The $\mathfrak{sl}_2$-actions on the symmetric polynomials and on Young diagrams
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916358378225664 |
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| author | Bedratyuk, Leonid |
| author_facet | Bedratyuk, Leonid |
| contents | In the article, two implementations of the representation of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $Λ_n$ by differential operators are proposed. The realizations of irreducible subrepresentations, both finite-dimensional and infinite-dimensional, are described, and the decomposition of $Λ_n$ is found. The actions on the Schur polynomials is also determined. By using an isomorphism between $Λ_n$ and the vector space of Young diagrams $\mathbb{Q}\mathcal{Y}_n$ with no more than $n$ rows, these representations are transferred to $\mathbb{Q}\mathcal{Y}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $\mathfrak{sl}_2$-actions on the symmetric polynomials and on Young diagrams Bedratyuk, Leonid Combinatorics In the article, two implementations of the representation of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $Λ_n$ by differential operators are proposed. The realizations of irreducible subrepresentations, both finite-dimensional and infinite-dimensional, are described, and the decomposition of $Λ_n$ is found. The actions on the Schur polynomials is also determined. By using an isomorphism between $Λ_n$ and the vector space of Young diagrams $\mathbb{Q}\mathcal{Y}_n$ with no more than $n$ rows, these representations are transferred to $\mathbb{Q}\mathcal{Y}_n$. |
| title | The $\mathfrak{sl}_2$-actions on the symmetric polynomials and on Young diagrams |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.07777 |