Symmetric Tensors of the Family

Abstract

This chapter focuses on the symmetrie tensors of the α family. The latest achievement concerning solvable models is particularly important. Sato's theory of the universal Grassmann manifold revealed the intrinsic mechanism of exact solvability of the soliton equations. Exact solvability in statistical mechanics and quantum field theory is also of primary importance. As for lattice models, an extrinsic criterion is known for exact solvability. That is the commutativity of transfer matrices, or its local version called the Yang-Baxter equation or the star-triangle relation. Its role is similar to that of the Lax pair in the soliton theory. The chapter discusses fusion, symmetrization, and restriction. The proof of the star-triangle relation is given by showing the finiteness of the unwanted terms. Various propositions and lemmas are also discussed in the chapter.