Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams
UDC 512.81 We propose two realizations of representations of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $\Lambda_n$ by differential operators. For each of these realizations, the actions on Schur polynomials are determined and the decomposition of $\Lambda_n$...
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| Datum: | 2026 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/8860 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513340363636736 |
|---|---|
| author | Bedratyuk, L. Бедратюк, Леонід |
| author_facet | Bedratyuk, L. Бедратюк, Леонід |
| author_sort | Bedratyuk, L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T11:04:14Z |
| description | UDC 512.81
We propose two realizations of representations of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $\Lambda_n$ by differential operators. For each of these realizations, the actions on Schur polynomials are determined and the decomposition of $\Lambda_n$ into irreducible representations is obtained. By using the $\mathfrak{sl}_2$-isomorphism between $\Lambda_n$ and the vector space of Young diagrams $\mathbb{Q}\mathcal{Y}_n$ with at most $n$ rows, these representations are transferred to $\mathbb{Q}\mathcal{Y}_n.$ |
| doi_str_mv | 10.3842/umzh.v78i1-2.8860 |
| first_indexed | 2026-03-24T03:43:07Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-8860 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:43:07Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f8/9ea6027b9e967c0c345c378e47a8f5f8 |
| spelling | umjimathkievua-article-88602026-03-21T11:04:14Z Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams Дiї алгебри Лi $\mathfrak{sl}_2$ на симетричних многочленах і на дiаграмах Юнга Bedratyuk, L. Бедратюк, Леонід Lie algebra $\mathfrak{sl}_2$ representations of $\mathfrak{sl}_2$ symmetric polynomials Schur polynomials Young diagrams Алгебра Лі $\mathfrak{sl}_2$ зображення $\mathfrak{sl}_2$ симетричні многочлени многочлени Шура діаграми Юнга UDC 512.81 We propose two realizations of representations of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $\Lambda_n$ by differential operators. For each of these realizations, the actions on Schur polynomials are determined and the decomposition of $\Lambda_n$ into irreducible representations is obtained. By using the $\mathfrak{sl}_2$-isomorphism between $\Lambda_n$ and the vector space of Young diagrams $\mathbb{Q}\mathcal{Y}_n$ with at most $n$ rows, these representations are transferred to $\mathbb{Q}\mathcal{Y}_n.$ УДК 512.81 Запропоновано дві реалізації диференціальними операторами зображень комплексної алгебри Лі $\mathfrak{sl}_2$ на алгебрі симетричних многочленів $\Lambda_n.$ Для кожної з цих реалізацій знайдено дії на многочленах Шура та отримано розклад $\Lambda_n$ на незвідні зображення. За допомогою $\mathfrak{sl}_2$-ізоморфізму між $\Lambda_n$ і векторним простором діаграм Юнга $\mathbb{Q}\mathcal{Y}_n$ із не більш ніж $n$ рядками ці зображення перенесено на $\mathbb{Q}\mathcal{Y}_n.$ Institute of Mathematics, NAS of Ukraine 2026-03-02 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/8860 10.3842/umzh.v78i1-2.8860 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 1-2 (2026); 3–21 Український математичний журнал; Том 78 № 1-2 (2026); 3–21 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/8860/10612 Copyright (c) 2026 L. Bedratyuk |
| spellingShingle | Bedratyuk, L. Бедратюк, Леонід Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams |
| title | Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams |
| title_alt | Дiї алгебри Лi $\mathfrak{sl}_2$ на симетричних многочленах і на дiаграмах Юнга |
| title_full | Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams |
| title_fullStr | Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams |
| title_full_unstemmed | Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams |
| title_short | Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams |
| title_sort | actions of lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on young diagrams |
| topic_facet | Lie algebra $\mathfrak{sl}_2$ representations of $\mathfrak{sl}_2$ symmetric polynomials Schur polynomials Young diagrams Алгебра Лі $\mathfrak{sl}_2$ зображення $\mathfrak{sl}_2$ симетричні многочлени многочлени Шура діаграми Юнга |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8860 |
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