Generating functions of $W_{1+\infty}$ action on symmetric functions

TL;DR

This paper explores how the infinite-dimensional Lie algebra $W_{1+ abla}$ acts on symmetric functions, revealing a self-duality property through formal distributions.

Contribution

It provides a formal distribution framework to describe the $W_{1+ abla}$ action on Schur and Schur Q-functions, highlighting a novel self-duality property.

Findings

Explicit formulas for $W_{1+ abla}$ action on symmetric functions

Identification of a self-duality property in these formulas

Framework applicable to B-type analogue and symmetric functions

Abstract

We describe the action of the infinite-dimensional Lie algebra $W_{1+\infty}$ and its B-type analogue on Schur and Schur Q-functions, respectively, using formal distributions framework. We observe an interesting self-duality property possessed by these compact formulas.