Rank 2 cluster algebra

introduction

• cluster algebra defined by a 2x2 matrix
• Laurent phenomenon
• Positivity conjecture
• finite classification

 

 

cluster variables and exchange relations

• Fix two positive integers b and c.
• Let $y_1$ and $y_2$ be variable in the field \(F=\mathbb{Q}(y_1,y_2)\)
• Define a sequence $\{y_n\}$ by

$$ y_{m-1}y_{m+1}= \begin{cases} y_m^b+1, & \text{if $m$ is odd}\\ y_m^c+1, & \text{if $m$ is even} \\ \end{cases} $$

• We call this 'exchange relation'
• \(y_m\)'s are called 'cluster variable'
• \(\{y_i,y_{i+1}\}\) "cluster"
• \(\{y_m^py_{m+1}^q\}\) "cluster monomials" (supported on a given cluster)
• Note : we can use the exchange relation any $y_m$ in terms of arbitrary cluster \(\{y_i,y_{i+1}\}\) (rational expression)

 

matrix formulation

\[B=\begin{bmatrix} 0 & -b\\ c &\,0 \end{bmatrix}\] \[\mu_{1}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}\] \[\mu_{2}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}\]

• \(x_1x_1'=x_2^c+1\) call $x_1'=x_3$
• \(x_2x_2'=x_1^b+1\) call $x_2'=x_4$

 

observations

theorem (FZ)

For any $b,c$, $y_m$ is a Laurent polynomial.

• Positivity conjecture: coefficients of these Laurent polynomials are positive (numerator and denomonator always have )
• In this example, \(bc\leq 3\) iff the recurrence is periodic

 

 

cluster algebra associated to Cartan matrices

• Finite type classification \(A(b,c)\) related to root systems of Cartan matrix

\[ \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}\]

• Say \(A(b,c)\) is of finite/affine/indefinite type if \(bc\leq 3\), \(bc=4\), \(bc>4\)
• when \(bc\leq 3\), \(y_m=y_n\) if and only if \(m\equiv n \mod (h+2)\) where h is Coxeter number
• $bc=1, h=2$
• $bc=2, h=4$
• $bc=3, h=6$
• $bc\geq 4, h=\infty$
• If $bc\geq 4$, all $y_m$ are distinct

 

algebraic structure

• By "Laurent phenomenon" each element in $A(b,c)$ can be uniquely expressed as Laurent polynomial in $y_m$ and $y_{m+1}$ for any $m$

theorem (Berenstein, Fomin and Zelevinsky)

\[A(b,c)=\cap_{m\in\mathbb{Z}}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}] =\cap_{m=0}^{2}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}]\]

• standard monomial basis : the following set  is a \(\mathbb{Z}\)-basis of \(A(b,c)\)

\[\{y_0^{a_0}y_1^{a_1}y_2^{a_2}y_3^{a_3} : a_{m}\in\mathbb{Z}_{\geq 0}, a_0a_2=a_1a_3=0\}\]

• Here support of any such monomial is

\[\{y_0,y_1\},\{y_1,y_2\},\{y_2,y_3\},\{y_0,y_3\}\]

• \(A(b,c)\) is finitely generated. In fact,

\[A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/\langle y_0y_2-y_1^b-1,y_1y_3-y_2^c-1\rangle\]

 

related items

• Rank 2 cluster algebra examples 

 

articles

• [SZ2003]Sherman, Paul, and Andrei Zelevinsky. 2003. Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. math/0307082 (July 7). http://arxiv.org/abs/math/0307082.