A simple counting argument of the irreducible representations of SU(N) on mixed product spaces

That the number of irreducible representations of the special unitary group SU ( N ) on V ⊗ k (which is also the number of Young tableaux with k boxes) is given by the number of involutions in S k is a well-known result (see, e.g., Knuth in The art of computer programming, volume 3—sorting and searching, 2nd ed, Addison-Wesley, Boston, 1998 and other standard textbooks). In this paper, we present an alternative proof for this fact using a basis of projection and transition operators (Alcock-Zeilinger and Weigert J Math Phys 58(5):051702, 2017 , J Math Phys 58(5):051703, 2017 ) of the algebra of invariants of SU ( N ) on V ⊗ k . This proof is shown to easily generalize to the irreducible representations of SU ( N ) on mixed product spaces V ⊗ m ⊗ V ∗ ⊗ n , implying that the number of irreducible representations of SU ( N ) on a product space V ⊗ m ⊗ V ∗ ⊗ n remains unchanged if one exchanges factors V for V ∗ and vice versa, as long as the total number of factors remains unchanged, c.f. Corollary 1 .