"Rank 2 cluster algebra"의 두 판 사이의 차이
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| (같은 사용자의 중간 판 2개는 보이지 않습니다) | |||
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* finite classification | * finite classification | ||
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==cluster variables and exchange relations== | ==cluster variables and exchange relations== | ||
* Fix two positive integers b and c. | * Fix two positive integers b and c. | ||
* Let <math>y_1</math> and <math>y_2</math> be variable in the field <math>F=\mathbb{Q}(y_1,y_2)</math> | * Let <math>y_1</math> and <math>y_2</math> be variable in the field <math>F=\mathbb{Q}(y_1,y_2)</math> | ||
| − | * Define a | + | * Define a sequence <math>\{y_n\}</math> by |
:<math> | :<math> | ||
y_{m-1}y_{m+1}= | y_{m-1}y_{m+1}= | ||
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===matrix formulation=== | ===matrix formulation=== | ||
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:<math>\mu_{1}(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>\mu_{2}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}</math> | ||
| − | * <math>x_1x_1'=x_2^c+1</math> | + | * <math>x_1x_1'=x_2^c+1</math> call <math>x_1'=x_3</math> |
| − | * <math>x_2x_2'=x_1^b+1</math> | + | * <math>x_2x_2'=x_1^b+1</math> call <math>x_2'=x_4</math> |
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==observations== | ==observations== | ||
;theorem (FZ) : For any <math>b,c</math>, <math>y_m</math> is a Laurent polynomial. | ;theorem (FZ) : For any <math>b,c</math>, <math>y_m</math> is a Laurent polynomial. | ||
| − | * Positivity conjecture: coefficients of these Laurent polynomials are | + | * Positivity conjecture: coefficients of these Laurent polynomials are positive (numerator and denomonator always have ) |
| − | * In this example, | + | * In this example, <math>bc\leq 3</math> iff the recurrence is periodic |
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==cluster algebra associated to Cartan matrices== | ==cluster algebra associated to Cartan matrices== | ||
| 53번째 줄: | 53번째 줄: | ||
:<math> \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}</math> | :<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> | * 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> | ||
| − | * when <math>bc\leq 3</math>, <math>y_m=y_n</math> | + | * 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]] |
* <math>bc=1, h=2</math> | * <math>bc=1, h=2</math> | ||
* <math>bc=2, h=4</math> | * <math>bc=2, h=4</math> | ||
| 59번째 줄: | 59번째 줄: | ||
* <math>bc\geq 4, h=\infty</math> | * <math>bc\geq 4, h=\infty</math> | ||
* If <math>bc\geq 4</math>, all <math>y_m</math> are distinct | * If <math>bc\geq 4</math>, all <math>y_m</math> are distinct | ||
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==algebraic structure== | ==algebraic structure== | ||
| 65번째 줄: | 65번째 줄: | ||
;theorem (Berenstein, Fomin and Zelevinsky) : | ;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> | :<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 | + | * 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> | :<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 | * Here support of any such monomial is | ||
| 72번째 줄: | 72번째 줄: | ||
:<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> | :<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]] | + | * [[Rank 2 cluster algebra examples]] |
| − | + | ||
==articles== | ==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. | + | * '''[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. |
[[분류:cluster algebra]] | [[분류:cluster algebra]] | ||
| 85번째 줄: | 85번째 줄: | ||
[[분류:math]] | [[분류:math]] | ||
[[분류:migrate]] | [[분류:migrate]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q944095 Q944095] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'kaluza'}, {'OP': '*'}, {'LOWER': 'klein'}, {'LEMMA': 'theory'}] | ||
2021년 2월 17일 (수) 03:01 기준 최신판
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 <math>m\) is even} \\
\end{cases} </math>
- 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 phenomenon" 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.
메타데이터
위키데이터
- ID : Q944095
Spacy 패턴 목록
- [{'LOWER': 'kaluza'}, {'OP': '*'}, {'LOWER': 'klein'}, {'LEMMA': 'theory'}]