## Math 8680, Topics in Combinatorics: Cluster Algebras and Quiver Representations (Fall 2016)

Lectures: MWF 11:15-12:05 in Vincent Hall 213.

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

Office Hours: MWF 2:30-3:20 in Vincent 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 topics 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 combinatorial and representation theoretic aspects of cluster algebras, thereby providing a concrete introduction to this rapidly-growing field.

We begin with an introduction to quiver representations, which provide a class of representations (of finite dimensional associative algebras) that share some similar properties with representations of finite groups but also illuminating contrasts. We will follow a hands on approach with lots of examples when introducing these objects. Secondly, we turn our attention to our other main topic, cluster algebras, defining them algebraically and illustrating how they show up in a variety of combinatorial and geometric contexts. Finally, we will explore how cluster algebra theory can be studied via the language of tilting theory, cluster categories, and Auslander Reiten Translations coming full circle with the quiver representation theoretic approach presented earlier on.

While there is no required textbook, there are three recommended textbooks. We will cover Chapters 1-3 of "Quiver Representations" by Ralf Schiffler (and later chapters as time allows) and the majority of "Lecture Notes on Cluster Algebras" by Robert Marsh. Additionally, the first three chapters of Sergey Fomin, Lauren Williams, and Andrei Zelevinsky's text "Introduction to Cluster Algebras" was recently posted on the arXiv and this text will be used as a third recommended textbook. We will also supplement these three books with lecture notes, as well as survey and research articles that may be found online.

Parts of this course will cover material similar to my Spring 2011 Course on Cluster Algebras and Quiver Representations.

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:
Quiver Representations by Ralf Schiffler (2014, Springer CMS Books in Mathematics).
Lecture Notes on Cluster Algebras by Robert Marsh (2013, EMS Zurich Lectures in Advanced Mathematics).
On reserve in the math library
Introduction to representation theory (Section 5 covers Quiver Reps) by Pavel Etingof

Recommended Survey Articles:
• [FWZ16] Introduction to Cluster Algebras (Chaps 1-3 Prelim Version) (Just posted on the arxiv in August 2016! Highly Recommended)

• [W10] Cluster algebras: an introduction (by Lauren Williams)
• [FR04] Root systems and generalized associahedra (by Sergey Fomin and Nathan Reading, IAS/Park City 2004)
• [R10] Tilting theory and cluster algebras (by Idun Reiten)
• [K08] Cluster algebras, quiver representations and triangulated categories (by Bernhard Keller 2008)

• Lecture Notes on Quiver Representations by Harm Derksen (Lectures 1-5 Relevant for us)
• Introduction to representation theory (Section 5 covers Quiver Reps) by Pavel Etingof
• Representations of Quivers by M. Barot

• 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)
• Y-Systems and Generalized Associahedra (by Sergey Fomin and Andrei Zelevinsky 2003)
• Polytopal Realizations of Generalized Associahedra (by Frederic Chapoton, Sergey Fomin and Andrei Zelevinsky 2002)

• More articles available at the Cluster Algebras Portal.

• ## Computer Software:

We will occasionally demo Sage Software during this course for cluster algebraic and quiver representation theoretic computations.

You may easily access these via the Sage Math Cloud

Cloud worksheets will be listed here later in the course. For now see the (outdated) Compendium on the Cluster Algebra and Quiver Package in SAGE (with Christian Stump) for a tutorial.

Also, check out Keller's Quiver Applet in Java
.

There will be no exams, but registered students are expected to attend, and should hand in the homework assignments. There will be homework once a month with 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.

