| 1. |
- Hultman, Axel, et al.
(författare)
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From Bruhat intervals to intersection lattices and a conjecture of Postnikov
- 2009
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 116:3, s. 564-580
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Tidskriftsartikel (refereegranskat)abstract
- We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w is an element of (sic)(n). is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.
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| 2. |
- Hultman, Axel
(författare)
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Supersolvability and the Koszul property of root ideal arrangements
- 2016
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 144, s. 1401-1413
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Tidskriftsartikel (refereegranskat)abstract
- A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.
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| 3. |
- 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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| 4. |
- Hansson, Mikael, et al.
(författare)
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A word property for twisted involutions in Coxeter groups
- 2019
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Ingår i: Journal of combinatorial theory. Series A (Print). - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0097-3165 .- 1096-0899. ; 161, s. 220-235
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Tidskriftsartikel (refereegranskat)abstract
- Given an involutive automorphism theta of a Coxeter system (W, S), let J(theta) subset of W denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced S-expressions (also known as admissible sequences, reduced I-theta-expressions, or involution words) for any given w is an element of J(theta). This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg. (C) 2018 Elsevier Inc. All rights reserved.
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| 5. |
- 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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