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 $q = 1$ 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 $\mathbb C[N]$ 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.

Week		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 $n+3$-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 $\mathfrak g$-partitions to BZ-data & MV polytopes	BZ, Ka1
4	June 20	${\mathbf i}$-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

Specs

• 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

Homework

• Please read Survey article #1 and Survey article #2 by Geiss, Leclerc and Schroer. Explore David Speyer's material for an old workshop on this page to supplement.
• Please submit your homework in person, or by email
• Assignment #1 due June 17
• Assignment #2 due July 18 by 5 PM

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. $\text{GL}(n,\mathbb C)$ 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