Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property

Abstract

We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra $U_q(\hat{\g})$ for any simple Lie algebra g, generalizing the simply-laced case treated in [Kedem 2007]. We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the ``initial cluster seeds'', including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of generalized T-systems with appropriate boundary conditions.

arXiv comment27 pages. Appendices added to include (1) a discussion of generalized T-systems and polynomiality of solutions and (2) associated cluster algebras with coefficients. References added

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• Functional Relations in Solvable Lattice Models I: Functional Relations and Representation Theory
• Cluster algebra structures and semicanonical bases for unipotent groups
• Proof of the combinatorial Kirillov-Reshetikhin conjecture
• Bethe Equation at q=0, Moebius Inversion Formula, and Weight Multiplicities: II. X_n case

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