so the minimal $j$ such that $x_{\gamma}(-j)v_{\Lambda_{\ell-1}} \neq 0$ is $j=2$. This implies $x_{\gamma}(-1)v_{{\Lambda_{\ell-1}}}=0$. On the other hand, for $ \gamma \in \{ \gamma_{\ell}, \gamma_{\underline{\ell-1}}, \dots, \gamma_{\underline{2}} \}$ we have $\langle \gamma, \omega_{\ell-1} \rangle = 0$ so $x_{\gamma}(-1)v_{{\Lambda_{\ell-1}}} \neq 0$. This proves c) and d) is similar. It remains to show e). For all $\gamma \in \Gamma$, we have $\langle \gamma, \omega \rangle =1$ so again the minimal $j$ such that $x_{\gamma}(-j)v_{\Lambda_1} \neq 0$ must be $j=2$. Therefore $x_{\gamma}(-1)v_{\Lambda_1} = 0$ as claimed.

We will say that monomial $x(\pi) \in S({\tilde{\mathfrak{g}}}^{-}_1)$ satisfies *the initial conditions* for $W(\Lambda)$ (or on $v_{\Lambda}$) if it contains neither the $x_{\gamma}(-1)$ elements which annihilate $v_{\Lambda}$ nor the factors of the $(-1)(-1)$ shape which can be expressed as a sum of greater monomials on $v_{\Lambda}$.

Lemma \[poc\_uvj\_pom\] gives us the initial conditions for Feigin-Stoyanovsky’s subspaces of level 1 modules:

-   for $\Lambda = \Lambda_0$, a monomial $x(\pi)$ satisfies the initial conditions if it either doesn’t contain $x_{\gamma}(-1)$ elements or is of the form:

    -   $x(\pi) = \dots (-j)x_{\gamma}(-1)$,  $j \geq 2$, or

    -   $x(\pi) = \dots (-j)x_{\gamma_{\underline{2}}}(-1) x_{\gamma_2}(-1)$,  $j \geq 2$,

-   for $\Lambda = {\Lambda_{\ell-1}}$, a monomial $x(\pi)$ satisfies the initial conditions if it either 