Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams
DOI:
https://doi.org/10.3842/umzh.v78i1-2.8860Keywords:
Lie algebra $\mathfrak{sl}_2$, representations of $\mathfrak{sl}_2$, symmetric polynomials, Schur polynomials, Young diagramsAbstract
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 $\mathbb{Q}\mathcal{Y}_n.$
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