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Cluster algebras I: Foundations

Sergey Fomin, Andrei Zelevinsky

TL;DR

Cluster algebras provide a new combinatorial framework for dual canonical bases and total positivity by organizing generators into clusters connected by exchange relations. The paper develops an axiomatic setup of exchange patterns on trees, proves the Laurent phenomenon for cluster variables, and analyzes the exponents and coefficients governing mutations. It treats the rank-2 case in depth, revealing a close connection to root systems and Cartan data, and studies the exchange graph, including finite-type cycles and infinite-type trees. These foundations pave the way for integrating cluster algebras with representation theory and algebraic geometry, highlighting links to Grassmannians and double Bruhat cells.

Abstract

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

Cluster algebras I: Foundations

TL;DR

Cluster algebras provide a new combinatorial framework for dual canonical bases and total positivity by organizing generators into clusters connected by exchange relations. The paper develops an axiomatic setup of exchange patterns on trees, proves the Laurent phenomenon for cluster variables, and analyzes the exponents and coefficients governing mutations. It treats the rank-2 case in depth, revealing a close connection to root systems and Cartan data, and studies the exchange graph, including finite-type cycles and infinite-type trees. These foundations pave the way for integrating cluster algebras with representation theory and algebraic geometry, highlighting links to Grassmannians and double Bruhat cells.

Abstract

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

Paper Structure

This paper contains 7 sections, 20 theorems, 118 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main definitions
  3. The Laurent phenomenon
  4. Exchange relations: the exponents
  5. Exchange relations: the coefficients
  6. The rank $2$ case
  7. The exchange graph

Key Result

Proposition 2.6

Let $\mathcal{A}=\mathcal{A}_{\mathbb{A}} (\mathcal{M})$ be a cluster algebra of rank $n$. The $\mathbb{A}$-subalgebra of $\mathcal{A}$ generated by $\cup_{t \in \mathbb{T}_m} {\bf x} (t)$ is naturally identified with the cluster algebra $\mathcal{A}_{\mathbb{A}'} (\mathcal{M}')$, where $\mathbb{A}'

Figures (6)

  • Figure 1: The Grassmannian $Gr_{2,5}$
  • Figure 2: The "caterpillar" tree $\mathbb{T}_{n,m}$, for $n=4$, $m=8$
  • Figure 3: The exchange graph for a cluster algebra of type $A_2$
  • Figure 4: The two-layer brick wall
  • Figure 5: Close-up of a brick
  • ...and 1 more figures

Theorems & Definitions (40)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 30 more