Isotropical Linear Spaces and Valuated Delta-Matroids
International audience The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ skew-symmetric matrix. Its points correspond to $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space. In this paper we tropicalize this picture, and...
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ftccsdartic:oai:HAL:hal-01215074v1 2023-05-15T16:51:16+02:00 Isotropical Linear Spaces and Valuated Delta-Matroids Rincón, Felipe Lawrence Berkeley National Laboratory Berkeley (LBNL) Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel Reykjavik, Iceland 2011 https://hal.inria.fr/hal-01215074 https://hal.inria.fr/hal-01215074/document https://hal.inria.fr/hal-01215074/file/dmAO0170.pdf en eng HAL CCSD Discrete Mathematics and Theoretical Computer Science DMTCS hal-01215074 https://hal.inria.fr/hal-01215074 https://hal.inria.fr/hal-01215074/document https://hal.inria.fr/hal-01215074/file/dmAO0170.pdf info:eu-repo/semantics/OpenAccess ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) https://hal.inria.fr/hal-01215074 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.801-812 spinor variety isotropic subspace tropical linear space valuated matroid delta-matroid matroid subdivision [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 2020-12-25T18:15:03Z International audience The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ skew-symmetric matrix. Its points correspond to $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type $D$. La variété spinorielle est decoupée par les relations quadratiques de Wick parmi les Pfaffiens principaux d'une matrice antisymétrique $n \times n$. Ses points correspondent aux sous-espaces isotropes à $n$ dimensions d'un espace vectoriel de dimension $2n$. Dans cet article nous tropicalisons cette description, et nous développons une théorie combinatoire de vecteurs tropicaux de Wick et d'espaces linéaires tropicaux qui sont tropicalement isotropes. Nous caractérisons des vecteurs tropicaux de Wick en termes de subdivisions des polytopes Delta-matroïde, et nous étudions dans quelle mesure les relations de Wick forment une base tropicale. Notre théorie généralise plusieurs résultats pour les espaces linéaires tropicaux et évaluait des matroïdes à la classe des matroïdes de Coxeter du type $D$. 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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English |
topic |
spinor variety isotropic subspace tropical linear space valuated matroid delta-matroid matroid subdivision [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
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spinor variety isotropic subspace tropical linear space valuated matroid delta-matroid matroid subdivision [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] Rincón, Felipe Isotropical Linear Spaces and Valuated Delta-Matroids |
topic_facet |
spinor variety isotropic subspace tropical linear space valuated matroid delta-matroid matroid subdivision [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
description |
International audience The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ skew-symmetric matrix. Its points correspond to $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type $D$. La variété spinorielle est decoupée par les relations quadratiques de Wick parmi les Pfaffiens principaux d'une matrice antisymétrique $n \times n$. Ses points correspondent aux sous-espaces isotropes à $n$ dimensions d'un espace vectoriel de dimension $2n$. Dans cet article nous tropicalisons cette description, et nous développons une théorie combinatoire de vecteurs tropicaux de Wick et d'espaces linéaires tropicaux qui sont tropicalement isotropes. Nous caractérisons des vecteurs tropicaux de Wick en termes de subdivisions des polytopes Delta-matroïde, et nous étudions dans quelle mesure les relations de Wick forment une base tropicale. Notre théorie généralise plusieurs résultats pour les espaces linéaires tropicaux et évaluait des matroïdes à la classe des matroïdes de Coxeter du type $D$. |
author2 |
Lawrence Berkeley National Laboratory Berkeley (LBNL) Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel |
format |
Conference Object |
author |
Rincón, Felipe |
author_facet |
Rincón, Felipe |
author_sort |
Rincón, Felipe |
title |
Isotropical Linear Spaces and Valuated Delta-Matroids |
title_short |
Isotropical Linear Spaces and Valuated Delta-Matroids |
title_full |
Isotropical Linear Spaces and Valuated Delta-Matroids |
title_fullStr |
Isotropical Linear Spaces and Valuated Delta-Matroids |
title_full_unstemmed |
Isotropical Linear Spaces and Valuated Delta-Matroids |
title_sort |
isotropical linear spaces and valuated delta-matroids |
publisher |
HAL CCSD |
publishDate |
2011 |
url |
https://hal.inria.fr/hal-01215074 https://hal.inria.fr/hal-01215074/document https://hal.inria.fr/hal-01215074/file/dmAO0170.pdf |
op_coverage |
Reykjavik, Iceland |
genre |
Iceland |
genre_facet |
Iceland |
op_source |
ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) https://hal.inria.fr/hal-01215074 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.801-812 |
op_relation |
hal-01215074 https://hal.inria.fr/hal-01215074 https://hal.inria.fr/hal-01215074/document https://hal.inria.fr/hal-01215074/file/dmAO0170.pdf |
op_rights |
info:eu-repo/semantics/OpenAccess |
_version_ |
1766041383026556928 |