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82번째 줄: |
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| − | <h5>canonical basis (1/25/2011)</h5> | + | <h5>classification</h5> |
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| − | Goal : define and construct 'canonical basis' B in A(b,c) for <math>bc\leq 4</math><br> 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<br> B.F.Zelevinsky 's result :
| + | Finite type classification : |
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| − | '''*cap인지 cup인지 확인 필요*'''<math>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}]</math><br> standard monomial basis : the set <math>\{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\}</math> is a <math>\mathbb{Z}</math>-basis of <math>A(b,c)</math>.<br> Here support of any such monomial is <math>\{y_0,y_1\},\{y_1,y_2\},\{y_2,y_3\},\{y_0,y_3\}</math>.<br><math>A(b,c)</math> is finitely generated. In fact,<br><math>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></math>
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| − | Finite type classification :
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| | <math>A(b,c)</math> related to root systems of Cartan matrix | | <math>A(b,c)</math> related to root systems of Cartan matrix |
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| − | <math> \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}</math>
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| − | Say <math>A(b,c)</math> is of finite/affine/indefinite type if <math>bc\leq 3</math>, <math>bc=4</math>, <math>bc>4</math>
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| − | when <math>bc\leq 3</math>
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| − | <math>y_m=y_n</math> if and only if <math>m\equiv n \mod (h+2)</math> where h is [[Coxeter number|coxeter number]]
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| − | bc=1, h=2
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| − | bc=2, h=4
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| − | bc=3, h=6
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| − | bc\geq 4, h=\infity
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| − | If bc\geq 4, all y_m distinct
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| + | <math> \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}</math> |
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| − | <h5>positive elements</h5>
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| − | \def
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| − | A nonzero element <math>y\in A(b,c)</math> is positive if for any <math>m\in \mathbb{Z}</math>, all coefficients in the expansion of y as Laurent polynomial in y_{m} and y_{m+1} are positive.
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| − | <math>A^{+}(b,c) </math>= set of positive elements in <math>A(b,c)</math> semiring
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| − | \thm ('''[SZ2003]''') | |
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| − | Suppose <math>bc\leq 4</math>. Then there exists a unique basis B of <math>A(b,c)</math> such that <math>A^{+}(b,c) </math> = set of positive integer linear combinations of elements of B.
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| − | \remark
| + | Say <math>A(b,c)</math> is of finite/affine/indefinite type if <math>bc\leq 3</math>, <math>bc=4</math>, <math>bc>4</math> |
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| − | If exists, then the uniqueness follows : B must consist of all indecomposable elements of <math>A^{+}(b,c)</math> i.e. those one cannot be written as positive sum of 2 elements in <math>A^{+}(b,c) </math>.
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| − | Let <math>Q=\mathbb{Z}^2</math> be rank2 lattice with basis <math>\{\alpha_1, \alpha_2\}</math>. <math>\alpha=a_1\alpha_1+a_2\alpha_2</math> corresponds to <math>(a_1,a_2)</math>
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| − | \Theorem *
| + | when <math>bc\leq 3</math> |
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| − | When <math>bc\leq 4</math>, for each <math>\alpha=(a_1,a_2)\in Q</math>, there exists unique basis element <math>\chi[\alpha]\in B </math> of form <math>\chi[\alpha]=\frac{N_{\alpha}(y_1,y_2)}{y_1^{a_1}y_2^{a_2}}</math> where <math>N_{\alpha}</math> is a polynomial with constant term 1.
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| − | Map <math>\alpha \to \chi[\alpha]</math> is bijection <math>\mathbb{Z}^2\to B</math>
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| − | When <math>bc\leq 3</math>, B is the set of all cluster monomials.
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| − | examples : b=c=1 case and b=c=2 case
| + | <math>y_m=y_n</math> if and only if <math>m\equiv n \mod (h+2)</math> where h is [[Coxeter number|coxeter number]] |
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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)
| + | bc=1, h=2 |
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| − | 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>.
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| − | <math><\sigma_i : i\in \mathbb{Z}> = <\sigma_p,\sigma_{p+1}></math> for any p<br> \thm
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| − | The bijection B<-> Q translates the action of each <math>\sigma_p</math> on B into piecewise linear transformation of Q ;
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| − | <math>\sigma_1(a_1,a_2) = (a_1 , c \max (a_1,0) -a_2)</math>
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| − | <math>\sigma_2(a_1,a_2) = (b \max (a_2,0)-a_1, a_2)</math>
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| + | bc=2, h=4 |
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| − | <h5>canonical basis in finite case</h5>
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| − | \thm 1 (finite case)<br> If bc\leq 3, then B is the set of all cluster monomials.
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| + | bc=3, h=6 |
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| − | <h5>canonical basis in affine case</h5>
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| − | In affine case, introduce z an element of A(b,c) by
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| − | <math>z=y_0y_3-y_1y_2</math> if (b,c)=(2,2) or
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| − | <math>z=y_0^2y_3-(y_1+2)y_2^2</math> if (b,c)=(1,4)
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| − | 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
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| − | Then set <math>z_n = T_n(z)</math>
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| − | \thm 2 (affine case) | + | bc\geq 4, h=\infity |
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| − | If bc=4, B = the set of all cluster monomials union <math>\{z_n : n\geq 1\}</math>
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| − | '''open problem''' : Give combinatorial formula for Laurent expansion of cluster variable's y_m when bc>4.
| + | If bc\geq 4, all y_m distinct |
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| − | For any b,c identify Q with root lattice such that \alpha_1,\alpha_2 correspond to simple roots
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| − | Initial cluster variables y_1,y_2 correspond to negative simple roots
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| − | Each cluster variable <math>y_m\neq y_1, y_2</math> has form
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| − | <math>\frac{N_{\alpha}(y_1,y_2)}{y_1^{a_1}y_2^{a_2}}</math> for a positive real root <math>a_1\alpha_1+a_2\alpha_2</math>.
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| − | In affine case bc=4, positive imaginary root are all positive integer multiples of root \delta given by
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| − | <math>\delta =\alpha_1+\alpha_2</math> if (b,c)=(2,2)
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| − | <math>\delta = \alpha_1+2\alpha_2</math> if (b,c)=(1,4)
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| − | \proposition
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| − | bc=4
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| − | Cluster monimials <-> root lattice - {imaginary roots}
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| − | (proof) FZ
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| − | Each cluster variable y_m = \chi[\alpha(m)] where m\neq 1,2 has denominator \alpha(m) which is positive root.
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| − | Set of all cluster variable's \{y_m, m=1,2 \} <-> \{positive real roots \}
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| − | To finish prop, it's enought to show
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| − | (1) For each m\in \mathbb{Z}, \alpha(m) and \alpha(m+1) form \mathbb{Z}-basis of Q
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| − | (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)
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| − | (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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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
classification
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
history
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