Y - systems and generalized associahedra

The goals of this paper are two-fold. First, we prove, for an arbitrary finite root system Φ, the periodicity conjecture of Al. B. Zamolodchikov [24] that concerns Y -systems, a particular class of functional relations playing an important role in the theory of thermodynamic Bethe ansatz. Algebraically, Y -systems can be viewed as families of rational functions defined by certain birational recurrences formulated in terms of the root system Φ. We obtain explicit formulas for these rational functions, which always turn out to be Laurent polynomials, and prove that they exhibit the periodicity property conjectured by Zamolodchikov. In a closely related development, we introduce and study a simplicial complex ∆(Φ), which can be viewed as a generalization of the Stasheff polytope (also known as associahedron) for an arbitrary root system Φ. In type A, this complex is the face complex of the ordinary associahedron, whereas in type B, our construction produces the Bott-Taubes polytope, or cyclohedron. We enumerate the faces of the complex ∆(Φ), prove that its geometric realization is always a sphere, and describe it in concrete combinatorial terms for the classical types ABCD. The primary motivation for this investigation came from the theory of cluster algebras, introduced in [9] as a device for studying dual canonical bases and total positivity in semisimple Lie groups. This connection remains behind the scenes in the text of this paper, and will be brought to light in a forthcoming sequel1 to [9].