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

TL;DR

This paper explores two differential operator-based implementations of the $rak{sl}_2$ Lie algebra on symmetric polynomials and Young diagrams, detailing their irreducible subrepresentations and decompositions.

Contribution

It introduces new realizations of $rak{sl}_2$ actions on symmetric polynomials and Young diagrams, expanding understanding of their representation theory.

Findings

Descriptions of finite and infinite-dimensional irreducible subrepresentations

Decomposition of the algebra of symmetric polynomials

Actions on Schur polynomials

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 $\mathbb{Q}\mathcal{Y}_n$.