{
  "id": "2408.07777",
  "version": "v1",
  "published": "2024-08-14T19:09:46.000Z",
  "updated": "2024-08-14T19:09:46.000Z",
  "title": "The $\\mathfrak{sl}_2$-actions on the symmetric polynomials and on Young diagrams",
  "authors": [
    "Leonid Bedratyuk"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In the article, two implementations of the representation of the complex Lie algebra $\\mathfrak{sl}_2$ on the algebra of symmetric polynomials $\\Lambda_n$ by differential operators are proposed. The realizations of irreducible subrepresentations, both finite-dimensional and infinite-dimensional, are described, and the decomposition of $\\Lambda_n$ is found. The actions on the Schur polynomials is also determined. By using an isomorphism between $\\Lambda_n$ and the vector space of Young diagrams $\\mathbb{Q}\\mathcal{Y}_n$ with no more than $n$ rows, these representations are transferred to $\\mathbb{Q}\\mathcal{Y}_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-14T19:09:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "young diagrams",
      "symmetric polynomials",
      "complex lie algebra",
      "vector space",
      "differential operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}