We decompose the adjoint representation of slr+1​=slr+1​(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 slr+1​ is given by the rth Fibonacci number. We
then obtain the exponents of slr+1​ from this point of view.Comment: 9 page