  It is a well known result that the number of irreducible representations of
SU(N) on a tensor product containing k factors of a vector space V is given by
the number of involutions in the symmetric group on k letters. In this paper,
we present an alternative proof for this fact using a basis of projection and
transition operators of the algebra of invariants of SU(N). This proof easily
generalizes to the irreducible representations of SU(N) on mixed tensor product
spaces (consisting of factors of V as well as its dual space). This implies
that the number of irreducible representation of SU(N) on such a space remains
unchanged if one exchanges factors V for its dual and vice versa, as long as
the total number of factors remains unchanged.
