an analysis of spectral properties of some integral matrices. We observe that Fibonacci polynomials are connected to minimal polynomials of certain integral matrices which encode the combinatorics of simple transitive $2$-representations. Using the factorization of Fibonacci polynomials over $\mathbb{Q}$, see [@Lev], and estimating the values of the maximal possible eigenvalues for the matrices which encode the action of Soergel bimodules corresponding to simple reflections in simple transitive $2$-representations, allows us to deduce that the only possibility for these matrices are the ones appearing in cell $2$-representations.

Apart from Theorem \[thmintro\], we also prove the following general result. We refer to Subsection \[s2.4\] and [@CM Subsection 3.2] for the definition of the apex and to Subsection \[s2.3\] and [@MM2 Subsection 4.2] for the definition of the abelianization $\overline{\mathbf{M}}$ of a $2$-representation $\mathbf{M}$.

\[thmmintrotwo\] Let $\mathbf{M}$ be a simple transitive $2$-representation of a fiat $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$ with apex $\mathcal{I}$. Then, for every $1$-morphism $\mathrm{F}\in \mathcal{I}$ and every object $X$ in any $\overline{\mathbf{M}}(\mathtt{i})$, the object $\mathrm{F}\,X$ is projective. Moreover, $\overline{\mathbf{M}}(\mathrm{F})$ is a projective functor.

Our proof of this statement is based crucially on the results from [@KM2]. Theorem \[thmmintrotwo\] will be very useful in further studies of simple transitive $2$-representations. It applies directly 