"Rank 2 cluster algebra"의 두 판 사이의 차이
| 68번째 줄: | 68번째 줄: | ||
<h5>observations</h5> | <h5>observations</h5> | ||
| − | (FZ) For any b,c, y_m is a Laurent polynomial. | + | (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 ) | Positivity conjecture: coefficients of these Laurent polynomials are positive (numerator and denomonator always have ) | ||
| 104번째 줄: | 106번째 줄: | ||
If bc\geq 4, all y_m distinct | If bc\geq 4, all y_m distinct | ||
| − | + | ||
| − | <h5> | + | <h5>algebraic structure</h5> |
| − | 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<br> B.F.Zelevinsky 's result : | + | 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<br> B.F.Zelevinsky 's result : |
<math>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}]</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> | <math>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}]</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> | ||
2011년 1월 28일 (금) 06:23 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"
\(\{y_m^py_{m+1}^q\}\) "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,
\(A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/<y_0y_2-y_1^b-1,y_1y_3-y_2^c-1>\)
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/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
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- http://ncatlab.org/nlab/show/HomePage
experts on the field