Andrei Zelevinsky

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Cluster algebras I: Foundations

2001-12-28

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. 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.

Induced representations of reductive ${\germ p}$-adic groups. I

1977-01-01

I. N. Bernstein , Andrei Zelevinsky

© Gauthier-Villars (Editions scientifiques et medicales Elsevier), 1977, tous droits reserves. L’acces aux archives de la revue « Annales scientifiques de l’E.N.S. » (http://www. elsevier.com/locate/ansens), implique l’accord avec les conditions … © Gauthier-Villars (Editions scientifiques et medicales Elsevier), 1977, tous droits reserves. L’acces aux archives de la revue « Annales scientifiques de l’E.N.S. » (http://www. elsevier.com/locate/ansens), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

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Cluster algebras II: Finite type classification

2003-05-16

Sergey Fomin , Andrei Zelevinsky

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Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$

1980-01-01

Andrei Zelevinsky

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Cluster algebras IV: Coefficients

2007-01-01

Sergey Fomin , Andrei Zelevinsky

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial … We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of ‘principal’ coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parameterizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parameterizations, some proven and some conjectural, suggest links to duality conjectures of Fock and Goncharov. The coefficient dynamics leads to a natural generalization of Zamolodchikov's -systems, previously known in finite type only, and sharpen the periodicity result from an earlier paper. For cluster algebras of finite type, we identify a canonical ‘universal’ choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.

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Cluster algebras III: Upper bounds and double Bruhat cells

2005-01-15

Arkady Berenstein , Sergey Fomin , Andrei Zelevinsky

We develop a new approach to cluster algebras, based on the notion of an upper cluster algebra defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established … We develop a new approach to cluster algebras, based on the notion of an upper cluster algebra defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [7], 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 an upper cluster algebra explicitly defined in terms of relevant combinatorial data.

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Quivers with potentials and their representations I: Mutations

2008-06-03

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

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Double Bruhat cells and total positivity

1999-01-01

Sergey Fomin , Andrei Zelevinsky

We study the totally nonnegative variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript greater-than-or-equal-to 0"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{\ge 0}</mml:annotation> </mml:semantics> … We study the totally nonnegative variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript greater-than-or-equal-to 0"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{\ge 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a semisimple algebraic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These varieties were introduced by G. Lusztig, and include as a special case the variety of unimodular matrices of a given order whose all minors are nonnegative. The geometric framework for our study is provided by intersecting <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript greater-than-or-equal-to 0"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{\ge 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with double Bruhat cells (intersections of cells of the two Bruhat decompositions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to opposite Borel subgroups).

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Quivers with potentials and their representations II: Applications to cluster algebras

2010-02-08

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. … We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the “Cluster algebras IV” paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors</italic>, and a family of integer polynomials called <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polynomials</italic>. In the case of skew-symmetric exchange matrices we find an interpretation of these <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polynomials made in loc. cit.

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Quantum cluster algebras

2004-09-30

Arkady Berenstein , Andrei Zelevinsky

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Tensor product multiplicities, canonical bases and totally positive varieties

2001-01-01

Arkady Berenstein , Andrei Zelevinsky

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Parametrizations of Canonical Bases and Totally Positive Matrices

1996-09-01

Arkady Berenstein , Sergey Fomin , Andrei Zelevinsky

The Laurent Phenomenon

2002-02-01

Sergey Fomin , Andrei Zelevinsky

Generalized Euler integrals and A-hypergeometric functions

1990-12-01

Israel M. Gelfand , Mikhail Kapranov , Andrei Zelevinsky

Polytopal Realizations of Generalized Associahedra

2002-12-01

Frédéric Chapoton , Sergey Fomin , Andrei Zelevinsky

Abstract We prove polytopality of the generalized associahedra introduced in [5]. Abstract We prove polytopality of the generalized associahedra introduced in [5].

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Total Positivity: Tests and Parametrizations

2000-12-01

Sergey Fomin , Andrei Zelevinsky

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Generalized associahedra via quiver representations

2003-06-10

Bethany Marsh , Markus Reineke , Andrei Zelevinsky

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 … 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.

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Total positivity in Schubert varieties

1997-03-31

Arkady Berenstein , Andrei Zelevinsky

We extend the results of [2] on totally positive matrices to totally positive elements in arbitrary semisimple groups. We extend the results of [2] on totally positive matrices to totally positive elements in arbitrary semisimple groups.

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Polyhedral Realizations of Crystal Bases for Quantized Kac–Moody Algebras

1997-10-01

Toshiki Nakashima , Andrei Zelevinsky

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Chow polytopes and general resultants

1992-07-01

Mikhail Kapranov , Bernd Sturmfels , Andrei Zelevinsky

Quotients of toric varieties

1991-03-01

Mikhail Kapranov , Bernd Sturmfels , Andrei Zelevinsky

On tropical dualities in cluster algebras

2012-01-01

Tomoki Nakanishi , Andrei Zelevinsky

We study two families of integer vectors playing a crucial part in the structural theory of cluster algebras: the g-vectors parameterizing cluster variables, and the c-vectors parameterizing the coefficients.We prove … We study two families of integer vectors playing a crucial part in the structural theory of cluster algebras: the g-vectors parameterizing cluster variables, and the c-vectors parameterizing the coefficients.We prove two identities relating these vectors to each other.The proofs depend on the sign-coherence assumption for c-vectors that still remains unproved in full generality.

