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

Abstract

Let $\phi_{\sqrt{2m}}(z)=\sum_{n\in\Z} a_n z^{-n-m}, m\in\N$ be bosonic vertex operator, $L$ some irreducible representation of the vertex algebra $\A_{(m)}$, associated with one-dimensional lattice $\Zl$, generated by vector $l$, $\bra l,l \ket=2m$. 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.