Y-systems and generalized associahedra

By Sergey Fomin, Andrei Zelevinsky. 2001

568 citations recorded by the source corpus.

The goals of this paper are two-fold. First, we prove, for an arbitrary finite root system D, 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 (. 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 A(b), which can be viewed as a generalization of the Stasheff polytope (also known as associahedron) for an arbitrary root system (D. 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 A(@), 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].

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0170a6589f3e5e53a7454202c559d8cbcfee3a14

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Citations in the corpus

An Update on the Hirsch Conjecture (2009, background): “Polytopes with special symmetries, such as regular ones and variations of them arise naturally from Coxeter groups and other algebraic structures [6, 25].”
Quivers with relations arising from clusters $(A_n$ case) (2004, unknown): “Let Φ≥−1 be the set of almost positive roots and τ+ and τ− the involutions on Φ≥−1 defined by Fomin and Zelevinsky in [FZ03b].”
Categorification of Skew-symmetrizable Cluster Algebras (2009, background): “Since their emergence, cluster algebras have generated a lot of interest, coming in particular from their links with many other subjects: combinatorics (see for instance [8, 22]), Poisson geometry [35, 36], integrable systems [ 25 ], Teichmuller spaces [20], and, last but not least, representations ”
Cluster algebras and triangulated surfaces. Part I: Cluster complexes (2006, background): “3 holds for any cluster algebra of finite type; its cluster complex is the generalized associahedron [19] of the corresponding type.”
Tilting theory and cluster combinatorics (2004, background): “... properties of the dual canonical basis of the quantised enveloping algebra of a semisimple Lie algebra over C, and that it should model the (classical and quantised) coordinate rings of varieties associated to algebraic groups (now shown in several cases — see [BFZ02], [Sco03].), with particular”
Periodicities of T-systems and Y-systems (2008, methodology): “Cr, F4, G2) with the enumeration I in Figure 1. This is the Y-system for nonsimply laced Xr considered in [ FZ3 , Kel2].”
Surveys in Combinatorics 2011: The cyclic sieving phenomenon: a survey (2010, methodology): “responding CSP remains elusive. There is another object enumerated by Cat(W;1) when W is a Coxeter group, namely facets of cluster complexes. Cluster complexes were introduced by Fomin and Zelevinsky [22] motivated by their theory of cluster algebras. To deﬁne them, we need some background on root s”
CLUSTER MUTATION VIA QUIVER REPRESENTATIONS (2004, methodology): “They have been used to provide the solution [ FZ3 ] of a conjecture of Zamolodchikov concerning Y -systems, a class of functional relations important in the theory of the thermodynamic Bethe Ansatz, as well as solution [FZ4] of various recurrence problems involving Laurent polynomials, including a c”
Higher Segal Spaces (2012, background): “Such transitive systems appear in the theory of cluster algebras (see [FZ03] [FG06] [GSV10]).”
Semicanonical bases and preprojective algebras (2004, methodology): “ntains all the information. 12.2. The graphs G◦ 2 and G3◦ have been determined by Berenstein and Zelevinsky [4]. They are respectively dual to an associahedron of type A1 and A3 in the terminology of [11, 19, 20]. In 20.5 we display the graph G◦ 4, which has 36 vertices (corresponding to the multise”
PERIODICITIES OF T AND Y-SYSTEMS, DILOGARITHM IDENTITIES, AND CLUSTER ALGEBRAS I: TYPE Br (2010, methodology): “r more generally, the exchange relations of the coeﬃcients in the tropical semiﬁeld (called principal coeﬃcients in [FZ3]) of a given cluster algebra. In fact, it was already used by Fomin-Zelevinsky [FZ2] as a main tool in the proof of the periodicity of the Y-systems in the simply laced case at le”
Coxeter Complexes and Graph-Associahedra (2004, background): “Remark. The generalized associahedra of Fomin and Zelevinsky [ 11 ] are fundamentally different than graph-associahedra.”
Defining an m-cluster category (2006, background): “ there must be an Mi of each colour. This theorem has some further consequences related to the topology of ∆m(Φ) which will be worked out in a subsequent paper. Interpretation intypeA As described in [FZ], clusters in type An correspond to triangulations of a convex (n+3)-gon, where by triangulation”
Universal geometric cluster algebras (2012, unknown): “The classification [9,11] of cluster algebras of finite type says in particular that B = [ 0 a b 0 ] is of finite type if and only if ab ≥ −3.”
Grassmannians and Cluster Algebras (2003, background): “expansion above. In particular, the variable x i of the Dynkin cluster is paired with the negative simple root −α i. This bijection relies on two piece-wise linear involutions τ+ and τ−, developed in [6], of the root lattice Q obtained by tropicalizing the exchange relations for the Dynkin cluster. ”
A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, background): “Cluster algebras quickly gained popularity in various branches of mathematics, for example Poisson geometry (e.g. [16]), higher Teichm¨uller theory (e.g. [8]), combinatorics (e.g. [31]), integrable systems (e.g. [13]), etc.”
Nested complexes and their polyhedral realizations (2005, background): “Most of the results are not new; our aim is to bring into focus a striking similarity between nested complexes and associated fans and polytopes on one side, and cluster complexes and generalized associahedra introduced and studied in [7, 2] on the other side.”
Cluster algebras IV: Coefficients (2006, background): “(This is well defined by Proposition 7.8.)”