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Tensor product multiplicities and convex polytopes in partition space

1988-01-01

Arkady Berenstein , Andrei Zelevinsky

Cluster algebras: Notes for the CDM-03 conference

2003-01-01

Sergey Fomin , Andrei Zelevinsky

We present an overview of the main definitions, results and applications of the theory of cluster algebras. We present an overview of the main definitions, results and applications of the theory of cluster algebras.

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Positivity and Canonical Bases in Rank 2 Cluster Algebras of Finite and Affine Types

2004-01-01

P.J. Sherman , Andrei Zelevinsky

The main motivation for the study of cluster algebras initiated in [4,6,1] was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups.In this … The main motivation for the study of cluster algebras initiated in [4,6,1] 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.ju-bi-lee 1 : a year of emancipation and restoration provided by ancient Hebrew law to be kept every 50 years by the emancipation of Hebrew slaves, restoration of alienated lands to their former owners, and omission of all cultivation of the land 2 a : a special anniversary; especially

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Maximal Minors and Their Leading Terms

1993-03-01

Bernd Sturmfels , Andrei Zelevinsky

String bases for quantum groups of type 𝐴ᵣ

1993-08-18

Arkady Berenstein , Andrei Zelevinsky

This is the quantum deformation (or q−deformation) of the algebra of polynomial functions on the group Nr+1 of upper unitriangular (r + 1) × (r + 1) matrices. In this … This is the quantum deformation (or q−deformation) of the algebra of polynomial functions on the group Nr+1 of upper unitriangular (r + 1) × (r + 1) matrices. In this paper we introduce and study a class of bases in Ar which we call string bases. The main example of a string basis is given as follows. Let U+ = U+,r be the quantized universal enveloping algebra of the Lie algebra nr+1 of Nr+1 (see e.g., [10]). Then Ar is seen to be the graded dual of U+, and the basis in Ar dual to the Lusztig’s canonical basis in U+ is a string basis. The string bases are defined by means of so called string axioms which we find easier to work with than the axioms imposed by Lusztig or those by Kashiwara. The string axioms seem to be rather strong, and it is even conceivable that the string basis is unique but we do not know this in general. We prove the uniqueness of a string basis for A2 and A3. The main advantage of string bases is that they seem to have nicer multiplicative properties than the canonical basis. We say that x, y ∈ Ar quasicommute if xy = qyx for some integer n. We conjecture that every string basis B has the following property: two elements b, b′ ∈ B quasicommute if and only if qNbb′ ∈ B for some integer N . We prove this for A2 and A3, and provide some supporting evidence for general Ar. Before giving precise formulations of the results we would like to put this work into historic context. Let g be a semisimple complex Lie algebra of rank r with fixed Cartan decomposition g = n− ⊕ h ⊕ n+. Our main motivation was to study “good bases” in irreducible g−modules. Good bases were introduced independently in [5] and [1]. Let P ⊂ h∗ denote the weight lattice of g, and P+ ⊂ P denote the semigroup of dominant integral weights, i.e., weights of the form n1ω1 + . . . + nrωr, where ω1, . . . , ωr are fundamental weights of g, and n1, . . . , nr are nonnegative integers. For λ ∈ P+ let Vλ denote the

Laurent Expansions in Cluster Algebras via Quiver Representations

2006-01-01

Ph. Caldero , Andrei Zelevinsky

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 … 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.

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Multigraded Resultants of Sylvester Type

1994-01-01

Bernd Sturmfels , Andrei Zelevinsky

Multiple Flag Varieties of Finite Type

1999-01-01

Péter Magyar , Jerzy Weyman , Andrei Zelevinsky

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Canonical bases for the quantum group of type Ar and piecewise-linear combinatorics

1996-03-01

Arkady Berenstein , Andrei Zelevinsky

This work was motivated by the following two problems from the classical representation theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal … This work was motivated by the following two problems from the classical representation theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal only with the Ar case, we formulate them in this generality). 1. Construct a “good” basis in every irreducible finite-dimensional slr+1-module Vλ, which “materializes” the Littlewood-Richardson rule. A precise formulation of this problem was given in [3]; we shall explain it in more detail a bit later. 2. Construct a basis in every polynomial representation of GLr+1, such that the maximal element w0 of the Weyl group Sr+1 (considered as an element of GLr+1) acts on this basis by a permutation (up to a sign), and explicitly compute this permutation. This problem is motivated by recent work by John Stembridge [10] and was brought to our attention by his talk at the Jerusalem Combinatorics Conference, May 1993.

A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence

1981-03-01

Andrei Zelevinsky

Nested Complexes and their Polyhedral Realizations

2006-01-01

Andrei Zelevinsky

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: … 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.

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Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors

2008-09-11

Shih-Wei Yang , Andrei Zelevinsky

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CLUSTER ALGEBRAS OF FINITE TYPE AND POSITIVE SYMMETRIZABLE MATRICES

2006-06-01

Michael Barot , Christof Geiß , Andrei Zelevinsky

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 … 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.

