Rank 2 cluster algebra

수학노트
http://bomber0.myid.net/ (토론)님의 2011년 1월 28일 (금) 05:33 판
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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. 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

 

 

canonical basis (1/25/2011)

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 :

*cap인지 cup인지 확인 필요*\(A(b,c) =\cup_{m\in\mathbb{Z}}\mathbb{Z}[y_n^{\pm 1},y_{m+1}^{\pm 1}] =\cup_{m=0}^{\alpha}\mathbb{Z}[y_n^{\pm 1},y_{m+1}^{\pm 1}]\)
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. In fact,
\(A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/<y_0y_2-y_1^b-1,y_1y_3-y_2^c-1>\)

 

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

 

 

positive elements

\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 ([SZ2003])

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 : B 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 \(\{\alpha_1, \alpha_2\}\). \(\alpha=a_1\alpha_1+a_2\alpha_2\) corresponds to \((a_1,a_2)\)

 

\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 is the set of all 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
\thm

The bijection B<-> Q translates the action of each \(\sigma_p\) on B into piecewise linear transformation of Q ;

\(\sigma_1(a_1,a_2) = (a_1 , c \max (a_1,0) -a_2)\)

\(\sigma_2(a_1,a_2) = (b \max (a_2,0)-a_1, a_2)\)

 

 

canonical basis in finite case

\thm 1 (finite case)
If bc\leq 3, then B is the set of all cluster monomials.

 

 

canonical basis in affine case

In affine case, introduce z an element of 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 (affine case)

If bc=4, B = the set of all cluster monomials union \(\{z_n : n\geq 1\}\)

 

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 simple roots

Initial cluster variables y_1,y_2 correspond to negative simple roots

 

Each cluster variable \(y_m\neq y_1, y_2\) 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}

(proof) FZ

Each cluster variable y_m = \chi[\alpha(m)] where m\neq 1,2 has denominator \alpha(m) which is positive root.

Set of all cluster variable's \{y_m, m=1,2 \} <-> \{positive real roots \}

To finish prop, it's enought to show

(1) For each m\in \mathbb{Z}, \alpha(m) and \alpha(m+1) form \mathbb{Z}-basis of Q

(2) For each m\in \mathbb{Z}, \alpha(m) and \alpha(m+1) are only positive real roots in additive semi group \mathbb{Z}_{\geq 0} \alpha(m)+\mathbb{Z}_{\geq 0} \alpha(m+1)

(3) The union \cup_{m\in\mathbb{Z}} [ \mathbb{Z}_{\geq 0} \alpha(m)+\mathbb{Z}_{\geq 0} \alpha(m+1)] = Q-\phi_{+}^{im} ■

 

 

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