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.
2 @cite_24 . Kerov . proposed and proved the type II formula for @math by the combinatorial method, where the bijection between the Littlewood-Richardson tableaux and the rigged configurations was constructed.
@cite_24
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Techniques developed in the realms of the quantum method of the inverse problem are used to analyze combinatorial problems (Young diagrams and rigged configurations).