Teaching / Cluster algebras and canonical bases

Course description

According to Lusztig, “the theory of quantum groups is what led to extremely rigid structure, in which the objects of the theory are provided with canonical bases with rather remarkable properties” specializing for q=1q = 1 to canonical bases for objects in the classical theory.

Meanwhile, cluster algebras were introduced in 2000 by Fomin and Zelevinsky as a tool for studying dual canonical bases and total positivity in semisimple Lie groups.

The precise relation between cluster algebras and dual canonical bases coming from the theory of quantum groups remains a subject of active research. It is expected that when an algebra has both a cluster structure and a perfect basis, like the coordinate ring C[N]\mathbb C[N] of a maximal unipotent subgroup of a semisimple group, then the cluster monomials should form a subset of the perfect basis.

Below is a tentative syllabus or a bumpy adventure path. It will be updated as we go along.

Topic References
1 May 23 Description & first examples Keller (Ke1)
May 25 Cluster algebras associated to quivers & exchange graphs Ke1
May 27 From triangulations of n+3n+3-gons to ice quivers & cluster algebras with coefficients Ke1-2
2 May 30-June 1 A glimpse into two additive categorifications Ke1
June 3 From mutations to reflection groups & root systems Fomin and Reading (FR)
June 6-10 LAWRGe break Knutson and Zinn-Justin
3 June 13 Dynkin diagrams & Coxeter groups FR, Serre (S), Humphreys (H)
June 15 Good bases, perfect bases, crystals bases Berenstein and Zelevinsky (BZ), Kamnitzer (Ka1)
June 17 From g\mathfrak g-partitions to BZ-data & MV polytopes BZ, Ka1
4 June 20 i{\mathbf i}-Lusztig data from Lusztig's (perfect) canonical basis Tingley (T), Kamnitzer (Ka2)
June 22 Combinatorial crystals & their geometric analogues Ka2, Berenstein-Kazhdan (BK)
June 24 Planar networks, elementary Jacobi matrices, and total positivity
5 June 27 More of the same
June 29 Pseudoline arrangements and the chamber ansatz
July 1 Generic bases for cluster algebras and the chamber ansatz
6 July 5 Siyang, Examples
July 6 Wenhan, Samuel
July 7 Zejing, Fan
7 July 11 Tianle, Robin
July 13 Jack
July 15 Haoyang, Haosen, Sanat


  • Instructor contact: My email (please put "Math 610" in the subject line)
  • Lectures: MWF 2-4 PM, starting May 23 and ending July 8 July 15, in KAP 245
  • Grading scheme: Two assignments
  • Office hours: By appointment in KAP 406J or My Zoom Room


Some references

  1. Generic bases for cluster algebras and the chamber ansatz by Geiss, Leclerc and Schroer
  2. Extension-orthogonal components of preprojective varieties by Geiss and Schroer
  3. Parametrizations of Canonical Bases and Totally Positive Matrices by Berenstein, Fomin and Zelevinsky
  4. Symmetric Functions 2001: Surveys of Developments and Perspectives edited by Fomin
  5. Cluster algebras I. Foundations by Fomin and Zelevinsky
  6. Geometric and unipotent crystals II: from unipotent bicrystals to crystal bases, and Parametrizations of Canonical Bases and Totally Positive Matrices by Berenstein and Kazhdan
  7. LAWRGe
  8. Elementary construction of Lusztig’s canonical basis by Tingley
  9. Root Systems and Generalized Associahedra by Fomin and Reading
  10. Paths and root operators in representation theory by Littelmann
  11. Tilting Modules and their Applications by Mathieu
  12. GL(n,C)\text{GL}(n,\mathbb C) for combinatorialists by Stanley
  13. Grassmannians and cluster algebras by Scott
  14. A geometric approach to standard monomial theory by Brion and Lakshmibai
  15. Perfect bases in representation theory by Kamnitzer (Ka1)
  16. The crystal structure on the set of Mirkovic–Vilonen polytopes by Kamnitzer (Ka2)
  17. Algèbres amassées et applications [d’après Fomin-Zelevinsky,...] by Keller (Ke1)
  18. Cluster algebras, quiver representations and triangulated categories by Keller (Ke2)
  19. Fomin, Williams, Zelevinsky
  20. Triangulated categories... by Caldero and Keller