Rank 2 cluster algebra
introduction
- cluster algebra defined by a 2x2 matrix
cluster variables and exchange relations
Fix two positive integers b and c.
Let y_1 and y_2 be variable. 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"
Note : we can use the exchange relation any y_m in terms of arbitrary cluster {y_i,y_{i+1}} (rational expression)
example 1
Put b=c=1
\(y_{m-1}y_{m+1}=y_m+1\)
Start with two variables \(y_1,y_2\).
\(y_3y_1=y_2+1\). so \(y_3=\frac{y_2+1}{y_1}\)
\(y_2y_4=y_3+1 \)implies \(y_4=\frac{y_3+1}{y_2}=\frac{y_1+y_2+1}{y_1y_2}\)
\(y_3y_5=y_4+1\) implies \(y_5=\frac{y_4+1}{y_3}= \frac{y_1+1}{y_2}\) we are getting Laurent polynomials
\(y_4y_6=y_5\) implies \(y_6=\frac{y_5+1}{y_4}= \frac{\frac{y_1+1}{y_2}+1}{\frac{y_1+y_2+1}{y_1y_2}}=\frac{y_1(y_1+1)+y_1y_2}{y_1+y_2+1}=y_1\)
example 2
Put b=1, c=3
y_1,y_2
\(y_3y_1=y_2^3+1\). so \(y_3=\frac{y_2^3+1}{y_1}\)
\(y_2y_4=y_3+1 \)implies \(y_4=\frac{y_3+1}{y_2}=\frac{y_1+y_2^3+1}{y_1y_2}\)
\(y_3y_5=y_4^3+1\) implies \(y_5=\frac{y_4^3+1}{y_3}= \frac{(y_1+1)^3+y_2^3(y_2^3+3y_1+2)}{y_1^2y_2^3}\)http://www.wolframalpha.com/input/?i=((x%2By^3%2B1)^3%2B(xy)^3)/(x^2y^3(y^3%2B1))
Note that we are getting Laurent polynomials.
\(y_6=\frac{(y_1+1)^2+y_2^3}{y_1y_2^2}\)
\(y_7=\frac{(y_1+1)^3+y_2^3}{y_1y_2^3}\)
\(y_8=\frac{y_1+1}{y_2}\)
\(y_9=y_1\)
\(y_{10}=y_2\)
example 3
Put b=c=2.
y_1
y_2
y_3=\frac{y_2^2+1}{y_1}
y_4=\frac{1+y_1^2+2y_2^2+y_2^4}{y_1^2y_2}
y_5 has denominator y_1^3y_2^2
y_6 has denominator y_1^4y_2^3
y_0=\frac{y_1^2+1}{y_2}
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
1/25/2011
lecture following http://arxiv.org/abs/math/0307082v2
y_m cluster variables
\{y_m,y_{m+1}\} clusters
\{y_m^py_{m+1}^q\} cluster monomials (supported on a given cluster)
Goal : define and construct 'canonical basis' B in A(b,c) for bc\leq 4
By "Leurant 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) =\cup_{m\in\mathbb{Z}\mathbb{Z}[y_n^{\pm 1,\y_{m+1}^{\pm 1]=\cup_{m=0}^{\alpha}\mathbb{Z}[y_n^{\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,
A(b,c)=\mathbb{y_0,y_1,y_2,y_3]/<y_0y_2-y_1^b-1,y_1y_3-y_2^c-1>
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
\def
A nonzero element y\in A(b,c) is positive if for any m\in \mathbb{Z}, all coefficients in the expansion of y as Laurent polynomial in y_m and y_{m+1} are positive.
A^{+}(b,c) = set of positive elements in A(b,c) semiring
\thm (Sherman/Zelevinsky)
Suppose bc\leq 4. Then there exists a unique basis B of A(b,c) such that A^{+}(b,c) = set of positive integer linear combinations of elements of B.
\remark
If exists, then the uniqueness follows : must consist of all indecomposable elements of A^{+}(b,c) i.e. those one cannot be written as positive sum of 2 elements in A^{+}(b,c).
Let Q=\mathbb{Z}^2 be rank2 lattice with basis \{\alpha_1, \alpha_2\}. \alpha=a_1\alpha_1+a_2\alpha_2 <-> (a_1,a_2)
\Theorem *
When bc\leq 4, for each \alpha=(a_1,a_2)\in Q, there exists unique basis element \chi[\alpha]\in B of form \chi[\alpha]=\frac{N_{\alpha}(y_1,y_2)}{y_1^{a_1}y_2^{a_2}} where N_{\alpha} is a polynomial with constant term 1.
Map \alpha \to \chi[\alpha] is bijection \mathbb{Z}^2\to B
when bc\leq 3, B=\{cluster monomials\}
examples : b=c=1 case and b=c=2 case
From definition, B is invariant under any automorphism of A(b,c) preserving A^{+}(b,c) (Call such a map positive)
For all p\in \mathbb{Z}, there exists a positive automorphism \sigma_{p} of A(b,c) defined by \sigma_p(y_m)=y_{2p-m}.
<\sigma_i : i\in \mathbb{Z}> = <\sigma_p,\sigma_{p+1}> for any p
Theorem : Bijection B<-> Q translates action of each \sigma_p on B into piecewise linear transformation of Q ;
\sigma_1(a_1,a_2) = (a_1 , c \max (a_1,0) -a_2)
\sigma_2(a_1,a_2) = (b \max (a_2,0)-a_1, a_2)
\thm 1
If bc\leq 3, then B =\{\text{cluster monomials} \}
In affine case, introduce Z\in A(b,c) by
z=y_0y_3-y_1y_2 if (b,c)=(2,2) or z=y_0^2y_3-(y_1+2)y_2^2 if (b,c)=(1,4)
Let T_0, T_1,\cdots, be Chebyshev polynomials defined by T_0=1, T_n(t+t^{-1}) = t^n+t^{-n} for n >0
Then set z_n = T_n(z)
\thm 2
If bc=4, B = \{cluster monomials\}\cup \{z_n : n\geq 1\}
open problem : Give combinatorial formula for Laurent expansion of cluster variable's y_m when bc>4.
For any b,c identify Q with root lattice such that \alpha_1,\alpha_2 correspond to siple roots
Initial cluster variables y_1,y_2 correspond to negative simple roots
Each cluster variable y_m\neq y_1, y_21 has form
\frac{N_{\alpha}(y_1,y_2)}{y_1^{a_1}y_2^{a_2}} for a
positive real root a_1\alpha_1+a_2\alpha_2.
In affine case bc=4, positive imaginary root are all positive integer multiples of root \delta given by
\delta =\alpha_1+\alpha_2 if (b,c)=(2,2)
\delta = \alpha_1+2\alpha_2 if (b,c)=(1,4)
\proposition
bc=4
Cluster monimials <-> root lattice - {imaginary roots}
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