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Newton Polytopes of Cluster Variables - Project O · PDF fileNewton Polytopes of Cluster Variables by Adam Michael Kalman A dissertation submitted in partial satisfaction of the requirements

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Newton Polytopes of Cluster Variables

by

Adam Michael Kalman

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Mathematics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Associate Professor Lauren K. Williams, ChairProfessor Bernd Sturmfels

Professor Alistair J. Sinclair

Fall 2014

Newton Polytopes of Cluster Variables

Copyright 2014

by

Adam Michael Kalman

1

Abstract

Newton Polytopes of Cluster Variables

by

Adam Michael Kalman

Doctor of Philosophy in Mathematics

University of California, Berkeley

Associate Professor Lauren K. Williams, Chair

We study Newton polytopes of cluster variables in type An cluster algebras, whose cluster

and coefficient variables are indexed by the diagonals and boundary segments of a polygon.

Our main results include an explicit description of the affine hull and facets of the Newton

polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In

particular, we show that every Laurent monomial in a Laurent expansion of a type A cluster

variable corresponds to a vertex of the Newton polytope. We also describe the face lattice

of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of

the associated snake graph. Other results include a geometric interpretation of the proper

Laurent property in type A based on Newton polytopes, and a proof that the Newton

polytope of a type A cluster variable has no relative interior lattice points. We also consider

extensions of these ideas, results, and methods to other cluster algebras.

i

To Isaac and Jonah

ii

Contents

Contents ii

1 Introduction 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Cluster Algebra Associated to a Quiver . . . . . . . . . . . . . . . . . . 31.3 Cluster Expansions from Matchings . . . . . . . . . . . . . . . . . . . . . . . 7

2 Newton Polytopes of Cluster Variables of Type A 102.1 Affine Hull, Facets, and Face Lattice Results . . . . . . . . . . . . . . . . . . 102.2 Proofs and Minor Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Other Results and Conjectures 333.1 Remarks for Type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Proper Laurent Property and No Interior Lattice Points for Type A . . . . . 343.3 Remarks and Conjectures for Other Cluster Algebras . . . . . . . . . . . . . 38

4 Code and Visualizations 404.1 Code and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Bibliography 47

iii

Acknowledgments

I would first like to thank my advisor, Lauren Williams, for her unwavering guidance and

support, and for introducing me to the study of cluster algebras. I feel privileged to have

had the opportunity to work with such a talented mathematician and extraordinary person,

and I could never have made it to this point without her.

I would also like to thank the other faculty and staff of the UC Berkeley Mathematics

Department, especially Mark Haiman, Bernd Sturmfels, and Paul Vojta, for teaching me

lots of wonderful mathematics over these last few years and for taking time to serve on my

committees.

Special thanks go to Gregg Musiker, Dylan Thurston, and Jim Propp for helpful advice

and enlightening conversations.

Over the past few years, I have had the privilege of meeting and working with many

colleagues and collaborators who have generously shared their time and ideas, including

Christian Hilaire, Steven Karp, Jose Rodriguez, and the dozens of other truly amazing

graduate students I met along the way.

Finally, I would like to thank my family. Anna, Isaac, and Jonah, you are my inspiration

and my strongest support. Its all for you.

1

Chapter 1

Introduction

1.1 Overview

Cluster algebras, introduced by Fomin and Zelevinsky in the early 2000s [9, 10, 11], are a

class of commutative rings equipped with a distinguished set of generators (cluster variables)

that are grouped into sets of constant cardinality n (the clusters). A cluster algebra may be

defined from an initial cluster (x1, ..., xm) and a quiver, which contains combinatorial data

for the process of mutation, in which new clusters and quivers are created recursively from

old ones. There may also be coefficients involved in the construction. The cluster algebra is

the algebra generated by all cluster variables, after mutation is repeated ad infinitum.

The original motivation for cluster algebras was to create a combinatorial framework for

studying total positivity and dual canonical bases in semisimple Lie groups. Since then,

cluster structures have been studied in various areas of mathematics.

Perhaps the most fundamental example of a cluster algebra is the cluster algebra associ-

ated with triangulations of a polygon. Cluster algebras of finite type (i.e. those with finitely

many cluster variables) are classified by Dynkin diagrams, and the cluster algebras coming

from triangulations of a polygon are precisely those of type A. In this model, diagonals

correspond to cluster variables, triangulations (i.e. maximal collections of non-intersecting

diagonals) correspond to clusters, boundary segments correspond to coefficient variables,

and mutation corresponds to a local move called a flip of the triangulation, in which one

diagonal is replaced with another one.

A consequence of the definition of cluster algebra is that every cluster variable is a

rational function in the initial cluster variables, but more strongly, the remarkable Laurent

Phenomenon [9] states that every cluster variable is in fact a Laurent polynomial in those

variables.

In the last ten years, much work has been done on Laurent expansion formulas for cluster

CHAPTER 1. INTRODUCTION 2

algebras. Carroll and Price (in unpublished results [2]) were the first to discover formulas

for Laurent expansions of cluster variables in the case of a triangulated polygon, writing one

formula in terms of paths and another in terms of perfect matchings of so-called snake graphs

[19]. Their formula was subsequently rediscovered and generalized in a series of works [20],

[21], [22], [16], [18], with [18] providing Laurent expansions of cluster variables associated to

cluster algebras from arbitrary surfaces.

In this thesis, we study the Newton polytope of the Laurent expansion of a cluster

variable in a type A cluster algebra with respect to an arbitrary cluster. The study of

Newton polytopes of Laurent expansions of cluster variables was initiated by Sherman and

Zelevinsky in their study of rank 2 cluster algebras, in which it was shown that the Newton

polygon of any cluster variable in a rank 2 cluster algebra of finite or affine type is a triangle

[23]. We will extend these results in type A by considering cluster algebras of arbitrary rank.

One motivation for this study is that the cluster algebra of type An is isomorphic to the

coordinate ring of the affine cone over the Grassmannian Gr(2, n). From this perspective,

exchange relations in the cluster algebra correspond to three-term Plucker relations, and

Laurent expansions of cluster variables give expressions for arbitrary Plucker coordinates in

terms of a fixed set of algebraically independent Plucker coordinates. Thus the results in this

paper can be interpreted as results about the Grassmannian. For example, a facet description

of the Newton polytope can be interpreted in terms of constraints on Plucker coordinates.

Another motivation for the study of these Newton polytopes is that understanding Newton

polytopes of cluster variables has been useful for understanding bases of cluster algebras [3],

[23].

In the remainder of this chapter, we review the definition of a cluster algebra associated

to a quiver, as well as the relevant formula for Laurent expansion of a cluster variable in

terms of perfect matchings. Our main results in chapter 2 are explicit descriptions of the

affine hull and facets of the Newton polytope of a Laurent expansion of any cluster variable

of type An, as well as a description of the face lattice of such a polytope via an isomorphism

with the lattice of elementary subgraphs of the associated snake graph. Our affine hull and

facet description can be read off the triangulation directly. We also show that every Laurent

monomial in a Laurent expansion of a type A cluster variable corresponds to a vertex of the

Newton polytope. In chapter 3, we present other results for type A, including a geometric

interpretation of the proper Laurent property based on Newton polytopes, and a proof that

the Newton polytope of a type A cluster variable contains no relative interior lattice points.

We also consider extensions of these ideas, results, and methods to other cluster algebras.

Chapter 4 contains computer code written for the purpose of making computations with

cluster variables and generating conjectures, such as the ones that became the theorems and

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