# 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

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

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 |
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 |
BZ, Ka1 | ||

4 | June 20 | 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

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