On the adjoint representation of 𝔰𝔩_n and the Fibonacci numbers

We decompose the adjoint representation of 𝔰𝔩_r+1=𝔰𝔩_r+1(ℂ) by a purely combinatorial approach based on the introduction of a certain subset of the Weyl group called the 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 𝔰𝔩_r+1 is given by the r^th Fibonacci number. We then obtain the exponents of 𝔰𝔩_r+1 from this point of view.