  In this paper, we first rework B. Kaufman's 1949 paper, "Crystal Statistics.
II. Partition Function Evaluated by Spinor Analysis", by using representation
theory. Our approach leads to a simpler and more direct way of deriving the
spectrum of the transfer matrix for the finite periodic Ising model. We then
determine formulas for the spin correlation functions that depend on the matrix
elements of the induced rotation associated with the spin operator in a basis
of eigenvectors for the transfer matrix. The representation of the spin matrix
elements is obtained by considering the spin operator as an intertwining map.
We exhibit the "new" elements V+ and V- in the Bugrij-Lisovyy formula as part
of a holomorphic factorization of the periodic and anti-periodic summability
kernels on the spectral curve associated with the induced rotation for the
transfer matrix.
