Rational smoothness and affine Schubert varieties of type A
International audience The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is t...
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ftccsdartic:oai:HAL:hal-01215081v1 2023-12-24T10:17:58+01:00 Rational smoothness and affine Schubert varieties of type A Billey, Sara Crites, Andrew Department of Mathematics Seattle University of Washington Seattle Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel Reykjavik, Iceland 2011 https://inria.hal.science/hal-01215081 https://inria.hal.science/hal-01215081/document https://inria.hal.science/hal-01215081/file/dmAO0116.pdf https://doi.org/10.46298/dmtcs.2900 en eng HAL CCSD Discrete Mathematics and Theoretical Computer Science DMTCS info:eu-repo/semantics/altIdentifier/doi/10.46298/dmtcs.2900 hal-01215081 https://inria.hal.science/hal-01215081 https://inria.hal.science/hal-01215081/document https://inria.hal.science/hal-01215081/file/dmAO0116.pdf doi:10.46298/dmtcs.2900 info:eu-repo/semantics/OpenAccess ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science Discrete Mathematics and Theoretical Computer Science (DMTCS) 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) https://inria.hal.science/hal-01215081 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.171-182, ⟨10.46298/dmtcs.2900⟩ pattern avoidance affine permutations Schubert varieties [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] info:eu-repo/semantics/conferenceObject Conference papers 2011 ftccsdartic https://doi.org/10.46298/dmtcs.2900 2023-11-26T01:48:29Z International audience The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the combinatorics of its corresponding Weyl group element. A celebrated example of this occurs when G/B is of type A, due to Lakshmibai and Sandhya. They showed that the smooth Schubert varieties are precisely those indexed by permutations that avoid the patterns 3412 and 4231. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations. L'étude des variétés de Schubert dans G/B a mené à plusieurs avancées en combinatoire algébrique. Ces variétés sont indexées soit par l'élément du groupe de Weyl correspondant, soit par un groupe de Weyl affine, soit par un de leurs quotients paraboliques. Souvent, le but est de déterminer quelles propriétés algébriques et topologiques des variétés de Schubert peuvent être décrites en termes des propriétés combinatoires des éléments du groupe de Weyl correspondant. Un exemple bien connu, dû à Lakshmibai et Sandhya, concerne le cas où G/B est de type A. Ils ont montré que les variétés de Schubert lisses sont exactement celles qui sont indexées par les permutations qui évitent les motifs 3412 et 4231. Notre résultat principal est une caractérisation des variétés de Schubert lisses et rationnelles qui correspondent à des permutations affines pour les motifs 4231 et 3412 et les permutations spirales tordues. Conference Object Iceland Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
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Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
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language |
English |
topic |
pattern avoidance affine permutations Schubert varieties [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
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pattern avoidance affine permutations Schubert varieties [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] Billey, Sara Crites, Andrew Rational smoothness and affine Schubert varieties of type A |
topic_facet |
pattern avoidance affine permutations Schubert varieties [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
description |
International audience The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the combinatorics of its corresponding Weyl group element. A celebrated example of this occurs when G/B is of type A, due to Lakshmibai and Sandhya. They showed that the smooth Schubert varieties are precisely those indexed by permutations that avoid the patterns 3412 and 4231. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations. L'étude des variétés de Schubert dans G/B a mené à plusieurs avancées en combinatoire algébrique. Ces variétés sont indexées soit par l'élément du groupe de Weyl correspondant, soit par un groupe de Weyl affine, soit par un de leurs quotients paraboliques. Souvent, le but est de déterminer quelles propriétés algébriques et topologiques des variétés de Schubert peuvent être décrites en termes des propriétés combinatoires des éléments du groupe de Weyl correspondant. Un exemple bien connu, dû à Lakshmibai et Sandhya, concerne le cas où G/B est de type A. Ils ont montré que les variétés de Schubert lisses sont exactement celles qui sont indexées par les permutations qui évitent les motifs 3412 et 4231. Notre résultat principal est une caractérisation des variétés de Schubert lisses et rationnelles qui correspondent à des permutations affines pour les motifs 4231 et 3412 et les permutations spirales tordues. |
author2 |
Department of Mathematics Seattle University of Washington Seattle Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel |
format |
Conference Object |
author |
Billey, Sara Crites, Andrew |
author_facet |
Billey, Sara Crites, Andrew |
author_sort |
Billey, Sara |
title |
Rational smoothness and affine Schubert varieties of type A |
title_short |
Rational smoothness and affine Schubert varieties of type A |
title_full |
Rational smoothness and affine Schubert varieties of type A |
title_fullStr |
Rational smoothness and affine Schubert varieties of type A |
title_full_unstemmed |
Rational smoothness and affine Schubert varieties of type A |
title_sort |
rational smoothness and affine schubert varieties of type a |
publisher |
HAL CCSD |
publishDate |
2011 |
url |
https://inria.hal.science/hal-01215081 https://inria.hal.science/hal-01215081/document https://inria.hal.science/hal-01215081/file/dmAO0116.pdf https://doi.org/10.46298/dmtcs.2900 |
op_coverage |
Reykjavik, Iceland |
genre |
Iceland |
genre_facet |
Iceland |
op_source |
ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science Discrete Mathematics and Theoretical Computer Science (DMTCS) 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) https://inria.hal.science/hal-01215081 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.171-182, ⟨10.46298/dmtcs.2900⟩ |
op_relation |
info:eu-repo/semantics/altIdentifier/doi/10.46298/dmtcs.2900 hal-01215081 https://inria.hal.science/hal-01215081 https://inria.hal.science/hal-01215081/document https://inria.hal.science/hal-01215081/file/dmAO0116.pdf doi:10.46298/dmtcs.2900 |
op_rights |
info:eu-repo/semantics/OpenAccess |
op_doi |
https://doi.org/10.46298/dmtcs.2900 |
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1786206450484248576 |