Non-commutative creation operators for symmetric polynomials

TL;DR

This paper revisits non-commutative creation operators for symmetric polynomials, connecting classical Pieri rules with modern algebraic frameworks like $W_{1+ ablafty}$ and affine Yangian algebras.

Contribution

It introduces a modern perspective on classical creation operators, extending their construction to matrix and Fock representations of advanced algebraic structures.

Findings

Constructed creation operators $ ilde B_m$ in $W_{1+ ablafty}$ algebra representations.

Extended the operators to the Fock representation of the affine Yangian $Y( ablag_1)$.

Demonstrated non-commutative nature of these operators in different algebraic contexts.

Abstract

We reconsider in modern terms the old discovery by A. Kirillov and M. Noumi, who devised peculiar operators adding columns to Young diagrams enumerating the Schur, Jack and Macdonald polynomials. In this sense, these are a kind of ``creation'' operators, representing Pieri rules in a maximally simple form, when boxes are added to Young diagrams in a regular way and not to arbitrary ``empty places'' around the diagram. Instead the operators do not commute, and one should add columns of different lengths one after another. We consider this construction in different contexts. In particular, we build up the creation operators $\hat B_m$ in the matrix and Fock representations of the $W_{1+\infty}$ algebra, and in the Fock representation of the affine Yangian algebra $Y(\widehat{gl}_1)$.