and $R$ the corresponding root system. We have a root space decomposition ${\mathfrak{g}}= {\mathfrak{h}}+ \sum_{\alpha \in R} {\mathfrak{g}}_{\alpha}$ and we fix root vectors $x_{\alpha}$. For a minuscule co-weight $\omega$ we denote by $\Gamma$ the set of all roots such that $\alpha(\omega) = 1$. Then

$$\begin{aligned}
 {\mathfrak{g}}= {\mathfrak{g}}_{-1}  \oplus {\mathfrak{g}}_0 \oplus {\mathfrak{g}}_1 \end{aligned}$$

is a ${\mathbb{Z}}$-gradation of ${\mathfrak{g}}$ for ${\mathfrak{g}}_{\pm 1} = \sum_{\alpha \in \Gamma} {\mathfrak{g}}_{\pm \alpha}$, ${\mathfrak{g}}_0 = {\mathfrak{h}}+ \sum_{\omega(\alpha)=0} {\mathfrak{g}}_{\alpha}$

To ${\mathfrak{g}}$ we associate the affine Lie algebra, ${\tilde{\mathfrak{g}}}= {\mathfrak{g}}\otimes {\mathbb{C}}c \otimes {\mathbb{C}}d $, with the canonical central element $c$, degree operator $d$ and fixed real root vectors $x_{\alpha}(n) = x_{\alpha} \otimes t^n$. Gradation of ${\mathfrak{g}}$ induces ${\mathbb{Z}}$-gradation of ${\tilde{\mathfrak{g}}}$:

$${\tilde{\mathfrak{g}}}= {\tilde{\mathfrak{g}}}_{-1}  \oplus {\tilde{\mathfrak{g}}}_0 \oplus {\tilde{\mathfrak{g}}}_1.$$

${\tilde{\mathfrak{g}}}_1 = {\mathfrak{g}}_1 \otimes {\mathbb{C}}[t, t^{-1}]$ has a basis

$$\begin{aligned}
 \{x_{\gamma}(j) \ \vert \ j \in {\mathbb{Z}}, \gamma \in \Gamma \ \}.\end{aligned}$$

Let $L(\Lambda)$ be a standard ${\tilde{\mathfrak{g}}}$-module of level $k=\Lambda(c)$ and let $v_{\Lambda}$ be its highest weight vector. A Feigin-Stoyanovsky’s type subspace is a ${\tilde{\mathfrak{g}}}_1$-submodule of $L(\Lambda)$ generated with $v_{\Lambda}$, $$\begin{aligned}
W(\Lambda) = U({\tilde{\mathfrak{g}}}_1) \cdot v_{\Lambda}.\end{aligned}$$ The goal is to find a combinatorial basis of $W(\Lambda)$ whose elements are of the form $x(\pi) v_{\Lambda}$ where $x(\pi)$ are *monomials* consisting of $\{ \ 