Generalized Polarization Modules (extended abstract)
This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2015-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2456/pdf |
| _version_ | 1797270201698353152 |
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| author | Héctor Blandin |
| author_facet | Héctor Blandin |
| author_sort | Héctor Blandin |
| collection | DOAJ |
| description | This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\ell$ sets of $n$ variables $x_{ij}$ with $1\le i \le \ell$ and $1 \le j \le n$. Given a $\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_{ij}$ the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$ is the polarization module generated by the family $F$. These polarization modules are all representation of the direct product $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of $\mathfrak{S}_n$-stable families of homogeneous polynomials in n variables, for every $n \ge 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$. |
| first_indexed | 2024-04-25T02:00:30Z |
| format | Article |
| id | doaj.art-57f94009995844528ab5dcad59b2e50d |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:00:30Z |
| publishDate | 2015-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-57f94009995844528ab5dcad59b2e50d2024-03-07T15:01:25ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502015-01-01DMTCS Proceedings, 27th...Proceedings10.46298/dmtcs.24562456Generalized Polarization Modules (extended abstract)Héctor Blandin0Laboratoire de combinatoire et d'informatique mathématique [Montréal]This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\ell$ sets of $n$ variables $x_{ij}$ with $1\le i \le \ell$ and $1 \le j \le n$. Given a $\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_{ij}$ the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$ is the polarization module generated by the family $F$. These polarization modules are all representation of the direct product $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of $\mathfrak{S}_n$-stable families of homogeneous polynomials in n variables, for every $n \ge 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$.https://dmtcs.episciences.org/2456/pdfalgebraic combinatoricssymmetric functionsdiagonally symmetric polynomialsrepresentation theorypolarization operators[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Héctor Blandin Generalized Polarization Modules (extended abstract) Discrete Mathematics & Theoretical Computer Science algebraic combinatorics symmetric functions diagonally symmetric polynomials representation theory polarization operators [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Generalized Polarization Modules (extended abstract) |
| title_full | Generalized Polarization Modules (extended abstract) |
| title_fullStr | Generalized Polarization Modules (extended abstract) |
| title_full_unstemmed | Generalized Polarization Modules (extended abstract) |
| title_short | Generalized Polarization Modules (extended abstract) |
| title_sort | generalized polarization modules extended abstract |
| topic | algebraic combinatorics symmetric functions diagonally symmetric polynomials representation theory polarization operators [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2456/pdf |
| work_keys_str_mv | AT hectorblandin generalizedpolarizationmodulesextendedabstract |