On biunitary permutation matrices and some subfactors of index 9

This paper is devoted to a study of the subfactors arising from vertex models constructed out of `biunitary' matrices which happen to be permutation matrices. After a discussion on the computation of the higher relative commutants of the associated subfactor (in the members of the tower of Jones' basic construction), we discuss the principal graphs of these subfactors for small sizes $N=k \leq 3$ of the vertex model. Of the 18 possibly inequivalent such biunitary matrices when $N = 3$, we compute the principal graphs completely in 15 cases, all of which turn out to be finite. In the last section, we prove that two of the three remaining cases lead to subfactors of infinite depth and discuss their principal graphs.