Spin Representations and Centralizer Algebras for the Spinor Groups

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 two kinds of linear basis in CPk (one comes from invariant theory and the other from representation theory), both of which are parameterized by the ’generalized Brauer diagrams’. We develop analogous argument to the original Brauer centralizer algebra for O(N) and determine the transformation matrices between the above two basis and give the multiplication rules of those basis. Finally we define the subspaces in ∆ ⊗⊗ k V , on which the symmetric group Sk and Pin(N) or Spin(N) act as a dual pair.