whose elements do not annihilate $V$. The two-sided cell $\mathcal{J}(V)$ is called the [*apex*]{} of $V$. The two-sided cell $\mathcal{J}(V)$ is [*idempotent*]{} in the sense that it contains some $a_i$, $a_j$ and $a_s$ (non necessarily different) such that $\gamma_{i,j}^s\neq 0$. The simple subquotient $L_{\mathcal{J}(V)}$ has composition multiplicity one in $V$ and is also called the [*special*]{} subquotient of $V$.

Decategorification {#s2.8}
------------------

For a finitary $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$, we denote by $A_{{\scc\mbox{C}\hspace{1.0pt}}}$ the complexification of the Grothendieck decategorification of ${\sc\mbox{C}\hspace{1.0pt}}$. The algebra $A_{{\scc\mbox{C}\hspace{1.0pt}}}$ is positively based with respect to the [*natural basis*]{} given by the classes of indecomposable $1$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}$, see [@Ma Subsection 1.2] or [@KM2 Subsection 2.5] for details.

Given a $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$, the Grothendieck decategorification of $\mathbf{M}$ is, naturally, a positively based $A_{{\scc\mbox{C}\hspace{1.0pt}}}$-module. Moreover, the latter is transitive if and only if $\mathbf{M}$ is transitive. This allows us to speak about the [*apex*]{} of $\mathbf{M}$ in the obvious way, see [@CM Subsection 3.2] for details.

We denote by $\mathrm{F}_{\mathrm{tot}}$ a multiplicity free direct sum of all indecomposable $1$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}$. Given a $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$, we set $\mathtt{M}_{\mathrm{tot}}:=\Lparen \mathrm{F}_{\mathrm{tot}}\Rparen$. Then $\mathbf{M}$ is transitive if and only if all coefficients of $\mathtt{M}_{\mathrm{tot}}$ are non-zero (and thus 