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Introduction to Cluster Algebras. Chapters 1-3

2016-01-01

Sergey Fomin , Lauren Williams , Andrei Zelevinsky

This is a preliminary draft of Chapters 1-3 of our forthcoming textbook "Introduction to Cluster Algebras." This installment contains: Chapter 1. Total positivity Chapter 2. Mutations of quivers and matrices … This is a preliminary draft of Chapters 1-3 of our forthcoming textbook "Introduction to Cluster Algebras." This installment contains: Chapter 1. Total positivity Chapter 2. Mutations of quivers and matrices Chapter 3. Clusters and seeds

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Singularities of hyperdeterminants

1996-01-01

Jerzy Weyman , Andrei Zelevinsky

We study the singular locus of the variety of degenerate hypermatrices of an arbitrary format. Our main result is a classification of irreducible components of the singular locus. Equivalently, we … We study the singular locus of the variety of degenerate hypermatrices of an arbitrary format. Our main result is a classification of irreducible components of the singular locus. Equivalently, we classify irreducible components of the singular locus for the projectively dual variety of a product of several projective spaces taken in the Segre embedding.

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Cluster algebras I: Foundations

2001-01-01

Sergey Fomin , Andrei Zelevinsky

Newton polytopes of the classical resultant and discriminant

1990-12-01

Israel M. Gelfand , Mikhail Kapranov , Andrei Zelevinsky

Symplectic Multiple Flag Varieties of Finite Type

2000-08-01

Péter Magyar , Jerzy Weyman , Andrei Zelevinsky

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Greedy elements in rank 2 cluster algebras

2012-12-16

Kyungyong Lee , Li Li , Andrei Zelevinsky

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A correction to the paper ?hypergeometric functions and toric varieties?

1994-01-01

I. M. Gel'fand , Mikhail Kapranov , Andrei Zelevinsky

Littlewood-Richardson semigroups

1999-01-01

Andrei Zelevinsky

This note is an extended abstract of my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. We discuss the problem of finding an explicit … This note is an extended abstract of my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. We discuss the problem of finding an explicit description of the semigroup $LR_r$ of triples of partitions of length $\leq r$ such that the corresponding Littlewood-Richardson coefficient is non-zero. After discussing the history of the problem and previously known results, we suggest a new approach based on the ``polyhedral'' combinatorial expressions for the Littlewood-Richardson coefficients.

Cluster algebras IV: Coefficients

2006-01-01

Sergey Fomin , Andrei Zelevinsky

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial … We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of "principal" coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parametrizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parametrizations, some proven and some conjectural, suggest links to duality conjectures of V.Fock and A.Goncharov [math.AG/0311245]. The coefficient dynamics leads to a natural generalization of Al.Zamolodchikov's Y-systems. We establish a Laurent phenomenon for such Y-systems, previously known in finite type only, and sharpen the periodicity result from [hep-th/0111053]. For cluster algebras of finite type, we identify a canonical "universal" choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.

Semicanonical Basis Generators of the Cluster Algebra of Type $A_1^{(1)}$

2007-01-19

Andrei Zelevinsky

We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these … We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these elements was given by P.Caldero and the author. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp. The original argument by P.Caldero and the author used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton. This note provides a quick, self-contained and completely elementary alternative proof of the same results.

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Triangular Bases in Quantum Cluster Algebras

2012-12-17

Arkady Berenstein , Andrei Zelevinsky

Journal Article Triangular Bases in Quantum Cluster Algebras Get access Arkady Berenstein, Arkady Berenstein 1Department of Mathematics, University of Oregon, Eugene, OR 97403, USA Search for other works by this … Journal Article Triangular Bases in Quantum Cluster Algebras Get access Arkady Berenstein, Arkady Berenstein 1Department of Mathematics, University of Oregon, Eugene, OR 97403, USA Search for other works by this author on: Oxford Academic Google Scholar Andrei Zelevinsky Andrei Zelevinsky 2Department of Mathematics, Northeastern University, Boston, MA 02115, USA Correspondence to be sent to: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2014, Issue 6, 2014, Pages 1651–1688, https://doi.org/10.1093/imrn/rns268 Published: 17 December 2012 Article history Received: 12 November 2012 Accepted: 14 November 2012 Published: 17 December 2012

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Y-systems and generalized associahedra

2001-01-01

Sergey Fomin , Andrei Zelevinsky

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, … 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.

Multiplicative properties of projectively dual varieties

1994-12-01

Jerzy Weyman , Andrei Zelevinsky

Totally nonnegative and oscillatory elements in semisimple groups

2000-06-07

Sergey Fomin , Andrei Zelevinsky

We generalize the well known characterizations of totally nonnegative and oscillatory matrices, due to F. R. Gantmacher, M. G. Krein, A. Whitney, C. Loewner, M. Gasca, and J. M. Peña … We generalize the well known characterizations of totally nonnegative and oscillatory matrices, due to F. R. Gantmacher, M. G. Krein, A. Whitney, C. Loewner, M. Gasca, and J. M. Peña to the case of an arbitrary complex semisimple Lie group.

