The \mathfraksl₂-actions on the symmetric polynomials and on Young diagrams

2024/08/14 by Leonid Bedratyuk, Bedratyuk, Leonid
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2408.07777

openalex publication_date 2024/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the article, two implementations of the representation of the complex Lie algebra \mathfraksl₂ 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 ℚY_n with no more than n rows, these representations are transferred to ℚY_n.

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