Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics

DOI10.1112/blms/bdv070zbMath1330.05035arXiv1503.05049OpenAlexW3100652330MaRDI QIDQ3458418

Sophie Morier-Genoud

Publication date: 18 December 2015

Published in: Bulletin of the London Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1503.05049

zbMATH Keywords

projective geometryquiver representationsCoxeter numberLaurent polynomialcluster algebrasDynkin quivertriangulations of polygonsFrieze patternssuperperiodic difference equations

Mathematics Subject Classification ID

Combinatorial aspects of representation theory (05E10) Polyhedra and polytopes; regular figures, division of spaces (51M20) Grassmannians, Schubert varieties, flag manifolds (14M15) Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics (82B20) Representations of quivers and partially ordered sets (16G20) Difference operators (39A70) Designs and configurations (05B99) Cluster algebras (13F60)

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Cites Work

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• Lecture notes on cluster algebras
• Categorification of a frieze pattern determinant
• \(T\)-systems and the pentagram map
• Coxeter's frieze patterns and discretization of the Virasoro orbit
• Friezes
• Friezes and a construction of the Euclidean cluster variables
• Linear recurrence relations for cluster variables of affine quivers.
• The Laurent phenomenon
• 2-frieze patterns and the cluster structure of the space of polygons
• Frieze patterns for punctured discs
• The geometry of frieze patterns
• Cluster algebras. II: Finite type classification
• \(Y\)-systems and generalized associahedra
• The projective geometry of the Gale transform.
• Cluster-additive functions on stable translation quivers.
• Arithmetics of 2-friezes
• Geometry in Grassmannians and a generalization of the dilogarithm
• The periodicity conjecture for pairs of Dynkin diagrams
• Graded quiver varieties, quantum cluster algebras and dual canonical basis
• \(\mathrm{SL}_{2}\)-tilings and triangulations of the strip
• Positivity of the \(T\)-system cluster algebra
• Reflection groupoids of rank two and cluster algebras of type \(A\)
• Generalized frieze pattern determinants and higher angulations of polygons
• Cyclic sieving of increasing tableaux and small Schröder paths
• On the periodicity conjecture for \(Y\)-systems
• Cluster algebras as Hall algebras of quiver representations.
• Tilting theory and cluster combinatorics.
• Quiver representations.
• Generalized frieze patterns
• FUNCTIONAL RELATIONS IN SOLVABLE LATTICE MODELS I: FUNCTIONAL RELATIONS AND REPRESENTATION THEORY
• Cluster algebras I: Foundations
• Linear difference equations, frieze patterns, and the combinatorial Gale transform
• Quantum frieze patterns in quantum cluster algebras of type A
• Cluster algebras, quiver representations and triangulated categories
• FRIEZES, STRINGS AND CLUSTER VARIABLES
• Positive Definite Matrices and Catalan Numbers, Revisited
• GRASSMANNIANS AND CLUSTER ALGEBRAS
• Restricted solid-on-solid models connected with simply laced algebras and conformal field theory
• Additive frieze patterns and multiplication tables
• Triangulated polygons and frieze patterns
• Representation Theory of Artin Algebras II
• On Tropical Friezes Associated with Dynkin Diagrams
• Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object
• Frieze patterns
• The pentagram map: a discrete integrable system
• \(\mathrm{SL}_2(\mathbb{Z})\)-tilings of the torus, Coxeter-Conway friezes and Farey triangulations

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