On the adjoint representation of $\mathfrak{sl}_n$ and the Fibonacci numbers

We decompose the adjoint representation of $\mathfrak{sl}_{r+1}=\mathfrak {sl}_{r+1}(\mathbb C)$ by a purely combinatorial approach based on the introduction of a certain subset of the Weyl group called the \emph{Weyl alternation set} associated to a pair of dominant integral weights. The cardinality of the Weyl alternation set associated to the highest root and zero weight of $\mathfrak {sl}_{r+1}$ is given by the $r^{th}$ Fibonacci number. We then obtain the exponents of $\mathfrak {sl}_{r+1}$ from this point of view.