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<p>We present two realizations of the 𝔰𝔩<sub>2</sub>-action of a complex Lie algebra 𝔰𝔩<sub>2</sub> on the algebra of symmetric polynomials Λ<sub><i>n</i></sub> via differential operators. For each realization, we determine its action on Schur polynomials and obtain a decomposition of Λ<sub><i>n</i></sub> into irreducible representations. By using an 𝔰𝔩<sub>2</sub>-isomorphism between Λ<sub><i>n</i></sub> and the vector space of Young diagrams ℚ<i>Y</i><sub><i>n</i></sub> with at most <i>n</i> rows, we transfer these representations to ℚ<i>Y</i><sub><i>n</i></sub><i>.</i></p>

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Actions of the Lie Algebra 𝔰𝔩2 Upon Symmetric Polynomials and Young Diagrams

  • 期刊论文

  • 发表日期：

    2026年5月12日

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  • Leonid Bedratyuk

摘要

We present two realizations of the 𝔰𝔩2-action of a complex Lie algebra 𝔰𝔩2 on the algebra of symmetric polynomials Λn via differential operators. For each realization, we determine its action on Schur polynomials and obtain a decomposition of Λn into irreducible representations. By using an 𝔰𝔩2-isomorphism between Λn and the vector space of Young diagrams ℚYn with at most n rows, we transfer these representations to ℚYn.