Non-commutative creation operators for symmetric polynomials

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 $Y(\widehat{gl}_1)$.