Using Kostant's weight multiplicity formula, we describe and enumerate the
terms contributing a nonzero value to the multiplicity of a positive root μ
in the adjoint representation of slr+1​(C), which we
denote L(α~), where α~ is the highest root of
slr+1​(C). We prove that the number of terms
contributing a nonzero value in the multiplicity of the positive root
μ=αi​+αi+1​+⋯+αj​ with 1≤i≤j≤r in
L(α~) is given by the product Fi​⋅Fr−j+1​, where Fn​
is the nth Fibonacci number. Using this result, we show that the
q-multiplicity of the positive root
μ=αi​+αi+1​+⋯+αj​ with 1≤i≤j≤r in the
representation L(α~) is precisely qr−h(μ), where
h(μ)=j−i+1 is the height of the positive root μ. Setting q=1 recovers
the known result that the multiplicity of a positive root in the adjoint
representation of slr+1​(C) is one.Comment: 16 pages, 0 figure