The ${\mathfrak{s}\mathfrak{l}}_2$-actions on the symmetric polynomials and on Young diagrams

In the article, two implementations of the representation of the complex Lie algebra ${\mathfrak{s}\mathfrak{l}}_2$ on the algebra of symmetric polynomials ${\mathrm{\Lambda }}_n$ by differential operators are proposed. The realizations of irreducible subrepresentations, both finite-dimensional and infinite-dimensional, are described, and the decomposition of ${\mathrm{\Lambda }}_n$ is found. The actions on the Schur polynomials is also determined. By using an isomorphism between ${\mathrm{\Lambda }}_n$ and the vector space of Young diagrams $\mathbb{Q}{\mathcal{Y}}_n$ with no more than $n$ rows, these representations are transferred to $\mathbb{Q}{\mathcal{Y}}_n$.