a fixed basis {v1,...,vn} of V , define vj = vj1 ⊗···⊗vjr , any j = (j1,...,jr ) ∈ I(n,r). Then the set {vj : j ∈ I(n,r)} is a basis of V ⊗r . For i = (i1,...,ir ), j = (j1,...,jr ) in I(n,r) write ci,j = ci1,j1 ···cir ,jr . Then Sr acts by place-permutation on I(n,r) and thus on I(n,r)×I(n,r). We have the equality rule: ci,j = ck,l ⇐⇒ (i,j), (k,l) lie in the same Sr -orbit of I(n,r)×I(n,r). If Ω is a set of Sr -orbit representatives then {ci,j : (i,j) ∈ Ω} is a basis of A k (n,r). Let {ξi,j : (i,j) ∈ Ω} be the dual basis of S k (n,r) = A k (n,r)∗.