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Subword complexes in Coxeter groups.(English)Zbl 1069.20026

One of the fundamental results in the theory of combinatorial-topological Coxeter groups is that Björner has shown that intervals in the weak Bruhat ordering are homotopy equivalent to balls or spheres. This work is analogous, or even sharper, where the authors demonstrate that the set of all subwords of an ordered list is homeomorphic to balls or spheres. The main result here depends on shellability of subword complexes. The authors raise some serious open questions at the end, and they might be answered soon. In brief, the article is a creative addition to the subject.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
06A11 Algebraic aspects of posets
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E25 Group actions on posets, etc. (MSC2000)
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes

Cite

References:

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[3]Billey, S. C.; Jockusch, W.; Stanley, R. P., Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., 2, 4, 345-374 (1993) ·Zbl 0790.05093
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[5]A. Björner, Orderings of coxeter groups, Combinatorics and Algebra (Boulder, Co., 1983), American Mathematical Society, Providence, RI, 1984, pp. 175-195.; A. Björner, Orderings of coxeter groups, Combinatorics and Algebra (Boulder, Co., 1983), American Mathematical Society, Providence, RI, 1984, pp. 175-195. ·Zbl 0594.20029
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[10]S. Fomin, A.N. Kirillov, Grothendieck polynomials and the Yang-Baxter equation, Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183-190.; S. Fomin, A.N. Kirillov, Grothendieck polynomials and the Yang-Baxter equation, Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183-190.
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[15]Lascoux, A.; Schützenberger, M.-P., Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C.R. Acad. Sci. Paris Sér. I Math., 295, 11, 629-633 (1982) ·Zbl 0542.14030
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[17]I.G. Macdonald, Notes on Schubert polynomials, Publications du LACIM, Universitè du Québec à Montréal, 1991.; I.G. Macdonald, Notes on Schubert polynomials, Publications du LACIM, Universitè du Québec à Montréal, 1991. ·Zbl 0784.05061
[18]E. Miller, Resolutions and duality for monomial ideals, Ph.D. Thesis, University of California, Berkeley, 2000.; E. Miller, Resolutions and duality for monomial ideals, Ph.D. Thesis, University of California, Berkeley, 2000.
[19]E. Miller, D. Perkinson, in: A.V. Geramita, (Ed.), Eight Lectures on Monomial Ideals, COCOA VI: Proceedings of the International School, May-June, 1999. Queens Papers in Pure and Applied Mathematics, Vol. 120, 2001 pp. 3-105.; E. Miller, D. Perkinson, in: A.V. Geramita, (Ed.), Eight Lectures on Monomial Ideals, COCOA VI: Proceedings of the International School, May-June, 1999. Queens Papers in Pure and Applied Mathematics, Vol. 120, 2001 pp. 3-105.
[20]Ramanathan, A., Schubert varieties are arithmetically Cohen-Macaulay, Invent. Math., 80, 2, 283-294 (1985) ·Zbl 0541.14039
[21]R.P. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, 2nd Edition, Vol. 41, Birkhäuser Boston Inc., Boston, MA, 1996.; R.P. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, 2nd Edition, Vol. 41, Birkhäuser Boston Inc., Boston, MA, 1996. ·Zbl 0838.13008
[22]Terai, N., Alexander duality theorem and Stanley-Reisner rings, Sūrikaisekikenkyūsho Kōkyūroku, 1078, 174-184 (1999), (Free resolutions of coordinate rings of projective varieties and related topics (Kyoto, 1998)) (Japanese) ·Zbl 0974.13019
[23]Yanagawa, K., Alexander duality for Stanley-Reisner rings and squarefree \(N^n\)-graded modules, J. Algebra, 225, 630-645 (2000) ·Zbl 0981.13011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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