Communications in Mathematical Physics

, Volume 227, Issue 1, pp 155–190

The Canonical Solutions of the Q-Systems¶ and the Kirillov–Reshetikhin Conjecture


  • Atsuo Kuniba
    • Institute of Physics, University of Tokyo, Tokyo 153-8902, Japan. E-mail:
  • Tomoki Nakanishi
    • Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan.¶E-mail:
  • Zengo Tsuboi
    • Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan.¶E-mail:

DOI: 10.1007/s002200200631

Cite this article as:
Kuniba, A., Nakanishi, T. & Tsuboi, Z. Commun. Math. Phys. (2002) 227: 155. doi:10.1007/s002200200631


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 $Q$-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.

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© Springer-Verlag Berlin Heidelberg 2002