Powers of the Vandermonde determinant, Schur functions, and the dimension game
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...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2011-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2893/pdf |
| _version_ | 1797270385314496512 |
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| author | Cristina Ballantine |
| author_facet | Cristina Ballantine |
| author_sort | Cristina Ballantine |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-25T02:03:26Z |
| format | Article |
| id | doaj.art-dec0edd19c7e460a9c5a5e347311676d |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:03:26Z |
| publishDate | 2011-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-dec0edd19c7e460a9c5a5e347311676d2024-03-07T14:49:33ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502011-01-01DMTCS Proceedings vol. AO,...Proceedings10.46298/dmtcs.28932893Powers of the Vandermonde determinant, Schur functions, and the dimension gameCristina Ballantine0Department of Mathematics and Computer ScienceSince 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.https://dmtcs.episciences.org/2893/pdfschur functionsvandermonde determinantyoung diagramssymmetric functionsquantum hall effect[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Cristina Ballantine Powers of the Vandermonde determinant, Schur functions, and the dimension game Discrete Mathematics & Theoretical Computer Science 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] |
| title | 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_short | Powers of the Vandermonde determinant, Schur functions, and the dimension game |
| title_sort | powers of the vandermonde determinant schur functions and the dimension game |
| 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] |
| url | https://dmtcs.episciences.org/2893/pdf |
| work_keys_str_mv | AT cristinaballantine powersofthevandermondedeterminantschurfunctionsandthedimensiongame |