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arXiv:2106.02160 (Published 2021-06-03)
Introduction to Cluster Algebras. Chapter 7
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 has been posted as arXiv:1707.07190. Chapter 6 has been posted as arXiv:2008.09189. This installment contains: Chapter 7. Plabic graphs
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arXiv:2008.09189 (Published 2020-08-20)
Introduction to Cluster Algebras. Chapter 6
Comments: 43 pagesThis 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 has been posted as arXiv:1707.07190. This installment contains: Chapter 6. Cluster structures in commutative rings
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arXiv:1707.07190 (Published 2017-07-22)
Introduction to Cluster Algebras. Chapters 4-5
Comments: 68 pagesSubjects: 13F60This 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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arXiv:1608.05735 (Published 2016-08-19)
Introduction to Cluster Algebras. Chapters 1-3
Comments: 69 pagesSubjects: 13F60This 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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arXiv:1303.5806 (Published 2013-03-23)
Positivity and tameness in rank 2 cluster algebras
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.
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Greedy elements in rank 2 cluster algebras
Comments: Minor editorial changes and corrections; figures changed to black and white; final version, to appear in Selecta MathSubjects: 13F60A 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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arXiv:math/0606775 (Published 2006-06-30)
Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$
Comments: 4 pages, no figuresSubjects: 16S99We 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.
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Laurent expansions in cluster algebras via quiver representations
Comments: 19 pages; v2: added Remark 5.7; v3: minor editorial changes, final version to appear in Moscow Math. Journal; v4: more minor editorial changesWe 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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Cluster algebras IV: Coefficients
Comments: 59 pages, 9 tables; minor editorial changes. Final version, to appear in Compos. MathWe 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.
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Nested complexes and their polyhedral realizations
Comments: v2: 14 pages, 1 figure; historical account in the introduction modified, Section 5 reorganized, Corollary 5.2 supplied with a self-contained proof; v3: minor editorial changes, Proposition 7.2 supplied with a proof; v4: a few typos fixed, one reference updated, final version, to appear in Pure and Applied Mathematics QuarterlyJournal: Pure and Applied Mathematics Quarterly, 2 (2006), no. 3, 1-17Categories: math.COKeywords: nested complexes, alex postnikovs paper math, dual polyhedral realizations, complete simplicial fans, cluster complexesTags: journal articleThis 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 and positive symmetrizable matrices
Comments: 20 pages. In version 3, some new material is added in the end of section 2, discussing the classification and characterizations of positive quasi-Cartan matrices. In final version 4, Proposition 2.9 is corrected and its proof expanded. To appear in J. London Math. Soc. In version 5 only typos in the arXiv data fixedThe 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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Cluster algebras: Notes for the CDM-03 conference
Comments: The paper has been completely rewritten and greatly expanded compared to the previous version. Extensive bibliography added. 33 pages, 11 figuresTags: conference paperThis 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.
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Cluster algebras II: Finite type classification
Comments: 50 pages, 18 figures. Version 2: new introduction; final version, to appear in Invent. MathSubjects: 14M99Keywords: cluster algebra, finite type classification, finite root systems, essential combinatorial ingredient, semisimple lie algebrasTags: journal articleThis 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.
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arXiv:math/0202004 (Published 2002-02-01)
Polytopal realizations of generalized associahedra
Comments: 25 pagesSubjects: 05E15In 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.
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arXiv:hep-th/0111053 (Published 2001-11-06)
Y-systems and generalized associahedra
Comments: 35 pagesWe 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.
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arXiv:math/0104241 (Published 2001-04-25)
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.
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arXiv:math/9912128 (Published 1999-12-15)
Total positivity: tests and parametrizations
Comments: 20 pages, with color figuresAn introduction to total positivity (TP), with the emphasis on efficient TP criteria and parametrizations of TP matrices. Intended for general mathematical audience.
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arXiv:math/9912012 (Published 1999-12-02)
Tensor product multiplicities, canonical bases and totally positive varieties
Comments: Latex, 42 pages. For viewing the table of contents, LaTeX the file 3 timesWe obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here "polyhedral" means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of $\ii$-trails which resemble Littelmann's paths but seem to be more tractable. We also study combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approaches were shown by Lusztig to be equivalent to each other, they lead to different combinatorial parametrizations of the canonical bases. One of our main results is an explicit description of the relationship between these parametrizations. Our approach to the above problems is based on a remarkable observation by G. Lusztig that combinatorics of the canonical basis is closely related to geometry of the totally positive varieties. We formulate this relationship in terms of two mutually inverse transformations: "tropicalization" and "geometric lifting."
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arXiv:math/9906203 (Published 1999-06-29)
Simply-laced Coxeter groups and groups generated by symplectic transvections
Comments: LaTeX, 18 pages, 4 figuresLet 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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arXiv:math/9811100 (Published 1998-11-17)
Totally nonnegative and oscillatory elements in semisimple groups
Comments: 10 pagesWe 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.Pena to the case of an arbitrary complex semisimple Lie group.