Representations of Quivers of TypeAand the Multisegment Duality

1996-02-01

H. Knight , Andrei Zelevinsky

Introduction to Cluster Algebras. Chapter 7

2021-01-01

Sergey Fomin , Lauren Williams , Andrei Zelevinsky

This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. … This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. Chapter 6 has been posted as arXiv:2008.09189. This installment contains: Chapter 7. Plabic graphs

Introduction to Cluster Algebras. Chapter 6

2020-01-01

Sergey Fomin , Lauren Williams , Andrei Zelevinsky

This is a preliminary draft of Chapter 6 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. … This is a preliminary draft of Chapter 6 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. This installment contains: Chapter 6. Cluster structures in commutative rings

Introduction to Cluster Algebras. Chapters 4-5

2017-01-01

Sergey Fomin , Lauren Williams , Andrei Zelevinsky

This is a preliminary draft of Chapters 4-5 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. This installment contains: Chapter 4. New patterns … This is a preliminary draft of Chapters 4-5 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. This installment contains: Chapter 4. New patterns from old Chapter 5. Finite type classification

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The existence of greedy bases in rank 2 quantum cluster algebras

2016-04-05

Kyungyong Lee , Li Li , Dylan Rupel +1 more

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Introduction to Cluster Algebras. Chapters 1-3

2016-01-01

Sergey Fomin , Lauren Williams , Andrei Zelevinsky

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Strongly primitive species with potentials I: mutations

2015-07-13

Daniel Labardini-Fragoso , Andrei Zelevinsky

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Greedy bases in rank 2 quantum cluster algebras

2014-06-30

Kyungyong Lee , Li Li , Dylan Rupel +1 more

We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial … We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky.

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Positivity and tameness in rank 2 cluster algebras

2014-03-20

Kyungyong Lee , Li Li , Andrei Zelevinsky

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Strongly primitive species with potentials I: Mutations

2013-06-14

Daniel Labardini-Fragoso , Andrei Zelevinsky

Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials … Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.

Positivity and tameness in rank 2 cluster algebras

2013-03-23

Kyungyong Lee , Li Li , Andrei Zelevinsky

We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a … We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a rank 2 cluster algebra has a basis of indecomposable positive elements if and only if it is of finite or affine type. This statement disagrees with a conjecture by Fock and Goncharov.

Israel Moiseevich Gelfand, Part I

2013-01-01

Vladimir Retakh , I. M. Singer , David Kazhdan +4 more

compared by Henri Cartan to Poincaré and Hilbert, was born on September 2, 1913, in the small town of Okny (later Red Okny) near Odessa in the Ukraine and died … compared by Henri Cartan to Poincaré and Hilbert, was born on September 2, 1913, in the small town of Okny (later Red Okny) near Odessa in the Ukraine and died in New Brunswick, New Jersey, USA, on October 5, 2009.Nobody guided Gelfand in his studies.He attended the only school in town, and his mathematics teacher could offer him nothing except encouragement-and this was very important.In Gelfand's own words: "Offering encouragement is a teacher's most important job."In 1923 the family moved to another place and Gelfand entered a vocational school for chemistry lab technicians.However, he was expelled in the ninth grade as a son of a "bourgeois element" ("netrudovoi element" in Soviet parlance)-his father was a mill manager.After that Gelfand (he was sixteen and a half at that time) decided to go to Moscow, where he had some distant relatives.Until his move to Moscow in 1930, Gelfand lived in total mathematical isolation.The only books available to him were secondary school texts and several community college textbooks.The most advanced of these books claimed that there are three kinds of functions: analytical, defined by formulas; empirical, defined by tables; and correlational.Like Ramanujan, he was experimenting a lot.Around

Strongly primitive species with potentials I: Mutations

2013-01-01

Daniel Labardini-Fragoso , Andrei Zelevinsky

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Positivity and tameness in rank 2 cluster algebras

2013-01-01

Kyungyong Lee , Li Li , Andrei Zelevinsky

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Triangular Bases in Quantum Cluster Algebras

2012-12-17

Arkady Berenstein , Andrei Zelevinsky

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Greedy elements in rank 2 cluster algebras

2012-12-16

Kyungyong Lee , Li Li , Andrei Zelevinsky

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Greedy elements in rank 2 cluster algebras

2012-08-12

Kyungyong Lee , Li Li , Andrei Zelevinsky

A lot of recent activity in the theory of cluster algebras has been directed towards various constructions of natural bases in them. One of the approaches to this problem was … A lot of recent activity in the theory of cluster algebras has been directed towards various constructions of natural bases in them. One of the approaches to this problem was developed several years ago by P.Sherman - A.Zelevinsky who have shown that the indecomposable positive elements form an integer basis in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. Here we go around this difficulty by constructing a new basis in any rank 2 cluster algebra that we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of K.Lee - R.Schiffler and D.Rupel, we give explicit combinatorial expressions for greedy elements using the language of Dyck paths.

Triangular bases in quantum cluster algebras

2012-06-15

Arkady Berenstein , Andrei Zelevinsky

A lot of recent activity has been directed towards various constructions of natural bases in cluster algebras. We develop a new approach to this problem which is close in spirit … A lot of recent activity has been directed towards various constructions of natural bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig's construction of a basis, and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra. The key ingredient of our approach is a new version of Lusztig's Lemma that we apply to all acyclic quantum cluster algebras. As a result, we construct the canonical basis in every such algebra that we call the triangular basis.

