where $\Sigma \subset \{1, \dots, \ell \}$. The irreducible ${\mathfrak{g}}$-modules of weight $\omega_{\ell-1}$ and $\omega_{\ell}$ are on the top of the ${\tilde{\mathfrak{g}}}$-modules $L({\Lambda_{\ell-1}})$ and $L({\Lambda_{\ell}})$ modules with respect to the gradation . Using we can easily deduce that in the lattice construction weight vectors $w_{\Sigma}$ are of the form $1 \otimes e^{\mu} = e^{\mu}$ where $\mu$ denotes their weight, $\mu = \frac{1}{2} \left( \sum_{i \in \Sigma} \epsilon_i - \sum_{j \notin \Sigma} \epsilon_j \right)$.

A simple calculation shows that spinor representations weights satisfy the commutativity condition if and only if $1 \in \Sigma$, i.e. if $$\mu = \frac{1}{2} \left(  \epsilon_1 + \dots \right)$$ Therefore vertex operators $\mathcal{Y}(1 \otimes e^{\mu},z) = \mathcal{Y}(w_{\Sigma},z)$ commute with the action of ${\tilde{\mathfrak{g}}}_1$ if and only if $1 \in \varGamma$ and from now on we will assume $1 \in  \Sigma$ and replace $\Sigma$ with $\Gamma_1$.

Vertex operator relations {#uvjeti_razlike}
=========================

In this section we find the relations between vector fields. We first observe the identities we have on vacuum vector, i.e. $v_{\Lambda_0}$. Using we see that

\[vakuum0\] $$\begin{aligned}
&  \label{vakuum1} x_{\delta}(-1)x_{\gamma}(-1) v_{\Lambda_0} = 0 \qquad \textrm{if} \quad \delta \neq \underline{\gamma} , \\
&  \label{vakuum2} x_{\underline{\gamma}}(-1) x_{\gamma}(-1) v_{\Lambda_0} = C \cdot x_{\underline{\delta}}(-1) x_{\delta}(-1) v_{\Lambda_0} = C' \cdot e^{2 