Actions of Lie algebra $\mathfrak{sl}_2$ on symmetric polynomials and on Young diagrams - Bedratyuk, L.; Бедратюк, Леонід (Institute of Mathematics, NAS of Ukraine, 2026)
UDC 512.81 We propose two realizations of representations of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $Λ_n$ by differential operators. For each of these realizations, the actions on Schur polynomials are determined and the decomposition of $Λ_n$ into irreducible representations is obtained. By using the $\mathfrak{sl}_2$-isomorphism between $Λ_n$ and the vector space of Young diagrams $\mathbb{Q}\mathcal{Y}_n$ with at most $n$ rows, these representations are transferred to $\mathbb{Q}\mathcal{Y}_n.$