Simplified bifurcation analysis of the sixteen vertex model, based on the standard representation of equivalence classes of models

Abstract

The search for bifurcation points in the parameter space of sixteen vertex models as described in three papers by Gaaff and Hijmans1) and Hijmans and Schram2,3), canbe considerably simplified and clarified by representing the equivalence classes of models having the same partition function by their “standard models”, instead of by their “normal models”. The preference for the standard model arises from the fact that it is characterized by parameters belonging to the irreducible representations of the symmetry group of the partition function, whereas the parameters defining the “normal” model belong to a mixed representation. The standard model defined in terms of 10 characteristic parameters is unique apart from a finite “stargroup” of 96 operations arising from the freedom to permute the eigenvalues and eigenvectors of a characteristic (4 X 4)-matrix, Qss. Bifurcations occur whenever the set of 10 characteristic parameters is invariant with respect to a non-trivial “little group” of operations from the “stargroup”.