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For a class of multiparameter statistical models based on $N^2\times N^2$braid matrices the eigenvalues of the transfer matrix ${\bf T}^{(r)}$ areobtained explicitly for all $(r,N)$. Our formalism yields them as solutions ofsets of linear equations with simple constant coefficients. The role ofzero-sum multiplets constituted in terms of roots of unity is pointed out andtheir origin is traced to circular permutations of the indices in the tensorproducts of basis states induced by our class of ${\bf T}^{(r)}$ matrices. Therole of free parameters, increasing as $N^2$ with $N$, is emphasizedthroughout. Spin chain Hamiltonians are constructed and studied for all $N$.Inverse Cayley transforms of Yang-Baxter matrices corresponding to our braidmatrices are obtained for all $N$. They provide potentials for factorizable$S$-matrices. Main results are summarized and perspectives are indicated in theconcluding remarks.

Multiparameter statistical models from $N^2\times N^2$ braid matrices:  Explicit eigenvalues of transfer matrices ${\bf T}^{(r)}$, spin chains,  factorizable scatterings for all $N$