References

Introduction to Cluster Algebras


  • Lauren K.Williams: Introduction to Cluster Algebras http://people.math.harvard.edu/~williams/book.html

  • Chapter 1~3:http://people.math.harvard.edu/~williams/papers/Chapters1-3.pdf

  • Chapter 4~5:http://people.math.harvard.edu/~williams/papers/Chapters4-5.pdf

  • Chapter 6: http://people.math.harvard.edu/~williams/papers/chapter6.pdf

  • Chapter 7: http://people.math.harvard.edu/~williams/papers/chapter7.pdf

Cluster algebraic formulation of (untwisted) Q-systems


  • Kedem: Q-systems as cluster algebras arXiv

  • Di Francesco Kedem: Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property arXiv

Combinatorial objects tied to Q-systems (and T-systems), and its applications to integrability


  • Di Francesco Kedem: Q-systems, Heaps, Paths and Cluster Positivity

https://arxiv.org/pdf/0811.3027.pdf

  • Di Francesco Kedem: Noncommutative integrability, paths and quasi-determinants https://arxiv.org/pdf/1006.4774.pdf

  • Di Francesco Kedem: Positivity of the T-system cluster algebra https://arxiv.org/pdf/0908.3122.pdf

  • Di Francesco Kedem: T-systems with boundaries from network solutions https://arxiv.org/pdf/1208.4333.pdf

  • Tomoki Nakanishi: T-systems, Y-systems, and cluster algebras: Tamely laced case

https://arxiv.org/abs/1003.1180

Poisson structure, networks, factorization dynamics, pentagram maps


  • Gekhtmann Shapiro Vainshtein: Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective

https://arxiv.org/pdf/0906.1364v4.pdf

https://projecteuclid.org/download/pdf_1/euclid.acta/1485892547

  • Gekhtmann Shapiro Tabachnikov Vainshtein: Higher pentagram maps, weighted directed networks, and cluster dynamics

https://arxiv.org/pdf/1110.0472.pdf

https://www.sciencedirect.com/science/article/pii/S0001870816001274

  • Harold Williams: Q-Systems, Factorization Dynamics, and the Twist Automorphism

(Actually, this paper also contains the cluster algebraic formulation of certain twisted Q-systems)

https://arxiv.org/pdf/1310.6624.pdf

Statistical physics, dimers etc.


  • Summer school with videos:

http://www.math.lsa.umich.edu/~speyer/DimerConference2020/

  • Kenyon Pemantle: Double-dimers, the Ising model and the hexahedron recurrence

https://arxiv.org/abs/1308.2998

  • Di Francesco: T-systems, networks and dimers

https://arxiv.org/abs/1307.0095

  • Speyer: Perfect Matchings and the Octahedron Recurrence

https://arxiv.org/abs/math/0402452

  • Kuo: Applications of Graphical Condensation for Enumerating Matchings and Tilings

https://arxiv.org/abs/math/0304090

  • Kenyon: Lectures on Dimers

https://arxiv.org/abs/0910.3129

  • Bousquet-Mélou (LaBRI), James Propp, Julian West: Perfect matchings for the three-term Gale-Robinson sequences

https://arxiv.org/abs/0906.3125

  • Baxter: Exactly solved models in statistical mechanics (Free textbook and a good read)

https://physics.anu.edu.au/theophys/_files/Exactly.pdf

  • Sacha Friedli and Yvan Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction (also free textbook, same subject as Baxter's but more examples and exercises)

https://www.unige.ch/math/folks/velenik/smbook/

Integrable systems


  • Ovsienko Schwartz Tabachnikov: The Pentagram map: a discrete integrable system

https://arxiv.org/pdf/0810.5605.pdf

  • Glick: The pentagram map and Y-patterns

https://arxiv.org/abs/1005.0598

  • Kenyon Goncharov: Dimers and cluster integrable systems

https://arxiv.org/pdf/1107.5588.pdf

Symmetric Polynomials


  • Pavel Etingof, Alexander A. Kirillov, Jr. : Macdonald's polynomials and representation of quantum groups

https://arxiv.org/pdf/hep-th/9312103.pdf

  • Ian G Macdonald : Symmetric Functions and Orthogonal Polynomials (AMS University Lecture Series)

Quantum/NC cluster algebra


  • Berenstein Zelevinsky

https://arxiv.org/pdf/math/0404446.pdf

  • Berenstein Retakh: NC Catalan numbers

https://arxiv.org/pdf/1708.03316.pdf

  • NC surfaces https://arxiv.org/pdf/1510.02628.pdf

  • Glick Rupel: Introduction to cluster algebras https://arxiv.org/pdf/1803.08960.pdf


Quantum Dilogarithm


  • Rinat M. Kashaev, Tomoki Nakanishi: Classical and Quantum Dilogarithm Identities

https://arxiv.org/abs/1104.4630

  • Bernhard Keller: On Cluster theory and quantum dilogarithm identities

https://arxiv.org/abs/1102.4148


Snake Graphs


  • Ilke Canakci, Ralf Schiffler: Snake Graph Calculus and cluster algebras from surfaces

https://arxiv.org/pdf/1209.4617.pdf

  • Ilke Canakci, Ralf Schiffler: Snake Graph Calculus and cluster algebras from surfaces II: Self-crossing snake graphs

https://arxiv.org/pdf/1407.0500.pdf