On tropical dualities in cluster algebras

2012-01-01

Tomoki Nakanishi , Andrei Zelevinsky

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Triangular bases in quantum cluster algebras

2012-01-01

Arkady Berenstein , Andrei Zelevinsky

A lot of recent activity has been directed towards various constructions of "natural" bases in cluster algebras. We develop a new approach to this problem which is close in spirit … A lot of recent activity has been directed towards various constructions of "natural" bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig's construction of a canonical basis, and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra. The key ingredient of our approach is a new version of Lusztig's Lemma that we apply to all acyclic quantum cluster algebras. As a result, we construct the "canonical" basis in every such algebra that we call the canonical triangular basis.

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Greedy elements in rank 2 cluster algebras

2012-01-01

Kyungyong Lee , Li Li , Andrei Zelevinsky

A lot of recent activity in the theory of cluster algebras has been directed towards various constructions of "natural" bases in them. One of the approaches to this problem was … A lot of recent activity in the theory of cluster algebras has been directed towards various constructions of "natural" bases in them. One of the approaches to this problem was developed several years ago by P.Sherman - A.Zelevinsky who have shown that the indecomposable positive elements form an integer basis in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. Here we go around this difficulty by constructing a new basis in any rank 2 cluster algebra that we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of K.Lee - R.Schiffler and D.Rupel, we give explicit combinatorial expressions for greedy elements using the language of Dyck paths.

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On tropical dualities in cluster algebras

2011-01-01

Tomoki Nakanishi , Andrei Zelevinsky

We study two families of integer vectors playing a crucial part in the structural theory of cluster algebras: the $\gg$-vectors parameterizing cluster variables, and the $\cc$-vectors parameterizing the coefficients. We … We study two families of integer vectors playing a crucial part in the structural theory of cluster algebras: the $\gg$-vectors parameterizing cluster variables, and the $\cc$-vectors parameterizing the coefficients. We prove two identities relating these vectors to each other. The proofs depend on the sign-coherence assumption for $\cc$-vectors that still remains unproved in full generality.

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Quivers with potentials and their representations II: Applications to cluster algebras

2010-02-08

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

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Quiver Grassmannians and their Euler characteristics: Oberwolfach talk, May 2010

2010-01-01

Andrei Zelevinsky

This is an extended abstract of my talk at the Oberwolfach Workshop "Interactions between Algebraic Geometry and Noncommutative Algebra" (May 10 - 14, 2010). We present some properties of quiver … This is an extended abstract of my talk at the Oberwolfach Workshop "Interactions between Algebraic Geometry and Noncommutative Algebra" (May 10 - 14, 2010). We present some properties of quiver Grassmannians and examples of explicit computations of their Euler characteristics.

Quivers with potentials and their representations II: Applications to cluster algebras

2009-04-04

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

Quivers with potentials and their representations II: Applications to cluster algebras

2009-01-01

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

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Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors

2008-09-11

Shih-Wei Yang , Andrei Zelevinsky

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Quivers with potentials and their representations I: Mutations

2008-06-03

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

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Cluster algebras of finite type via Coxeter elements and principal minors

2008-01-01

Shih-Wei Yang , Andrei Zelevinsky

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate … We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.

Semicanonical Basis Generators of the Cluster Algebra of Type $A_1^{(1)}$

2007-01-19

Andrei Zelevinsky

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Mutations for quivers with potentials: Oberwolfach talk, April 2007

2007-01-01

Andrei Zelevinsky

This is an extended abstract of my talk at the Oberwolfach Workshop "Algebraic Groups" (April 22 - 28, 2007). It is based on a joint work with H.Derksen and J.Weyman … This is an extended abstract of my talk at the Oberwolfach Workshop "Algebraic Groups" (April 22 - 28, 2007). It is based on a joint work with H.Derksen and J.Weyman (arXiv:0704.0649v2 [math.RA]).

Cluster algebras IV: Coefficients

2007-01-01

Sergey Fomin , Andrei Zelevinsky

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Quivers with potentials and their representations I: Mutations

2007-01-01

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at … We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.

CLUSTER ALGEBRAS OF FINITE TYPE AND POSITIVE SYMMETRIZABLE MATRICES

2006-06-01

Michael Barot , Christof Geiß , Andrei Zelevinsky

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Laurent expansions in cluster algebras via quiver representations

2006-01-01

Philippe Caldero , Andrei Zelevinsky

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 … 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.

Cluster algebras IV: Coefficients

2006-01-01

Sergey Fomin , Andrei Zelevinsky

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial … We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of "principal" coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parametrizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parametrizations, some proven and some conjectural, suggest links to duality conjectures of V.Fock and A.Goncharov [math.AG/0311245]. The coefficient dynamics leads to a natural generalization of Al.Zamolodchikov's Y-systems. We establish a Laurent phenomenon for such Y-systems, previously known in finite type only, and sharpen the periodicity result from [hep-th/0111053]. For cluster algebras of finite type, we identify a canonical "universal" choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.

Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$

2006-01-01

Andrei Zelevinsky

We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these … We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these elements was obtained by P.Caldero and the author in math.RT/0604054. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp in math.CO/0602408. The arguments in math.RT/0604054 used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton (math.RT/0410184). This note provides a quick, self-contained and completely elementary alternative proof of the same results.

Laurent Expansions in Cluster Algebras via Quiver Representations

2006-01-01

Ph. Caldero , Andrei Zelevinsky

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Nested Complexes and their Polyhedral Realizations

2006-01-01

Andrei Zelevinsky

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Visibles Revisited

2005-09-01

Mark Bridger , Andrei Zelevinsky

Cluster algebras III: Upper bounds and double Bruhat cells

2005-01-15

Arkady Berenstein , Sergey Fomin , Andrei Zelevinsky

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Quantum cluster algebras: Oberwolfach talk, February 2005

2005-01-01

Andrei Zelevinsky

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 … 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).

Nested complexes and their polyhedral realizations

2005-01-01

Andrei Zelevinsky

Cluster Algebras of finite type and symmetrizable matrices

2004-11-15

Michael Barot , Christof Geiß , Andrei Zelevinsky

Quantum cluster algebras

2004-09-30

Arkady Berenstein , Andrei Zelevinsky

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

2004-01-01

Andrei Zelevinsky

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 … 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.

Quantum cluster algebras

2004-01-01

Arkady Berenstein , Andrei Zelevinsky

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 … 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.

Positivity and Canonical Bases in Rank 2 Cluster Algebras of Finite and Affine Types

2004-01-01

P.J. Sherman , Andrei Zelevinsky

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Generalized associahedra via quiver representations

2003-06-10

Bethany Marsh , Markus Reineke , Andrei Zelevinsky

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Cluster algebras II: Finite type classification

2003-05-16

Sergey Fomin , Andrei Zelevinsky

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

2003-01-01

Paul Sherman , Andrei Zelevinsky

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 … 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.

Coauthor	Papers Together
Sergey Fomin	28
Arkady Berenstein	14
Jerzy Weyman	13
Mikhail Kapranov	13
Israel M. Gelfand	9
Kyungyong Lee	8
Li Li	8
Harm Derksen	5
Péter Magyar	4
Bernd Sturmfels	4
Bethany Marsh	3
Markus Reineke	3
Daniel Labardini-Fragoso	3
Lauren Williams	3
Toshiki Nakashima	3
Frédéric Chapoton	2
Michael Barot	2
I. M. Gel'fand	2
Péter Magyar	2
Alek Vainshtein	2
Dylan Rupel	2
Tomoki Nakanishi	2
Boris Shapiro	2
Christof Geiß	2
Shih-Wei Yang	2
Michael Shapiro	2
P.J. Sherman	1
Vera Serganova	1
T. A. Springer	1
Simon Gindikin	1
David Kazhdan	1
Paul Sherman	1
Joachim Rosenthal	1
I. M. Singer	1
Ph. Caldero	1
Mark Bridger	1
Peter D. Lax	1
M. G. Kogan	1
Prakash Santhanakrishnan	1
Andrew Vince	1
H. Knight	1
Vladimir Retakh	1
I. N. Bernstein	1
Lauren Williams	1
Philippe Caldero	1
Bertram Kostant	1

Cluster algebras I: Foundations

2001-12-28

Sergey Fomin , Andrei Zelevinsky

Double Bruhat cells and total positivity

1999-01-01

Sergey Fomin , Andrei Zelevinsky

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Parametrizations of Canonical Bases and Totally Positive Matrices

1996-09-01

Arkady Berenstein , Sergey Fomin , Andrei Zelevinsky

Cluster algebras III: Upper bounds and double Bruhat cells

2005-01-15

Arkady Berenstein , Sergey Fomin , Andrei Zelevinsky

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Y -systems and generalized associahedra

2003-11-01

Sergey Fomin , Andre Zelevinsky

To the memory of To the memory of

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Introduction to Quantum Groups

2010-01-01

G. Lusztig

According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping … According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping algebra of a semisimple

Total positivity in Schubert varieties

1997-03-31

Arkady Berenstein , Andrei Zelevinsky

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Cluster algebras II: Finite type classification

2003-05-16

Sergey Fomin , Andrei Zelevinsky

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Generalized associahedra via quiver representations

2003-06-10

Bethany Marsh , Markus Reineke , Andrei Zelevinsky

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Total Positivity: Tests and Parametrizations

2000-12-01

Sergey Fomin , Andrei Zelevinsky

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The Laurent Phenomenon

2002-02-01

Sergey Fomin , Andrei Zelevinsky

From triangulated categories to cluster algebras II

2006-11-01

Ph. Caldero , Barbara Keller

In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to … In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the Calabi–Yau property of the cluster category. Pour le cas des carquois acycliques, nous établissons une correspondance biunivoque entre les objets basculants de la catégorie amassée et les amas de l'algèbre amassée associée. Cette correspondance nous permet de résoudre des conjectures sur les algèbres amassées. Nous prouvons un théorème de multiplication, un théorème de dénominateurs, ainsi que certaines conjectures sur les propriétés du graphe de mutation. Comme dans l'article précédent, les démonstrations reposent sur la propriété de Calabi–Yau de la catégorie amassée.