## Tentative Lecture Schedule

• (Sep 7) Lecture 1: Introduction to the Course and Quiver Representations (Sections 1.1 of Schiffler)
• (Sep 9) Lecture 2: Isomorphisms, Direct Sums and Subrepresentations (1.1-1.2 of Schiffler)
• (Sep 12) Lecture 3: Krull-Schmidt Theorem and Short Exact Sequences and (1.2-1.3 of Schiffler)
• (Sep 14) Lecture 4: Auslander-Reiten Quivers and Introduction to the Path Algebra (1.5 and Chapter 2 preamble of Schiffler)
• (Sep 16) Lecture 5: Simple, Projective, and Injective Representations (2.1 of Schiffler)
• (Sep 19) Lecture 6: Projective Reslutions I: Maps from Projectives (2.1-2.2 of Schiffler)
• (Sep 21) Lecture 7: Projective Reslutions II: Proof of Exactness (2.2 of Schiffler)
• (Sep 23) Lecture 8: Projective Reslutions III: Radicals and Hereditary Categories (2.2 of Schiffler)
• (Sep 26) Lecture 9: Auslander-Reiten Translation (2.3 of Schiffler)
• (Sep 28) Lecture 10: Ext Groups (glimpse) and AR Quivers of Type A_n (2.4 (glimpse) and 3.1 of Schiffler)
• (Sep 30) Lecture 11: AR Quivers of Type A_n II: Indecomposables and Triangulation Interpretation (5.2 of Etingof and 3.1 of Schiffler)
• (Oct 3) Lecture 12: AR Quivers of Type A_n III: Coxeter Interpretation (3.1 of Schiffler)
• (Oct 5) Lecture 13: AR Quivers of Type D_n (3.3 of Schiffler)
• (Oct 7) Lecture 14: Gabriel's Theorem I and the Three Subspaces Problem (3.2 of Schiffler and 5.3 of Etingof)
• (Oct 10) Lecture 15: Gabriel's Theorem II and the AR Quiver for the Kronecker Quiver
• (Oct 12) Lecture 16: Gabriel's Theorem III: Quiver Varieties (8.1, 8.4 of Schiffler, 5.5-5.6 of Etingof) (Vic Reiner guest lectures)
• (Oct 14) Lecture 17: Gabriel's Theorem IV: BGP Reflection Functors (8.2-8.3 of Schiffler)
• (Oct 17) Lecture 18: Gabriel's Theorem V: Finishing the Proof and Kac's Theorem
• (Oct 19) Lecture 19: Cluster Algebras I: Labelled seeds and general definition of a cluster algebra (Chapter 2 of Marsh, Section 3.6 of Fomin-Williams-Zelevinsky)
• (Oct 21) Lecture 20: Cluster Algebras II: Cluster Algebras of Geometric Type (Chapter 2 of Marsh or Sections 3.1-3.2 of Fomin-Williams-Zelevinsky )
• (Oct 24) Lecture 21: Cluster Algebras III: Quiver mutation and Cluster Algebras of Type An (Sections 2.1-2.2 of Fomin-Williams-Zelevinsky or Section 3 of Fomin-Reading)
• (Oct 26) Lecture 22: Cluster Algebras IV: Cluster Complexes and Associahedra (Section 4 of Fomin-Reading)
• (Oct 28) Lecture 23: Root Systems and Cluster Algebras of Finite Type (Section 4 of Fomin-Reading and Lecture 4 from 2011)
• (Oct 31) Lecture 24: More on Root Systems
• (Nov 2) Lecture 25: Towards the proof of the finite type classification
• (Nov 4) Lecture 26: Proof of the finite type classification II (Lecture 7 from 2011)
• (Nov 7) Lecture 27: Polytopal Realizations of Generalized Associahedra (Section 4 of Fomin-Reading, Lecture 6 from 2015, "Y-systems and Generalized Associahedra" and "Polytopal Realizations of Generalized Associahedra")
• (Nov 9) Lecture 28: The Laurent Phenomenon and the Caterpillar Lemma (Lecture 8 from 2011)
• (Nov 11) Lecture 29: The Laurent Phenomenon and the Caterpillar Lemma II
• (Nov 14) Lecture 30: Applications of the Laurent Phenomenon to Somos Sequences ( Lecture 9 from 2011)
• (Nov 16) Lecture 31: Cluster Algebras from Surfaces ( Lecture 21 from 2011)
• (Nov 18) Lecture 32: Cluster Algebras from Surfaces II ( MSRI Lecture Day 1 from 2011)
• (Nov 21) Lecture 33: Cluster Algebras from Surfaces III ( Lecture 22 from 2011)
• (Nov 23) Lecture 34: Cluster Algebras from Surfaces IV ( MSRI Lecture Day 2 from 2011)
• Other Resources for Cluster Algebras from Surfaces:
Slides from 2012
Slides from 2010
Lecture 23 from 2011
MSRI Lecture Day 3 from 2011
• (Nov 25) Happy Thanksgiving!
• (Nov 28) Lecture 35: Hyperbolic Geometry and Lambda Lengths ( Lecture 26 2011)
• (Nov 30) Lecture 36: Teichmuller Theory and Cluster Variables ( Lecture 27 from 2011)
• (Dec 2) Lecture 37: Cluster Variable formulas, Matrices, and Snake Graphs ( MSRI Lecture Day 3 from 2011, MSRI Lecture Day 5 from 2011)
• Further resources:
Matrix Formuale and Skein Relations (with Lauren Williams)
Total Positivity by S. Fomin
• (Dec 5) Lecture 38: Quiver Grassmannians and the Caldero-Chapoton Formula ( Lecture 17 from 2011 starting with page 9)
• (Dec 7) Lecture 39: The Bounded Derived Category and Cluster Category (Lecture 24 from 2011, Section 2.1 of Schiffler's notes, Section 2 of Reiten's survey article)
• (Dec 9) Lecture 40: Cluster Tilted Algebras - Acyclic Case (Section 6.1 of Schiffler, Sections 2.2, 2.3 of Schiffler's notes)
• (Dec 12) Lecture 41: Cluster Tilted Algebras - Non-Acyclic Case (Sections 3.4, 5.3, 5.5, 6.4 of Schiffler, Section 3 of Schiffler's notes)
• (Dec 14) Lecture 42: Cluster Algebras and Brane Tilings (Slides 1, Slides 2)
• ## Homework assignments

Assignment Due date
Homework 1 Friday 9/30
Homework 2 Monday 11/21 (postponed)
Homework 3 Monday 12/12