  We pursue an analogy of the Schur-Weyl reciprocity for the spinor groups and
pick up the irreducible spin representations in the tensor space $\Delta
\textstyle{\bigotimes \bigotimes^k V}$. Here $\Delta$ 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 $\mathbf{CP_k} =
Pin(N)(\Delta \textstyle{\bigotimes \bigotimes^k V})$ for $Pin(N)$, the double
covering group of O(N) and define two kinds of linear basis in $\mathbf{CP_k}$
(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
$\Delta \textstyle{\bigotimes \bigotimes^k V}$, on which the symmetric group
$\frak{S}_k$ and $Pin(N)$ or $Spin(N)$ act as a dual pair.