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Total Positivity in Reductive Groups

1994-01-01

G. Lusztig

An invertible n×n matrix with real entries is said to be totally ≥0 (resp. totally >0) if all its minors are ≥0 (resp. >0). This definition appears in Schoenberg’s 1930 … An invertible n×n matrix with real entries is said to be totally ≥0 (resp. totally >0) if all its minors are ≥0 (resp. >0). This definition appears in Schoenberg’s 1930 paper [S] and in the 1935 note [GK] of Gantmacher and Krein. (For a recent survey of totally positive matrices, see [A].)

Polytopal Realizations of Generalized Associahedra

2002-12-01

Frédéric Chapoton , Sergey Fomin , Andrei Zelevinsky

Abstract We prove polytopality of the generalized associahedra introduced in [5]. Abstract We prove polytopality of the generalized associahedra introduced in [5].

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Homotopy associativity of 𝐻-spaces. II

1963-01-01

James Stasheff

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Infinite-Dimensional Lie Algebras

1990-09-20

Victor G. Kač

This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is … This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is based on courses given over a number of years at MIT and in Paris, and is sufficiently self-contained and detailed to be used for graduate courses. Each chapter begins with a motivating discussion and ends with a collection of exercises, with hints to the more challenging problems.

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Groupes et algèbres de Lie

1971-01-01

Nicolás Bourbaki

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes … Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.

Positivity and Canonical Bases in Rank 2 Cluster Algebras of Finite and Affine Types

2004-01-01

P.J. Sherman , Andrei Zelevinsky

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Totally nonnegative and oscillatory elements in semisimple groups

2000-06-07

Sergey Fomin , Andrei Zelevinsky

Tensor product multiplicities, canonical bases and totally positive varieties

2001-01-01

Arkady Berenstein , Andrei Zelevinsky

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Cluster algebras as Hall algebras of quiver representations

2006-09-30

Philippe Caldero , Frédéric Chapoton

Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type A-D-E can … Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type A-D-E can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.

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From triangulated categories to cluster algebras

2008-01-28

Philippe Caldero , Bernhard Keller

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Canonical bases arising from quantized enveloping algebras

1990-01-01

G. Lusztig

0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call … 0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call them (or, rather, a slight modification of them, see ?2) bases of PBW type, since for v = 1, they specialize to bases of U+ of the type provided by the Poincare see however ? 12.)

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COXETER FUNCTORS AND GABRIEL'S THEOREM

1973-04-30

I. N. Bernstein , I. M. Gel'fand , В. А. Пономарев

It has recently become clear that a whole range of problems of linear algebra can be formulated in a uniform way, and in this common formulation there arise general effective … It has recently become clear that a whole range of problems of linear algebra can be formulated in a uniform way, and in this common formulation there arise general effective methods of investigating such problems. It is interesting that these methods turn out to be connected with such ideas as the Coxeter—Weyl group and the Dynkin diagrams. We explain these connections by means of a very simple problem. We assume no preliminary knowledge. We do not touch on the connections between these questions and the theory of group representations or the theory of infinite—dimensional Lie algebras. For this see [3]—[5]. Let Γ be a finite connected graph; we denote the set of its vertices by Γο and the set of its edges by ΓΊ (we do not exclude the cases where two vertices are joined by several edges or there are loops joining a vertex to itself). We fix a certain orientation Λ of the graph Γ; this means that for each edge / e Γι we distinguish a starting-point a(/) e Γο and an end-point

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The invariant theory of binary forms

1984-01-01

Joseph P. S. Kung , Gian‐Carlo Rota

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Cluster ensembles, quantization and the dilogarithm

2003-01-01

V. V. Fock , A. B. Goncharov

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to … Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.

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String bases for quantum groups of type 𝐴ᵣ

1993-08-18

Arkady Berenstein , Andrei Zelevinsky

On the self-linking of knots

1994-10-01

Raoul Bott , Clifford Henry Taubes

This note describes a subcomplex F of the de Rham complex of parametrized knot space, which is combinatorial over a number of universal ‘‘Anomaly Integrals.’’ The self-linking integrals of Guadaguini, … This note describes a subcomplex F of the de Rham complex of parametrized knot space, which is combinatorial over a number of universal ‘‘Anomaly Integrals.’’ The self-linking integrals of Guadaguini, Martellini, and Mintchev [‘‘Perturbative aspects of Chern–Simons field theory,’’ Phys. Lett. B 227, 111 (1989)] and Bar-Natan [‘‘Perturbative aspects of the Chern–Simons topological quantum field theory,’’ Ph.D. thesis, Princeton University, 1991; also ‘‘On the Vassiliev Knot Invariants’’ (to appear in Topology)] are seen to represent the first nontrivial element in H0(F)—occurring at level 4, and are anomaly free. However, already at the next level an anomalous term is possible.

