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
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)
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