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Introduction to Cluster Algebras. Chapters 1-3
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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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Cited by 10 Pith papers
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Gr\"obner Cones for Finite Type Cluster Algebras
Compatibility-degree weights of cluster variables lie in the Gröbner cone of any finite-type cluster algebra, yielding explicit circular term orders and, for classical types, complete ray and lineality descriptions.
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Cluster Algebras, Cube Roots, and Energy Correlators in $\mathcal{N}=4$ SYM
Infinite mutation sequences of the cluster quiver Q3 generate the six cubic algebraic symbol letters of the near-collinear four-point energy correlator in N=4 super-Yang-Mills theory.
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Electrical networks, Grassmannians, and cluster algebras
For odd n the circular-minor cluster algebra CM_n is isomorphic to the Grassmannian cluster algebra A_{n-1,2n} after freezing/trivializing n central variables, relating circular total positivity to Grassmannian positi...
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On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges
Every Markov triple has a unique q-deformed polynomial solution to the q-Markov equation, and these polynomials count weighted perfect matchings of snake graphs.
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Representation theory of hereditary artin algebras of finite representation type
All hammocks and the full shape of the Auslander-Reiten quiver are determined from the ext-quiver for hereditary artin algebras of finite type, with applications to counting indecomposables and radical nilpotency.
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Okamoto's symmetry on the representation space of the sixth Painlev\'e equation
Okamoto's symmetry w2 of Painlevé VI is shown to act on monodromy representations as an explicit rational map made of six cluster X-mutations, equivalently a half-turn of colored hexagon triangulations.
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Frieze patterns with coefficients
Tame frieze patterns with coefficients are developed, with a complete classification of triangles realizable from classic Conway-Coxeter friezes and a finiteness theorem over discrete subsets.
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Curve integral formula for the M\"obius strip
A global Schwinger integral is constructed for Möbius-strip amplitudes and verified against the tropical limit of the type-I superstring Möbius amplitude.
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Legendrian doubles, twist spuns, and clusters
The authors construct cluster structures on sheaf moduli of twist-spun Legendrian surfaces and use them to produce new exact Lagrangian fillings and obstructions in contact R^5.
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Singularities of massless scattering and cluster algebras
Partial flag cluster algebras reproduce a large part of the two-loop five- and six-point massless scattering symbol alphabets, with the momentum twistor embedding stronger at six points after permutation completion.
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