We pursue an analogy of the Schur-Weyl reciprocity for the spinor groups and pick up the irreducible spin representations in the tensor space ∆ ⊗⊗ k V . Here ∆ is the fundamental representation of Pin(N) and V is the natural (vector) representation of the orthogonal group O(N). We consider the centralizer algebra CPk = EndPin(N)(∆ ⊗⊗k V ) for Pin(N), the double covering group of O(N) and define...