## Math 8680, Topics in Combinatorics: Cluster Algebras and Quiver Representations (Spring 2011)

Lectures: MW 4:00-5:15 in Vincent Hall 1.

Instructor: Gregg Musiker (musiker "at" math.umn.edu)

Office Hours: MWF 2:30-3:20 Vicent Hall 251. Also, by appointment, or feel free to knock. I usually keep my door open if I'm in.

## Course Description:

This is a graduate level course in algebraic combinatorics. The topic for this semester is cluster algebras and quiver representations. Cluster algebras are a class of combinatorially defined rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. A partial list of related areas includes quiver representations, statistical physics, and Teichmuller theory. This course will focus on the algebraic and combinatorial aspects of cluster algebras, thereby providing a concrete introduction to this rapidly-growing field. Besides providing background on the fundamentals of cluster theory, we will discuss complementary topics such as total positivity, quiver representations, the polyhedral geometry of cluster complexes, cluster algebras from surfaces, and connections to statistical physics.

Prerequisites: No prior knowledge of cluster algebras or representation theory will be assumed; although familiarity with groups, rings, and modules, as in Math 8202, will be helpful.

Recommended (but not required) Texts:
Cluster Algebras and Poisson Geometry by Michael Gekhtman, Michael Shapiro, and Alek Vainshtein (2010, AMS Monograph). On reserve in the math library

Elements of the Representation Theory of Associative Algebras, Vol. 1, by Ibarahim Assem, Daniel Simson, and Andrzej Skowronski. (2006, Cambridge University Press) On reserve in the math library

Recommended Articles:

Surveys for Cluster Algebras:
• Total Positivity and cluster algebras (by Sergey Fomin, ICM 2010)
• Root systems and generalized associahedra (by Sergey Fomin and Nathan Reading, IAS/Park City 2004)
• Total Positivity: Tests and Parametrizations (by Sergey Fomin and Andrei Zelevinsky 1999)

• Relevant Research articles
• Cluster Algebra I: Foundations (by Sergey Fomin and Andrei Zelevinsky 2002)
• Cluster Algebra II: Finite Type Classification (by Sergey Fomin and Andrei Zelevinsky 2003)
• The Laurent Phenomenon (by Sergey Fomin and Andrei Zelevinsky 2002)
• Cluster Algebras and triangulated surfaces (by Sergey Fomin, Michael Shapiro, and Dylan Thurston 2008)
• Positivity for Cluster Algebras from Surfaces (by Gregg Musiker, Ralf Schiffler, and Lauren Williams 2009)
• Laurent expansions in cluster algebras via quiver representations (by Philippe Caldero and Andrei Zelevinsky 2006)
• More articles available at the Cluster Algebras Portal.

• For Quiver Representations and later in the course:
• Lecture Notes on Quiver Representations (Lectures 1-5 Relevant for us)
• Introduction to representation theory (Section 5 covers Quiver Reps) by Pavel Etingof
• Cluster algebras, quiver representations and triangulated categories (by Bernhard Keller 2008)
• Representations of Quivers by M. Barot

• More articles to be listed later

SAGE Software for Cluster Algebras (with Christian Stump)
Sage-Combinat Server (Experimental)

There will be no exams, but registered students are expected to attend, and should hand in the homework assignments. There will be homework every three weeks or so, tentatively three assignments over the semester. I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page their collaborators. The first homework will be collected on Wednesday, February 23th (Please note the change of date).

## Tentative Lecture Schedule

• (Jan 19) Lecture 1: Introduction to the Course, including Cluster Algebra SAGE Package Demo (PDF of Demo)
• (Sections 1 - 4 of Fomin-Reading to be covered through Feb 7)
• (Jan 24) Lecture 2: Labelled seeds and general definition of a cluster algebra
• (Jan 26) Lecture 3: Cluster complexes and exchange matrices as quivers
• (Jan 31) Lecture 4: Crash Course on Finite Reflection Groups and Coxeter Diagrams
• (Feb 2) Lecture 5: Root systems and the Finite type classification
• (Feb 7) Lecture 6: Double-wiring Diagrams and connections between cluster algebras and total positivity (Guest Lecturer: Pasha)
• (Feb 9) Lecture 7: Rank two cluster algebras and sketch of the proof finite type classification
• (Feb 14) Lecture 8: The Laurent Phenomenon and the Caterpillar Lemma
• (Feb 16) Lecture 9: Applications of the Laurent Phenomenon to Somos Sequences
• (Feb 21) Lecture 10: More on rank two cluster algebras and Canonical Bases
• (Feb 23) Lecture 11: Rank Two Canonical Bases continued
• (Feb 28) Lecture 12: Introduction to Quiver Representations: Simples and Indecomposables
• (Mar 2) Lecture 13: The Path Algebra of a Quiver
• (Mar 7) Lecture 14: Introduction to Category Theory and the Krull-Remak-Schmidt Theorem
• (Mar 9) Lecture 15:Projectives and Injectives
• (Mar 14) Spring Break:
• (Mar 16) Spring Break:
• (Mar 21) Lecture 16: BGP Reflection Functors and Proof of Gabriel's Theorem begins
• (Mar 23) Lecture 17: The Caldero-Chapoton Formula for Cluster Variables in terms of Quiver Representations
• (Mar 28) Lecture 18: Gabriel's Theorem Part II and Part of Kac's Theorem
• (Mar 30) Lecture 19: More examples with imaginary roots
• (April 4) Lecture 20: Crash course on Group Actions and applications to Quivers
• (April 6) Lecture 21: Cluster algebras from surfaces: annulus and introductory examples
• (April 11) Lecture 22: Cluster algebras from surfaces with punctures: tagged arcs
• (April 13) Lecture 23: Laurent expansion formulas for cluster algebras from surfaces
• (April 18) Lecture 24: Auslander Reiten Quivers and the Cluster Category in the An Case
• (April 20) Lecture 25: Cluster Categories for Dn and Affine Cases
• (April 25) Lecture 26: Cluster Algebras and Teichmuller Theory: Shear Coordinates
• (April 27) Lecture 27: More on Teichmuller Theory: Lambda Lengths
• (May 2) Lecture 28: More on Cluster algebras from surfaces
• (May 4) Lecture 29: Cluster Algebras of infinite mutation type

• Homework assignments
Assignment Due date
Homework 1 Wednesday 2/23 (Please note the change of date)
Homework 2 Wednesday 3/30 (NOTE: 3/28: Some typos have been fixed and notes added in the attached pdf. )
Homework 3 Monday 5/2