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An introduction to representation-theoretic canonical bases of cluster algebras

Fan Qin

TL;DR

This paper develops a representation-theoretic lens on cluster algebras by introducing and analyzing the common triangular (Kazhdan-Lusztig-type) bases $\mathbf{L}$, connecting them to dual canonical bases $\mathbf{B}^*$ in Lie-theoretic contexts. It builds seeds from signed words, details cluster operations (freezing, base change), and proves how the KL algorithm and categorification ideas produce robust bases that are compatible across mutation sequences and extend to infinite ranks via colimits. The results demonstrate the existence and propagation of common triangular bases for many cluster algebras arising from Lie theory, including quasi-categorifications in symmetric cases, and provide a framework to transfer structures through extensions and reductions of seeds. The work equips researchers with concrete tools to study canonical bases in cluster algebras, with explicit constructions and examples from quantum groups, unipotent cells, and quantum affine algebras, and offers an overarching perspective on extending these bases to infinite-dimensional settings.

Abstract

This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds associated with signed words, cluster operations (freezing and base change), the extension technique from finite to infinite rank, which are convenient tools to study these algebras and their bases. We present recent results on the topic and provide examples. An appendix collects detailed examples of some cluster algebras from Lie theory, including those from Lie groups and representations of quantum affine algebras.

An introduction to representation-theoretic canonical bases of cluster algebras

TL;DR

This paper develops a representation-theoretic lens on cluster algebras by introducing and analyzing the common triangular (Kazhdan-Lusztig-type) bases , connecting them to dual canonical bases in Lie-theoretic contexts. It builds seeds from signed words, details cluster operations (freezing, base change), and proves how the KL algorithm and categorification ideas produce robust bases that are compatible across mutation sequences and extend to infinite ranks via colimits. The results demonstrate the existence and propagation of common triangular bases for many cluster algebras arising from Lie theory, including quasi-categorifications in symmetric cases, and provide a framework to transfer structures through extensions and reductions of seeds. The work equips researchers with concrete tools to study canonical bases in cluster algebras, with explicit constructions and examples from quantum groups, unipotent cells, and quantum affine algebras, and offers an overarching perspective on extending these bases to infinite-dimensional settings.

Abstract

This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds associated with signed words, cluster operations (freezing and base change), the extension technique from finite to infinite rank, which are convenient tools to study these algebras and their bases. We present recent results on the topic and provide examples. An appendix collects detailed examples of some cluster algebras from Lie theory, including those from Lie groups and representations of quantum affine algebras.

Paper Structure

This paper contains 36 sections, 36 theorems, 63 equations, 11 figures.

Table of Contents

  1. Introduction
  2. An informal introduction to cluster algebras
  3. Canonical bases of cluster algebras
  4. Contents.
  5. Convention
  6. Cluster algebras
  7. Definition of cluster algebras
  8. Tropical points
  9. Degrees and pointedness
  10. Injective reachability
  11. Valuations, partial compactification, and optimization
  12. Seeds associated with signed words
  13. Seeds construction
  14. Operations on signed words
  15. Injective reachable sequences and fundamental variables
  16. ...and 21 more sections

Key Result

Lemma 2.12

Assume that $z\in\boldsymbol{U}$ and $z'$ is a localized cluster monomial of a seed ${\bf s}'$. If $z$ and $z'$ are both $m$-pointed in $\boldsymbol{LP}({\bf s})$, and they are compatibly pointed at ${\bf s},{\bf s}',{\bf s}'[-1]$, then $z=z'$.

Figures (11)

  • Figure 2.1: The ice quiver for a seed of ${\mathbb C}[SL_{3}]$
  • Figure 2.2: The quiver for an infinite seed of type $A_{1}$ in geiss2024representations
  • Figure 3.1: The ice quiver for $\dot{{\bf s}}(\underline{\mathbf{j}})$, $\underline{\mathbf{j}}=(1,2,3)^{4}$.
  • Figure 3.2: The ice quiver for $\dot{{\bf s}}(\underline{\mathbf{i}})$, $\underline{\mathbf{i}}=(1,2,3)^{\infty}$.
  • Figure 3.3: The ice quiver for $\ddot{{\bf s}}(\underline{\mathbf{i}})$, $\underline{\mathbf{i}}=((1,2)^{\infty},1,-1,2,-2,1,-1,(2,1)^{\infty})$
  • ...and 6 more figures

Theorems & Definitions (99)

  • Remark 1.1: Different canonical bases
  • Remark 2.1
  • Example 2.2: A seed from ${\mathbb C}[SL_3]$
  • Example 2.3: An infinite seed of type $A_1$
  • Definition 2.5
  • Example 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9: Dominance order qin2017triangular cerulli2015caldero
  • Definition 2.10
  • ...and 89 more