terms are said to satisfy *the difference conditions* (or $DC$ in short) for $W(\Lambda)$.

These monomials can be described as: $$x_{\delta}(-i) x_{\gamma}(-j) \ \textrm{satisfies $DC$ if} \ \left\{ \begin{array}{lll} -i \leq -j-2, & \\ -i= -j-1 & \mbox{and $\delta < \gamma$ or} \\ & \mbox{$\delta = \gamma_{\ell}$ and $\gamma=\gamma_{\underline{\ell}},$} \\ -i=-j & \mbox{and $\delta = \gamma_{\underline{2}}$, $\gamma=\gamma_2$.} \end{array} \right.$$

Let $b_r$ (resp. $b_{\underline{r}}$) be the number of $x_{\gamma_r}(-j-1)$ (resp. $x_{\gamma_{\underline{r}}}(-j-1)$), and $a_r$ (resp. $a_{\underline{r}}$) of $x_{\gamma_r}(-j)$ (resp. $x_{\gamma_{\underline{r}}}(-j)$) factors in monomial $x(\pi)$. Then the difference conditions can be also written as:

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

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

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

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

-   $b_{\underline{r+1}} + \dots + b_{\underline{\ell}} + b_{\ell} + \dots +
    b_2 + a_{\underline{2}} + \dots + a_{\underline{r}} 