On a Quantum Analog of the Caldero-Chapoton Formula

2010-10-06

Dylan Rupel

Journal Article On a Quantum Analog of the Caldero–Chapoton Formula Get access Dylan Rupel Dylan Rupel Department of Mathematics, University of Oregon, Eugene, OR 97403, USA Correspondence to be sent … Journal Article On a Quantum Analog of the Caldero–Chapoton Formula Get access Dylan Rupel Dylan Rupel Department of Mathematics, University of Oregon, Eugene, OR 97403, USA Correspondence to be sent to: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2011, Issue 14, 2011, Pages 3207–3236, https://doi.org/10.1093/imrn/rnq192 Published: 01 January 2011 Article history Received: 26 April 2010 Revision received: 02 September 2010 Accepted: 03 September 2010 Published: 01 January 2011

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On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories

1991-01-01

Al.B. Zamolodchikov

Cluster Algebras and Poisson Geometry

2010-11-10

Michael Gekhtman , Michael Shapiro , Alek Vainshtein

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping … Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry. This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.|Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry. This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.

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Cluster algebras IV: Coefficients

2007-01-01

Sergey Fomin , Andrei Zelevinsky

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Combinatorics and total positivity

1995-08-01

Francesco Brenti

Proof of the Kontsevich non-commutative cluster positivity conjecture

2012-11-01

Dylan Rupel

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Cluster ensembles, quantization and the dilogarithm

2009-01-01

V. V. Fock , A. B. Goncharov

A cluster ensemble is a pair (𝒳,𝒜) of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group Γ. The space 𝒜 is closely … A cluster ensemble is a pair (𝒳,𝒜) of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group Γ. The space 𝒜 is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism p:𝒜→𝒳. The space 𝒜 is equipped with a closed 2-form, possibly degenerate, and the space 𝒳 has a Poisson structure. The map p is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role in the cluster ensemble structure. We define a non-commutative q-deformation of the 𝒳-space. When q is a root of unity the algebra of functions on the q-deformed 𝒳-space has a large center, which includes the algebra of functions on the original 𝒳-space. The main example is provided by the pair of moduli spaces assigned in [7] to a topological surface S with a finite set of points at the boundary and a split semisimple algebraic group G. It is an algebraic-geometric avatar of higher Teichmüller theory on S related to G. We suggest that there exists a duality between the 𝒜 and 𝒳 spaces. In particular, we conjecture that the tropical points of one of the spaces parametrise a basis in the space of functions on the Langlands dual space. We provide some evidence for the duality conjectures in the finite type case.

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Quivers with potentials and their representations II: Applications to cluster algebras

2010-02-08

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

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Infinite root systems, representations of graphs and invariant theory

1980-02-01

Victor G. Kač

Quivers with potentials and their representations I: Mutations

2008-06-03

Harm Derksen , Jerzy Weyman , Andrei Zelevinsky

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Generic Bases for Cluster Algebras From the Cluster Category

2012-04-11

Pierre‐Guy Plamondon

Inspired by recent work of Geiss–Leclerc–Schröer, we use Hom-finite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This … Inspired by recent work of Geiss–Leclerc–Schröer, we use Hom-finite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same index. If the matrix associated to the quiver is of full rank, then we prove that the elements in this set are linearly independent. If the cluster algebra arises from the setting of Geiss–Leclerc–Schröer, then we obtain the basis found by these authors. We show how our point of view agrees with the spirit of conjectures of Fock–Goncharov concerning the parametrization of a basis of the upper cluster algebra by points in the tropical -variety.

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A reduction theorem for totally positive matrices

1952-12-01

Anne Whitney

Quantum cluster algebras

2004-09-30

Arkady Berenstein , Andrei Zelevinsky

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Canonical bases for the quantum group of type Ar and piecewise-linear combinatorics

1996-03-01

Arkady Berenstein , Andrei Zelevinsky

Simply laced Coxeter groups and groups generated by symplectic transvections.

2000-01-01

Boris Shapiro , Michael Shapiro , Alek Vainshtein +1 more

Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of … Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i a subgroup Gamma_i in GL_m(Z) generated by symplectic transvections. We prove among other things that the subgroups corresponding to different reduced words for the same pair (u,v) are conjugate to each other inside GL_m(Z). We also generalize the enumeration result of the first three authors (see AG/9802093) by showing that, under certain assumptions on u and v, the number of Gamma_i(F_2)-orbits in F_2^m is equal to 3\times 2^s, where s is the number of simple reflections that appear in a reduced decomposition for u or v and F_2 is the two-element field.

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Quiver varieties and cluster algebras

2011-01-01

Hiraku Nakajima

Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop … Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig's dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [arXiv:math/0104151] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of `prime' simple ones according to the cluster expansion.

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Moduli spaces of local systems and higher Teichmüller theory

2006-06-01

V. V. Fock , A. B. Goncharov

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Graded quiver varieties, quantum cluster algebras and dual canonical basis

2014-06-05

Yoshiyuki Kimura , Qin Fan

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Chow polytopes and general resultants

1992-07-01

Mikhail Kapranov , Bernd Sturmfels , Andrei Zelevinsky

A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras

1994-12-01

Peter Littelmann

Cluster-tilted algebras

2006-07-21

Aslak Bakke Buan , Bethany Marsh , Idun Reiten

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation … We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

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Tilting theory and cluster combinatorics

2005-08-20

Aslak Bakke Buan , Bethany Marsh , Markus Reineke +2 more