& v_{\Lambda_0} = C_1 \cdot v_{\Lambda_1}, \\
e(\omega) & v_{\Lambda_1} = C_2 \cdot x_{\gamma_{\underline{2}}}(-1)x_{\gamma_2}(-1)v_{\Lambda_0}, \\
e(\omega) & v_{\Lambda_{\ell-1}} = C_3 \cdot x_{\gamma_{\underline{\ell}}}(-1) v_{\Lambda_{\ell}} \quad \textrm{and} \\
e(\omega) & v_{\Lambda_{\ell-1}} = C_4 \cdot x_{\gamma_{\ell}}(-1) v_{\Lambda_{\ell-1}}\end{aligned}$$ where $C_i,  \ i = 1,2,3,4$ are some constants. We will also use the operator $$e(n \omega) \cong e(\omega)^{n}, \qquad \textrm{for $n \in {\mathbb{N}}$}.$$ Further on, on tensor product we can define the tensor product of $e(\lambda)$ operators: $$\begin{aligned}
 e(\lambda) = e(\lambda) \otimes e(\lambda) \otimes \dots \otimes e(\lambda): V_P^{\otimes k} \longrightarrow V_P^{\otimes k}\end{aligned}$$ and equation is again valid for $\gamma \in \Gamma$.

Spinor representations of ${\mathfrak{g}}$ {#spinor}
==========================================

We will revise now in short spinor representations of the finite dimensional Lie algebra ${\mathfrak{g}}$ of type $D_{\ell}$ (see, for instance, [@FH] for details). Spinor representation are of fundamental weights $\omega_{\ell-1}$ and $\omega_{\ell}$ and can be realized via embedding of algebra ${\mathfrak{g}}$ into Clifford algebra $\textrm{Cliff}(V, B) = \textrm{C}(B)$ where $V={\mathbb{C}}^{2 \ell}$ and $B$ is a symmetric, bilinear form on $V$. If we set $V = W \oplus W'$ where $W$ and $W'$ are the maximal isotropic subspaces for the form $B$, then we have an isomorphism $$\textrm{C}(V,B) \cong \textrm{End} (\wedge^* W)$$ where $\wedge^* W$ denotes the exterior 