Powers of the Vandermonde determinant, Schur functions, and the dimension game
International audience Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will gi...
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ftccsdartic:oai:HAL:hal-01215094v1 2023-05-15T16:51:58+02:00 Powers of the Vandermonde determinant, Schur functions, and the dimension game Ballantine, Cristina Department of Mathematics and Computer Science (College of the Holy Cross) College of the Holy Cross Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel Reykjavik, Iceland 2011 https://hal.inria.fr/hal-01215094 https://hal.inria.fr/hal-01215094/document https://hal.inria.fr/hal-01215094/file/dmAO0109.pdf en eng HAL CCSD Discrete Mathematics and Theoretical Computer Science DMTCS hal-01215094 https://hal.inria.fr/hal-01215094 https://hal.inria.fr/hal-01215094/document https://hal.inria.fr/hal-01215094/file/dmAO0109.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-01215094 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.87-98 Schur functions Vandermonde determinant Young diagrams symmetric functions quantum Hall effect [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 2021-10-24T11:25:10Z International audience Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function $s_μ$ in the decomposition of an even power of the Vandermonde determinant in $n+1$ variables in terms of the coefficient of the Schur function $s_λ$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $μ$ is obtained from the Young diagram of $λ$ by adding a tetris type shape to the top or to the left. Comme toute puissance paire du déterminant de Vandermonde est un polynôme symétrique, nous voulons comprendre sa décomposition dans la base des fonctions de Schur. Nous allons étudier plusieurs propriétés combinatoires des coefficients de la décomposition. En particulier, nous allons donner une approche récursive pour le calcul du coefficient de la fonction de Schur $s_μ$ dans la décomposition d'une puissance paire du déterminant de Vandermonde en $n+1$ variables, en fonction du coefficient de la fonction de Schur $s_λ$ dans la décomposition de la même puissance paire du déterminant de Vandermonde en $n$ variables, lorsque le diagramme de Young de $μ$ est obtenu à partir du diagramme de Young de $λ$ par l'addition d'une forme de type tetris vers le haut ou vers la gauche. Conference Object Iceland Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Gauche ENVELOPE(-62.500,-62.500,-64.233,-64.233) |
institution |
Open Polar |
collection |
Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
op_collection_id |
ftccsdartic |
language |
English |
topic |
Schur functions Vandermonde determinant Young diagrams symmetric functions quantum Hall effect [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
spellingShingle |
Schur functions Vandermonde determinant Young diagrams symmetric functions quantum Hall effect [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] Ballantine, Cristina Powers of the Vandermonde determinant, Schur functions, and the dimension game |
topic_facet |
Schur functions Vandermonde determinant Young diagrams symmetric functions quantum Hall effect [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
description |
International audience Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function $s_μ$ in the decomposition of an even power of the Vandermonde determinant in $n+1$ variables in terms of the coefficient of the Schur function $s_λ$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $μ$ is obtained from the Young diagram of $λ$ by adding a tetris type shape to the top or to the left. Comme toute puissance paire du déterminant de Vandermonde est un polynôme symétrique, nous voulons comprendre sa décomposition dans la base des fonctions de Schur. Nous allons étudier plusieurs propriétés combinatoires des coefficients de la décomposition. En particulier, nous allons donner une approche récursive pour le calcul du coefficient de la fonction de Schur $s_μ$ dans la décomposition d'une puissance paire du déterminant de Vandermonde en $n+1$ variables, en fonction du coefficient de la fonction de Schur $s_λ$ dans la décomposition de la même puissance paire du déterminant de Vandermonde en $n$ variables, lorsque le diagramme de Young de $μ$ est obtenu à partir du diagramme de Young de $λ$ par l'addition d'une forme de type tetris vers le haut ou vers la gauche. |
author2 |
Department of Mathematics and Computer Science (College of the Holy Cross) College of the Holy Cross Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel |
format |
Conference Object |
author |
Ballantine, Cristina |
author_facet |
Ballantine, Cristina |
author_sort |
Ballantine, Cristina |
title |
Powers of the Vandermonde determinant, Schur functions, and the dimension game |
title_short |
Powers of the Vandermonde determinant, Schur functions, and the dimension game |
title_full |
Powers of the Vandermonde determinant, Schur functions, and the dimension game |
title_fullStr |
Powers of the Vandermonde determinant, Schur functions, and the dimension game |
title_full_unstemmed |
Powers of the Vandermonde determinant, Schur functions, and the dimension game |
title_sort |
powers of the vandermonde determinant, schur functions, and the dimension game |
publisher |
HAL CCSD |
publishDate |
2011 |
url |
https://hal.inria.fr/hal-01215094 https://hal.inria.fr/hal-01215094/document https://hal.inria.fr/hal-01215094/file/dmAO0109.pdf |
op_coverage |
Reykjavik, Iceland |
long_lat |
ENVELOPE(-62.500,-62.500,-64.233,-64.233) |
geographic |
Gauche |
geographic_facet |
Gauche |
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-01215094 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.87-98 |
op_relation |
hal-01215094 https://hal.inria.fr/hal-01215094 https://hal.inria.fr/hal-01215094/document https://hal.inria.fr/hal-01215094/file/dmAO0109.pdf |
op_rights |
info:eu-repo/semantics/OpenAccess |
_version_ |
1766042096462987264 |