right and two-sided orders and the right and two-sided cells are defined similarly (using multiplication from the right or from both sides) and denoted $\geq_R$, $\geq_J$, $\sim_R$ and $\sim_J$, respectively.

For each left cell $\mathcal{L}$, we have the corresponding [*left cell*]{} $A$-module $C_{\mathcal{L}}$. The module $C_{\mathcal{L}}$ contains a subquotient $L_{\mathcal{L}}$, called the [*special subquotient*]{}, which has composition multiplicity one in $C_{\mathcal{L}}$. This subquotient is defined as the unique subquotient of $C_{\mathcal{L}}$ which contains the Perron-Frobenius eigenvector of $C_{\mathcal{L}}$ for the linear operator $\sum_i a_i$. We have $L_{\mathcal{L}}\cong L_{\mathcal{L}'}$ if $\mathcal{L}$ and $\mathcal{L}'$ belong to the same two-sided cell. The latter justifies the notation $L_{\mathcal{J}}:=L_{\mathcal{L}}$, where $\mathcal{J}$ is the two-sided cell containing $\mathcal{L}$.

An $A$-module $V$ with a fixed basis $\{v_1,v_2,\dots,v_k\}$ is called [*positively based*]{} provided that each $a_i\cdot v_j$ is a linear combination of the $v_m$’s with non-negative real coefficients. A positively based module $V$ is called [*transitive*]{} provided that, for any $v_i$ and $v_j$, there is an $a_s$ such that $v_j$ appears in $a_s\cdot v_i$ with a non-zero coefficient. For each transitive $A$-module $V$, there is a unique two-sided cell $\mathcal{J}(V)$ which is the maximum element (with respect to the two-sided order) in the set of all two-sided cells 