"Rank 2 cluster algebra"의 두 판 사이의 차이
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From definition, B is invariant under any automorphism of A(b,c) preserving <math>A^{+}(b,c)</math> (Call such a map positive) | From definition, B is invariant under any automorphism of A(b,c) preserving <math>A^{+}(b,c)</math> (Call such a map positive) | ||
| − | For all <math>p\in \mathbb{Z}</math>, there exists a positive automorphism defined by \sigma_p(y_m)=y_{2p-m}. | + | For all <math>p\in \mathbb{Z}</math>, there exists a positive automorphism <math>\sigma_{p}</math> of A(b,c) defined by <math>\sigma_p(y_m)=y_{2p-m}</math>. |
| − | <\sigma_i : i\in \mathbb{Z}> = <\sigma_p,\sigma_{p+1}> | + | <math><\sigma_i : i\in \mathbb{Z}> = <\sigma_p,\sigma_{p+1}></math> for any p<br> \thm |
| − | + | The bijection B<-> Q translates the action of each <math>\sigma_p</math> on B into piecewise linear transformation of Q ; | |
\sigma_1(a_1,a_2) = (a_1 , c \max (a_1,0) -a_2) | \sigma_1(a_1,a_2) = (a_1 , c \max (a_1,0) -a_2) | ||
| 234번째 줄: | 234번째 줄: | ||
| − | \thm 1<br> If bc\leq 3, then B | + | \thm 1 (finite case)<br> If bc\leq 3, then B is the set of all cluster monomials. |
| − | In affine case, introduce | + | |
| + | |||
| + | |||
| + | |||
| + | In affine case, introduce z an element of A(b,c) by | ||
| − | z=y_0y_3-y_1y_2 | + | <math>z=y_0y_3-y_1y_2</math> if (b,c)=(2,2) or |
| − | + | <math>z=y_0^2y_3-(y_1+2)y_2^2</math> if (b,c)=(1,4) | |
| − | Then set z_n = T_n(z) | + | Let T_0, T_1,\cdots, be Chebyshev polynomials defined by <math>T_0=1</math>, <math>T_n(t+t^{-1}) = t^n+t^{-n}</math> for n >0 |
| + | |||
| + | Then set <math>z_n = T_n(z)</math> | ||
| − | \thm 2 | + | \thm 2 (affine case) |
| − | If bc=4, B = | + | If bc=4, B = the set of all cluster monomials + <math>\{z_n : n\geq 1\}</math> |
| 256번째 줄: | 262번째 줄: | ||
| − | For any b,c identify Q with root lattice such that \alpha_1,\alpha_2 correspond to | + | 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 | Initial cluster variables y_1,y_2 correspond to negative simple roots | ||
| 262번째 줄: | 268번째 줄: | ||
| − | Each cluster variable y_m\neq y_1, | + | Each cluster variable y_m\neq y_1, y_2 has form |
| − | |||
| − | |||
| − | positive real root a_1\alpha_1+a_2\alpha_2. | + | <math>\frac{N_{\alpha}(y_1,y_2)}{y_1^{a_1}y_2^{a_2}}</math> 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 | In affine case bc=4, positive imaginary root are all positive integer multiples of root \delta given by | ||
2011년 1월 26일 (수) 09:11 판
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"
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
canonical basis (1/25/2011)
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 :
*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
\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 <-> (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
\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)
\thm 1 (finite case)
If bc\leq 3, then B is the set of all cluster monomials.
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 + \(\{z_n : n\geq 1\}\)
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}
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
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.
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
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- http://dx.doi.org/
question and answers(Math Overflow)
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