algebra of W: $\wedge^* W = \wedge^0 W \oplus \dots \oplus \wedge^ {\ell} W$. The embedding of $D_{\ell}$ into Clifford algebra gives therefore a representation of $D_{\ell}$ on $\wedge^*W$ and this representation decomposes into two irreducible ones. The following holds:

-   if $\ell$ is even, then $\wedge^{(\textrm{even})} W$ and $\wedge^{(\textrm{odd})} W$ are the irreducible representations of weights $\omega_{\ell}$ and $\omega_{\ell-1}$ respectively,

-   if $\ell$ is odd, then $\wedge^{(\textrm{even})} W$ and $\wedge^{(\textrm{odd})} W$ are the irreducible representations of weights $\omega_{\ell-1}$ and $\omega_{\ell}$ respectively.

Let $e_I = e_{i_1} \wedge \dots \wedge e_{i_k}, \ I=\{ i_1, \dots, i_k\} \subset \{ 1, \dots, \ell \}$ be a vector of the standard basis for $\wedge^* W$. Then $e_I$ is of the weight $\frac{1}{2} \left(\sum_{i \in I} \epsilon_i - \sum_{j \notin I} \epsilon_j \right)$ and spans the associated one dimensional weight subspace. These are at the same time the only weights of spinor representations. It is clear that $e_{ \{ 1,2, \dots, \ell-1 \} }$ represents a highest weight vector of weigh $\omega_{\ell-1} $, while $e_{ \{1,2, \dots, \ell-1, \ell \} }$ is a highest weight vector of weight $\omega_{\ell}$.

In the reminder of the text we will denote the weight vectors with $e_{\Sigma} = w_{\Sigma}$ 