Singular vectors for the $W_N$ algebras

TL;DR

This paper derives explicit formulas for singular vectors in Fock modules of $W_N$ algebras using free field realizations, screening operators, and Jack symmetric functions, providing a detailed algebraic and combinatorial framework.

Contribution

It introduces a novel explicit construction of singular vectors for $W_N$ algebras via screening operators and Jack symmetric functions, expanding the computational tools in algebraic representation theory.

Findings

Explicit formulas for singular vectors in Fock modules

Connection between screening operators and Jack symmetric functions

Complete characterization of Fock modules via integer sequences

Abstract

In this paper, we use free field realisations of the A-type principal, or Casimir, $W_N$ algebras to derive explicit formulae for singular vectors in Fock modules. These singular vectors are constructed by applying screening operators to Fock module highest weight vectors. The action of the screening operators is then explicitly evaluated in terms of Jack symmetric functions and their skew analogues. The resulting formulae depend on sequences of pairs of integers that completely determine the Fock module as well as the Jack symmetric functions.