Critical Groups of Simplicial Complexes
International audience We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We sho...
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ftccsdartic:oai:HAL:hal-01215108v1 2023-05-15T16:51:01+02:00 Critical Groups of Simplicial Complexes Duval, Art M. Klivans, Caroline J. Martin, Jeremy L. Department of Mathematical Sciences (University of Texas at El Paso (UTEP)) The University of Texas Health Science Center at Houston (UTHealth) Department of Computer Science, University of Chicago University of Chicago Department of Mathematics (University of Kansas) University of Kansas Lawrence (KU) Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel Reykjavik, Iceland 2011 https://hal.inria.fr/hal-01215108 https://hal.inria.fr/hal-01215108/document https://hal.inria.fr/hal-01215108/file/dmAO0125.pdf en eng HAL CCSD Discrete Mathematics and Theoretical Computer Science DMTCS hal-01215108 https://hal.inria.fr/hal-01215108 https://hal.inria.fr/hal-01215108/document https://hal.inria.fr/hal-01215108/file/dmAO0125.pdf 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://hal.inria.fr/hal-01215108 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.269-280 graph simplicial complex critical group combinatorial Laplacian chip-firing game sandpile model spanning trees [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 We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups. Nous généralisons la théorie des groupes critiques des graphes aux complexes simpliciaux. Plus précisément, pour un complexe simplicial, nous définissons une famille de groupes abéliens en termes d'opérateurs de Laplace combinatoires, qui généralise la construction du groupe critique d'un graphe. Nous montrons comment réaliser ces groupes critiques explicitement comme conoyaux des opérateurs de Laplace réduits combinatoires, et montrons qu'ils sont finis. Leurs ordres sont obtenus en comptant (avec des poids) des arbres simpliciaux couvrants. Nous décrivons comment les groupes critiques d'un complexe représentent le flux le long de ses faces, et esquissons une autre interprétation potentielle comme analogues des groupes de Chow. Conference Object Iceland Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Laplace ENVELOPE(141.467,141.467,-66.782,-66.782) |
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Open Polar |
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Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
op_collection_id |
ftccsdartic |
language |
English |
topic |
graph simplicial complex critical group combinatorial Laplacian chip-firing game sandpile model spanning trees [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
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graph simplicial complex critical group combinatorial Laplacian chip-firing game sandpile model spanning trees [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] Duval, Art M. Klivans, Caroline J. Martin, Jeremy L. Critical Groups of Simplicial Complexes |
topic_facet |
graph simplicial complex critical group combinatorial Laplacian chip-firing game sandpile model spanning trees [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
description |
International audience We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups. Nous généralisons la théorie des groupes critiques des graphes aux complexes simpliciaux. Plus précisément, pour un complexe simplicial, nous définissons une famille de groupes abéliens en termes d'opérateurs de Laplace combinatoires, qui généralise la construction du groupe critique d'un graphe. Nous montrons comment réaliser ces groupes critiques explicitement comme conoyaux des opérateurs de Laplace réduits combinatoires, et montrons qu'ils sont finis. Leurs ordres sont obtenus en comptant (avec des poids) des arbres simpliciaux couvrants. Nous décrivons comment les groupes critiques d'un complexe représentent le flux le long de ses faces, et esquissons une autre interprétation potentielle comme analogues des groupes de Chow. |
author2 |
Department of Mathematical Sciences (University of Texas at El Paso (UTEP)) The University of Texas Health Science Center at Houston (UTHealth) Department of Computer Science, University of Chicago University of Chicago Department of Mathematics (University of Kansas) University of Kansas Lawrence (KU) Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel |
format |
Conference Object |
author |
Duval, Art M. Klivans, Caroline J. Martin, Jeremy L. |
author_facet |
Duval, Art M. Klivans, Caroline J. Martin, Jeremy L. |
author_sort |
Duval, Art M. |
title |
Critical Groups of Simplicial Complexes |
title_short |
Critical Groups of Simplicial Complexes |
title_full |
Critical Groups of Simplicial Complexes |
title_fullStr |
Critical Groups of Simplicial Complexes |
title_full_unstemmed |
Critical Groups of Simplicial Complexes |
title_sort |
critical groups of simplicial complexes |
publisher |
HAL CCSD |
publishDate |
2011 |
url |
https://hal.inria.fr/hal-01215108 https://hal.inria.fr/hal-01215108/document https://hal.inria.fr/hal-01215108/file/dmAO0125.pdf |
op_coverage |
Reykjavik, Iceland |
long_lat |
ENVELOPE(141.467,141.467,-66.782,-66.782) |
geographic |
Laplace |
geographic_facet |
Laplace |
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://hal.inria.fr/hal-01215108 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.269-280 |
op_relation |
hal-01215108 https://hal.inria.fr/hal-01215108 https://hal.inria.fr/hal-01215108/document https://hal.inria.fr/hal-01215108/file/dmAO0125.pdf |
op_rights |
info:eu-repo/semantics/OpenAccess |
_version_ |
1766041131190059008 |