Principle subspace for bosonic vertex operator ${\varphi }_{\sqrt{2m}}(z)$ and Jack polynomials

Abstract

Let ${\varphi }_{\sqrt{2m}}(z)={\sum }_{n\in \mathbb{Z}}{a}_n{z}^{-n-m},m\in \mathbb{N}$ be bosonic vertex operator, $L$ some irreducible representation of the vertex algebra $\A_{(m)}$, associated with one-dimensional lattice $\Zl$, generated by vector $l$, $⟨l|,l|=⟩2m$. Fix some extremal vector $v\in L$. We study the principle subspace $\C[a_i]_{i\in\Z}\cdot v$ and its finitization $\C[a_i]_{i>N}\cdot v$. We construct their bases and find characters. In the case of finitization basis is given in terms of Jack polynomials.

16 pages