"Rank 2 cluster algebra"의 두 판 사이의 차이
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==cluster variables and exchange relations== | ==cluster variables and exchange relations== | ||
| + | * Fix two positive integers b and c. | ||
| + | * Let $y_1$ and $y_2$ be variable in the field <math>F=\mathbb{Q}(y_1,y_2)</math> | ||
| + | * Define a sequence $\{y_n\}$ by | ||
| + | $$ | ||
| + | y_{m-1}y_{m+1}= | ||
| + | \begin{cases} | ||
| + | y_m^b+1, & \text{if $m$ is odd}\\ | ||
| + | y_m^c+1, & \text{if $m$ is even} \\ | ||
| + | \end{cases} | ||
| + | $$ | ||
| + | * We call this ''''exchange relation'''' | ||
| + | * <math>y_m</math>'s are called ''''cluster variable'''' | ||
| + | * <math>\{y_i,y_{i+1}\}</math> "'''cluster'''" | ||
| + | * <math>\{y_m^py_{m+1}^q\}</math> "'''cluster monomials'''" (supported on a given cluster) | ||
| + | * Note : we can use the exchange relation any $y_m$ in terms of arbitrary cluster <math>\{y_i,y_{i+1}\}</math> (rational expression) | ||
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| − | <math> | + | ===matrix formulation=== |
| + | :<math>B=\begin{bmatrix} 0 & -b\\ c &\,0 \end{bmatrix}</math> | ||
| + | :<math>\mu_{1}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}</math> | ||
| + | :<math>\mu_{2}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}</math> | ||
| + | * <math>x_1x_1'=x_2^c+1</math> call $x_1'=x_3$ | ||
| + | * <math>x_2x_2'=x_1^b+1</math> call $x_2'=x_4$ | ||
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==observations== | ==observations== | ||
| − | + | ;theorem (FZ) : For any $b,c$, $y_m$ is a Laurent polynomial. | |
| − | (FZ) | + | * Positivity conjecture: coefficients of these Laurent polynomials are positive (numerator and denomonator always have ) |
| − | + | * In this example, <math>bc\leq 3</math> iff the recurrence is periodic | |
| − | For any b,c, y_m is a Laurent polynomial. | ||
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| − | Positivity conjecture: coefficients of these Laurent polynomials are positive (numerator and denomonator always have ) | ||
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| − | In this example, | ||
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| − | <math>bc\leq 3</math> iff the recurrence is periodic | ||
| 85번째 줄: | 50번째 줄: | ||
==cluster algebra associated to Cartan matrices== | ==cluster algebra associated to Cartan matrices== | ||
| − | + | * Finite type classification <math>A(b,c)</math> related to root systems of Cartan matrix | |
| − | Finite type classification | + | :<math> \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}</math> |
| − | + | * 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> | |
| − | <math>A(b,c)</math> related to root systems of Cartan matrix | + | * when <math>bc\leq 3</math>, <math>y_m=y_n</math> if and only if <math>m\equiv n \mod (h+2)</math> where h is [[Coxeter number and dual Coxeter number|Coxeter number]] |
| − | + | * $bc=1, h=2$ | |
| − | <math> \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}</math> | + | * $bc=2, h=4$ |
| − | + | * $bc=3, h=6$ | |
| − | 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> | + | * $bc\geq 4, h=\infty$ |
| − | + | * If $bc\geq 4$, all $y_m$ are distinct | |
| − | 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 and dual 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=\ | ||
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| − | If bc\geq 4, all y_m distinct | ||
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==algebraic structure== | ==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 $m$ | |
| − | 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 | + | ;theorem (Berenstein, Fomin and Zelevinsky) : |
| − | + | :<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> | |
| − | + | * standard monomial basis : the following set is a <math>\mathbb{Z}</math>-basis of <math>A(b,c)</math> | |
| − | + | :<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> | |
| − | + | * 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> | |
| − | + | * <math>A(b,c)</math> is finitely generated. In fact, | |
| − | + | :<math>A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/\langle y_0y_2-y_1^b-1,y_1y_3-y_2^c-1\rangle</math> | |
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==related items== | ==related items== | ||
| − | + | * [[Rank 2 cluster algebra examples]] | |
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| − | * [[ | ||
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==articles== | ==articles== | ||
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* '''[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. | * '''[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. | ||
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[[분류:cluster algebra]] | [[분류:cluster algebra]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
2013년 10월 11일 (금) 08:23 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 in the field \(F=\mathbb{Q}(y_1,y_2)\)
- Define a sequence $\{y_n\}$ by
$$ y_{m-1}y_{m+1}= \begin{cases} y_m^b+1, & \text{if $m$ is odd}\\ y_m^c+1, & \text{if $m$ is even} \\ \end{cases} $$
- 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
\[B=\begin{bmatrix} 0 & -b\\ c &\,0 \end{bmatrix}\] \[\mu_{1}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}\] \[\mu_{2}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}\]
- \(x_1x_1'=x_2^c+1\) call $x_1'=x_3$
- \(x_2x_2'=x_1^b+1\) call $x_2'=x_4$
observations
- theorem (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
- $bc=1, h=2$
- $bc=2, h=4$
- $bc=3, h=6$
- $bc\geq 4, h=\infty$
- If $bc\geq 4$, all $y_m$ are 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 $m$
- theorem (Berenstein, Fomin and Zelevinsky)
\[A(b,c) =\cap_{m\in\mathbb{Z}}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}] =\cap_{m=0}^{2}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}]\]
- standard monomial basis : the following set is a \(\mathbb{Z}\)-basis of \(A(b,c)\)
\[\{y_0^{a_0}y_1^{a_1}y_2^{a_2}y_3^{a_3} : a_{m}\in\mathbb{Z}_{\geq 0}, a_0a_2=a_1a_3=0\}\]
- 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,
\[A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/\langle y_0y_2-y_1^b-1,y_1y_3-y_2^c-1\rangle\]
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.