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}.

\(y_{m-1}y_{m+1}=y_m^b+1\) if m odd

\(y_{m-1}y_{m+1}=y_m^c+1\) if m even

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

\(B=\begin{bmatrix} 0 & -b\\ c &\,0 \end{bmatrix}\)

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

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

For \(k\in \{1,2,\cdots, n\}\),  \(x_kx_k' = \prod_{b_{ik}>0} x_i^{b_{ik}}+\prod_{b_{ik}<0} x_i^{|b_{ik}|}\)

\(x_1x_1'=x_2^c+1\) call x_1'=x_3

\(x_2x_2'=x_1^b+1\) call x_2'=x_4

 

\(\mu_k(B)\)

\(-b_{ij}\) if k=i or j

\(b_{ij}\) if \(b_{ik}b_{kj}\leq 0\)

 

\(b_{ij}+b_{ik}b_{kj}\) if \(b_{ik}, b_{kj}>0\)

\(b_{ij}-b_{ik}b_{kj}\) if \(b_{ik},{b_{kj}< 0\)

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

 

 

observations

(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=\infity

If bc\geq 4, all y_m distinct

 

 

algebraic structure

By "Laurent phenomenen" each element in A(b,c) can be uniquely expressed as Laurent polynomial in y_m and y_{m+1} for any m
B.F.Zelevinsky 's result :

 \(A(b,c) =\cap_{m\in\mathbb{Z}}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}] =\cap_{m=0}^{2}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}]\)
standard monomial basis : the set \(\{y_0^{a_0}y_1^{a_1}y_2^{a_2}y_3^{a_3} : a_{m}\in\mathbb{Z}_{\geq 0}, a_0a_2=a_1a_3=0\}\) is a \(\mathbb{Z}\)-basis of \(A(b,c)\).
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,
\(A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/<y_0y_2-y_1^b-1,y_1y_3-y_2^c-1>\)

 

 

하위페이지

• rank 2 cluster algebra
  
  • rank 2 cluster algebra canonical basis
    
    • Sherman-Zelevinsky argument
      
  • rank 2 cluster algebra examples
    

 

 

history

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