\leq 1$, $r = \ell-1, \dots, 2$,

-   $b_{\underline{r}} + b_{\underline{r+1}} \dots +
     + b_{r+1} + b_{r-1} + \dots +
    b_2 + a_{\underline{2}} + \dots + a_{\underline{r}} \leq 1$, $r = \ell-1, \dots, 2$.

The proof is straightforward.

Initial conditions {#pocetni_uvjeti}
------------------

We further reduce the spanning set by observing that certain factors $x_{\delta}(-1)$ annihilate some highest weight vectors and as such can not be contained in the monomial basis. Further on, some monomials of the shape $(-1)(-1)$ can be expressed via greater monomials on highest weight vector $v_{\Lambda_0}$. The following lemma describes this cases:

\[poc\_uvj\_pom\] Let $v_{\Lambda_i}$ be the highest weight vector of module $L(\Lambda_i)$, $i=0,1,\ell-1, \ell$. We have:

-   $x_{\underline{\gamma}}(-1)x_{\gamma}(-1)v_{\Lambda_0} = C \cdot x_{\gamma_{\underline{2}}}(-1)x_{\gamma_2}(-1)v_{\Lambda_0} \quad$ for $\gamma \in \Gamma$,

-   $x_{\tau}(-1)x_{\delta}(-1)x_{\gamma}(-1)v_{\Lambda_0}=0 \quad$ for $\tau, \delta, \gamma \in \Gamma$,

-   $x_{\gamma}(-1)v_{\Lambda_{\ell-1}} = 0 \quad$ if and only if $\gamma \in \{\gamma_2, \dots, \gamma_{\ell-1}, \gamma_{\underline{\ell}} \}$,

-   $x_{\gamma}(-1)v_{\Lambda_{\ell}} = 0 \quad$ if and only if $\gamma \in \{\gamma_2, \dots, \gamma_{\ell-1}, \gamma_{\ell} \}$,

-   $x_{\gamma}(-1)v_{\Lambda_1}=0 \quad$ for $\gamma \in \Gamma$.

Claims a) and b) have been proved in Section \[uvjeti\_razlike\]. We show c) using formula . Let $\gamma \in \{\gamma_2, \dots, \gamma_{\ell-1}, \gamma_{\underline{\ell}} \}$. Then $\langle \gamma, \omega_{\ell-1} \rangle = 1$ 