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Matrix formulae and skein relations for cluster algebras from surfaces
 ARXIV:1108.3382. BASES FOR CLUSTER ALGEBRAS FROM SURFACES 53
, 2011
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A Compendium on the Cluster Algebra and Quiver Package in sage
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 65 (2011), ARTICLE B65D
, 2011
"... This is the compendium of the cluster algebra and quiver package for Sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to further develop the theory by working on examples, by gathering data, and by exhibiting and testing conjectures. I ..."
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This is the compendium of the cluster algebra and quiver package for Sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to further develop the theory by working on examples, by gathering data, and by exhibiting and testing conjectures. In this compendium, we include the relevant theory to introduce the reader to cluster algebras assuming no prior background. Throughout this compendium, we include examples that the user can run in the Sage notebook or command line, and then close with a detailed description of the data structures and methods in this package.
Categorical tinkertoys for N = 2 gauge theories
"... In view of classification of the quiver 4d N = 2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to aN = 2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abe ..."
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In view of classification of the quiver 4d N = 2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to aN = 2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abelian category rep(Q,W) of (finite–dimensional) representations of the Jacobian algebra CQ/(∂W) should enjoy what we call the Ringel property of type G; in particular, rep(Q,W) should contain a universal ‘generic ’ subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. More precisely, there is a family of ‘light ’ subcategories Lλ ⊂ rep(Q,W), indexed by points λ ∈ N, where N is a projective variety whose irreducible components are copies of P1 in one–to–one correspondence with the simple factors of G. If λ is the generic point of the i–th irreducible component, Lλ is the universal subcategory corresponding to the i–th simple factor of G. Matter, on the contrary, is encoded in the subcategories Lλa where {λa} is a finite set of closed points in N. In particular, for a Gaiotto theory there is one such family of subcategories, Lλ∈N, for each maximal degeneration of the corresponding surface Σ, and the index variety N may be identified with the degenerate Gaiotto surface itself: generic light subcategories correspond to cylinders, while closed–point subcategories to ‘fixtures ’ (spheres with three punctures of various kinds) and higher–order generalizations. The rules for ‘gluing ’ categories are more general that the geometric gluing of surfaces, allowing for a few additional exceptional N = 2 theories which are not of the Gaiotto class. We include several examples and some amusing consequence, as the characterization in terms of quiver combinatorics of asymptotically free theories.
Discrete integrable systems and Poisson algebras from cluster maps, arXiv: 1207.6072v2
, 2012
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Snake graph calculus and cluster algebras from surfaces II: Selfcrossing . . .
, 2014
"... Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula which is parametrized by the perfect matchings of a snake graph. In this paper, we continue our study of snake graphs from a combinatorial point of view. ..."
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Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula which is parametrized by the perfect matchings of a snake graph. In this paper, we continue our study of snake graphs from a combinatorial point of view. We introduce the notions of abstract snake graphs and abstract band graphs, their crossings and selfcrossings, as well as the resolutions of these crossings. We show that there is a bijection between the set of perfect matchings of (self)crossing snake graphs and the set of perfect matchings of the resolution of the crossing. In the situation where the snake and band graphs are coming from arcs and loops in a surface without punctures, we obtain a new proof of skein relations in the corresponding cluster algebra.
QUIVERS OF FINITE MUTATION TYPE AND SKEWSYMMETRIC MATRICES
, 905
"... Quivers of finite mutation type are certain directed graphs that first arised in FominZelevinsky’s theory of cluster algebras. It has been observed that these quivers are also closely related with different areas of mathematics. In fact, main examples of finite mutation type quivers are the quivers ..."
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Quivers of finite mutation type are certain directed graphs that first arised in FominZelevinsky’s theory of cluster algebras. It has been observed that these quivers are also closely related with different areas of mathematics. In fact, main examples of finite mutation type quivers are the quivers associated with triangulations
Cluster algebras: An introduction
, 2013
"... Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced ..."
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Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced
BPS quivers and spectra of complete N = 2 quantum field theories
 COMM. MATH. PHYS
, 2011
"... We study the BPS spectrum of N = 2 complete quantum field theories in four dimensions. For examples that can be described by a pair of M5 branes on a punctured Riemann surface we explain how triangulations of the surface fix a BPS quiver and superpotential for the theory. The BPS spectrum can then ..."
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We study the BPS spectrum of N = 2 complete quantum field theories in four dimensions. For examples that can be described by a pair of M5 branes on a punctured Riemann surface we explain how triangulations of the surface fix a BPS quiver and superpotential for the theory. The BPS spectrum can then be determined by solving the quantum mechanics problem encoded by the quiver. By analyzing the structure of this quantum mechanics we show that all asymptotically free examples, ArgyresDouglas models, and theories defined by punctured spheres and tori have a chamber with finitely many BPS states. In all such cases we determine the spectrum.
TUBULAR CLUSTER ALGEBRAS II: EXPONENTIAL GROWTH
"... Abstract. Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding ..."
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Abstract. Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding