[Paper Review] Canonically positive basis of cluster algebras of type $A_2^{(1)}$

Giovanni Cerulli Irelli|arXiv (Cornell University)|Apr 16, 2009

Advanced Topics in Algebra4 citations

TL;DR

This paper constructs a canonically positive basis for cluster algebras of type $A_2^{(1)}$ by solving T- and Y-system recurrence relations, proving that the basis elements are parameterized by $\mathbb{Z}^3$ via $\mathbf{g}$-vectors and that the denominator vector map is a bijection, enabling explicit decomposition of root lattice elements.

ABSTRACT

In this paper we study cluster algebras $\myAA$ of type $A_2^{(1)}$. We solve the recurrence relations among the cluster variables (which form a T--system of type $A_2^{(1)}$). We solve the recurrence relations among the coefficients of $\myAA$ (which form a Y--system of type $A_2^{(1)}$). In $\myAA$ there is a natural notion of positivity. We find linear $\BB$ of $\myAA$ such that positive linear combinations of elements of $\BB$ coincide with the cone of positive elements. We call these \emph{atomic bases} of $\myAA$. These are the analogue of the bases found by Sherman and Zelevinsky in type $A_{1}^{(1)}$. Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such in every cluster. We prove that the elements of $\BB$ are parameterized by $\ZZ^3$ via their $\mathbf{g}$--vectors in every cluster. We prove that the denominator vector map in every acyclic seed of $\myAA$ restricts to a bijection between $\BB$ and $\ZZ^3$. In particular this gives an explicit algorithm to determine the virtual canonical decomposition of every element of the root lattice of type $A_2^{(1)}$. We find explicit recurrence relations to express every element of $\myAA$ as linear combinations of elements of $\BB$.

Motivation & Objective

• To establish a canonically positive basis for cluster algebras of type $A_2^{(1)}$.
• To solve the T-system and Y-system recurrence relations governing cluster variables and coefficients.
• To identify a basis $\BB$ such that positive linear combinations of its elements form the full cone of positive elements in the algebra.
• To prove that the $\mathbf{g}$-vectors parameterize the basis elements bijectively via $\mathbb{Z}^3$ in every acyclic seed.
• To provide an algorithm for computing the virtual canonical decomposition of any element in the root lattice of type $A_2^{(1)}$.

Proposed method

• Solving the T-system of type $A_2^{(1)}$ to determine recurrence relations among cluster variables.
• Solving the Y-system of type $A_2^{(1)}$ to derive recurrence relations among coefficients.
• Defining an atomic basis $\BB$ consisting of cluster monomials and additional elements, ensuring positivity under linear combinations.
• Parameterizing the elements of $\BB$ by $\mathbb{Z}^3$ using their $\mathbf{g}$-vectors in each acyclic seed.
• Proving that the denominator vector map restricts to a bijection from $\BB$ to $\mathbb{Z}^3$ in every acyclic seed.
• Deriving explicit recurrence relations to express any element of the algebra as a linear combination of basis elements in $\BB$.

Experimental results

Research questions

• RQ1 What is the structure of a canonically positive basis for cluster algebras of type $A_2^{(1)}$?
• RQ2 How do the T-system and Y-system recurrence relations govern the cluster variables and coefficients in this algebra?
• RQ3 Can the basis elements be parameterized by $\mathbb{Z}^3$ via their $\mathbf{g}$-vectors in every acyclic seed?
• RQ4 Is the denominator vector map a bijection from the basis $\BB$ to $\mathbb{Z}^3$ in every acyclic seed?
• RQ5 What explicit recurrence relations allow the decomposition of any algebra element into the basis $\BB$?

Key findings

• The paper constructs an atomic basis $\BB$ for the cluster algebra of type $A_2^{(1)}$ such that positive linear combinations of its elements form the full cone of positive elements.
• The basis $\BB$ consists of cluster monomials and additional elements, with explicit expressions provided for each element in every cluster.
• The elements of $\BB$ are parameterized by $\mathbb{Z}^3$ via their $\mathbf{g}$-vectors in every acyclic seed.
• The denominator vector map is a bijection from $\BB$ to $\mathbb{Z}^3$ in every acyclic seed, enabling a canonical decomposition of root lattice elements.
• Explicit recurrence relations are derived that allow every element of the algebra to be expressed as a linear combination of basis elements in $\BB$.
• The results provide an algorithmic method to compute the virtual canonical decomposition of any element in the root lattice of type $A_2^{(1)}$.