We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite @math -systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland–Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov–Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems.
Abstract The factorized S -matrices as well as the eigenvalues of the transfer matrices for symmetrical degrees of fundamental representations of all Lie groups are presented in the form of the Bethe ansatz in terms of roots systems. As an application the explicit expressions of the S -matrices for the lowest dimension representations of some exceptional Lie groups and the solution of the principal chiral model are calculated.