvinok rank of a d x n matrix is the minimum number of points in Rd such that the tropical convex hull of the points contains all columns of the matrix. The concept originated in work by Barvinok and others on the travelling salesman problem. Our object of study is the space of real d x n matrices of Barvinok rank two. Let Bd,n denote this space modulo rescaling and translation. We show that Bd,n is a manifold, thereby settling a conjecture due to Develin. In fact, Bd,n is homeomorphic to the quotient of the product of spheres Sd-2 x Sn-2 under the involution which sends each point to its antipode simultaneously in both components. In addition, using discrete Morse theory, we compute the integral homology of Bd,n. Assuming d \ge n, for odd d the homology turns out to be isomorphic to that of Sd-2 x RPn-2. This is true also for even d up to degree d-3, but the two cases differ from degree d-2 and up. The homology computation straightforwardly extends to more general complexes of the form (Sd-2 x X)//Z2, where X is a finite cell complex of dimension at most d-2 admitting a free Z2-action.
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| 7. |
- 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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| 8. |
- 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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