On four-weight spin models and their gauge transformations

Jaeger, François

Journal of Algebraic Combinatorics, Tome 11 (2000) no. 3, pp. 241-268.

Résumé

Summary: We study the four-weight spin models (W $_{1}$, W $_{2}$, W $_{3}$, W $_{4}$) introduced by Eiichi and Etsuko Bannai (Pacific J. of Math, to appear). We start with the observation, based on the concept of special link diagram, that two such spin models yield the same link invariant whenever they have the same pair (W $_{1}$, W $_{3}$), or the same pair (W $_{2}$, W $_{4}$). As a consequence, we show that the link invariant associated with a four-weight spin model is not sensitive to the full reversal of orientation of a link. We also show in a similar way that such a link invariant is invariant under mutation of links. Next, we give an algebraic characterization of the transformations of four-weight spin models which preserve W $_{1}$, W $_{3}$ or preserve W $_{2}$, W $_{4}$. Such $ldquo$gauge transformations $rdquo$ correspond to multiplication of W $_{2}$, W $_{4}$ by permutation matrices representing certain symmetries of the spin model, and to conjugation of W $_{1}$, W $_{3}$ by diagonal matrices. We show for instance that up to gauge transformations, we can assume that W $_{1}$, W $_{3}$ are symmetric.

Détail

Keywords: spin model, link invariant, association scheme, Bose-mesner algebra