arXiv subjects

Andrei Zelevinsky

Publications and source records attributed to Andrei Zelevinsky.

At least 37 records · Page 2Linked to original sources

arXiv2006

Nested complexes and their polyhedral realizations

This note which can be viewed as a complement to Alex Postnikov's paper math.CO/0507163, presents a self-contained overview of basic properties of nested complexes and their two dual polyhedral realizations: as complete simplicial fans, and as simple polytopes. 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 hep-th/0111053, math.CO/0202004, on the other side.

math.CO↗

arXiv2006

Laurent expansions in cluster algebras via quiver representations

We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the elements of the canonical basis for the corresponding cluster algebra.

math.RT↗

arXiv2005

Cluster algebras of finite type and positive symmetrizable matrices

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.

math.CO↗

arXiv2005

Quantum cluster algebras: Oberwolfach talk, February 2005

This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (February 6 - 12, 2005). It gives self-contained and simplified definitions of quantum cluster algebras introduced and studied in a joint work with A.Berenstein (math.QA/0404446).

math.QA↗

arXiv2004

Cluster algebras: Notes for the CDM-03 conference

This is an expanded version of the notes of our lectures given at the conference "Current Developments in Mathematics 2003" held at Harvard University on November 21--22, 2003. We present an overview of the main definitions, results and applications of the theory of cluster algebras.

math.RT↗

arXiv2004

Cluster algebras: notes for 2004 IMCC (Chonju, Korea, August 2004)

This is an expanded version of the notes for the two lectures at the 2004 International Mathematics Conference (Chonbuk National University, August 4-6, 2004). The first lecture discusses the origins of cluster algebras, with the focus on total positivity and geometry of double Bruhat cells in semisimple groups. The second lecture introduces cluster algebras and discusses some basic results, open questions and conjectures.

math.RT↗

arXiv2004

Quantum cluster algebras

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in math.RT/0104151; their study continued in math.RA/0208229, math.RT/0305434. This is a family of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.

math.QA↗

arXiv2004

Cluster algebras III: Upper bounds and double Bruhat cells

We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.

math.RT↗

arXiv2003

Positivity and canonical bases in rank 2 cluster algebras of finite and affine types

The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups. In this paper, we introduce and explicitly construct the canonical basis for a special family of cluster algebras of rank 2.

math.RT↗

arXiv2003

Cluster algebras II: Finite type classification

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.

math.RA↗

arXiv2003

Generalized associahedra via quiver representations

We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties of these fans then become easy consequences of the known facts about tilting modules due to K. Bongartz, D. Happel and C. M. Ringel.

math.RT↗

arXiv2002

On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups

We provide an explicit description of symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as connected components of certain varieties. Our main tool is the machinery of twisted generalized minors. They also allow us to present several quasi-commuting coordinate systems on every symplectic leaf. As a consequence, we construct new completely integrable systems on some special symplectic leaves.

math.QA↗

arXiv2002

Polytopal realizations of generalized associahedra

In hep-th/0111053, a complete simplicial fan was associated to an arbitrary finite root system. It was conjectured that this fan is the normal fan of a simple convex polytope (a generalized associahedron of the corresponding type). Here we prove this conjecture by explicitly exhibiting a family of such polytopal realizations.

math.CO↗

arXiv2002

From Littlewood-Richardson coefficients to cluster algebras in three lectures

This is an expanded version of the notes of my three lectures at a NATO Advanced Study Institute ``Symmetric functions 2001: surveys of developments and perspectives" (Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; June 25-July 6, 2001). Lecture I presents a unified expression due to A. Berenstein and the author for generalized Littlewood-Richardson coefficients (= tensor product multiplicities) for any complex semisimple Lie algebra. Lecture II outlines a proof of this result; the main idea of the proof is to relate the LR-coefficients with canonical bases and total positivity. Lecture III introduces cluster algebras, a new class of commutative algebras introduced by S. Fomin and the author in an attempt to create an algebraic framework for canonical bases and total positivity.

math.RT↗

arXiv2001

Y-systems and generalized associahedra

We prove, for an arbitrary finite root system, the periodicity conjecture of Al.B.Zamolodchikov concerning Y-systems, a particular class of functional relations arising 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 underlying root system. In the course of proving periodicity, we obtain explicit formulas for all these rational functions, which turn out to always be Laurent polynomials. In a closely related development, we introduce and study a family of simplicial complexes that can be associated to arbitrary root systems. In type A, our construction produces Stasheff's associahedron, whereas in type B, it gives the Bott-Taubes polytope, or cyclohedron. We enumerate the faces of these complexes, prove that their geometric realization is always a sphere, and describe them in concrete combinatorial terms for the classical types ABCD.

hep-th↗

arXiv2001

The Laurent phenomenon

A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of D$.$Gale and R$.$Robinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by J$.$Propp, N$.$Elkies, and M$.$Kleber.

math.CO↗

arXiv2001

Cluster algebras I: Foundations

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

math.RT↗

arXiv2000

Connected components of real double Bruhat cells

Double Bruhat cells in a semisimple group are intersections of cells in two Bruhat decompositions corresponding to two opposite Borel subgroups. They form a geometric framework for the study of total positivity in semisimple groups; they are also closely related to symplectic leaves in the corresponding Poisson-Lie groups. The term "cells" might be misleading because their topology can be quite non-trivial. As a first step towards understanding this topology, we enumerate the connected components of real double Bruhat cells. This result extends (from the simply-laced case to the general one) and proves the conjecture made in a joint work with B.Shapiro-M.Shapiro-A.Vainshtein; it also extends earlier work by B.Shapiro-M.Shapiro-A.Vainshtein and K.Rietsch.

math.AG↗