as $\underline{\gamma} = \epsilon_1 \mp \epsilon_j$. We will also sometimes call $\gamma_{\underline{i}}$, $i=2, \dots, \ell$, *negative* and $\gamma_i$, $i=2, \dots, \ell$, *positive colors*. In further text we use the label $x(\pi)$ for the monomials from $U({\tilde{\mathfrak{g}}}_1)$.

Next we define *the shape* of the partition $\pi$. Partition $\pi : \{ x_{\delta}(-n) \ \vert \ \delta \in \Gamma, n \in {\mathbb{N}}\}$ (i.e. monomial $x(\pi)$) is of the shape $$\begin{aligned}
& s(\pi)  :  {\mathbb{Z}}\longrightarrow {\mathbb{Z}}_{+} \\
& s(\pi)(j)  =  \sum_{\delta \in \Gamma} \pi(x_{\delta}(-j))\end{aligned}$$ which can be graphically displayed.

Shape of the monomial $x(\pi)=x_{\delta_{4}}(-4)x_{\delta_{3}}(-4)x_{\delta_{2}}(-3)x_{\delta_{1}}(-2)$ looks like:

(100,50)(-30,-25) (0,0)(0,10)[3]{}[(1,0)[40]{}]{} (0,-10)(0,10)[1]{}[(1,0)[30]{}]{} (0,-20)(0,10)[1]{}[(1,0)[20]{}]{} (0,-20)(0,10)[1]{}[(0,1)[40]{}]{} (10,-20)(0,10)[1]{}[(0,1)[40]{}]{} (20,-20)(0,10)[1]{}[(0,1)[40]{}]{} (30,-10)(0,10)[1]{}[(0,1)[30]{}]{} (40,0)(0,10)[1]{}[(0,1)[20]{}]{}

We define that $s(\pi) < s(\pi')$ if there exist $j_0 \in {\mathbb{N}}$ such that $s(\pi)(j) = s(\pi')(j)$ for $j < j_0$ and $s(\pi)(j_0) < s(\pi')(j_0)$.

We further define order on the set of colors, i.e. on $\Gamma$. We set: $$\gamma_2 > \gamma_3 > \dots > \gamma_{\ell} > \gamma_{\underline{\ell}} > \dots > \gamma_{\underline{2}}.$$

Now, on the set of factors $x_{\gamma}(-j)$ we define

$$x_{\delta}(-i) < x_{\tau}(-j) \qquad \textrm{if} \qquad \left \{ \begin{array}{l}
-i<-j \textrm{\quad or}\\
i=j \textrm{\quad i \quad} \delta < \tau.
\end{array} \right.$$ Since elements from $U({\tilde{\mathfrak{g}}}_1) = S({\tilde{\mathfrak{g}}}_1)$ commute, we can assume from now on that elements in monomial 