"Rank 2 cluster algebra"의 두 판 사이의 차이

2020년 11월 13일 (금) 10:32 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 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,

\[A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/\langle y_0y_2-y_1^b-1,y_1y_3-y_2^c-1\rangle\]

related items

Rank 2 cluster algebra examples

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.

@@ 12번째 줄: / 12번째 줄: @@
 ==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>
+* 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 sequence $\{y_n\}$ by
+* Define a sequence <math>\{y_n\}</math> by
-$$
+:<math>
 y_{m-1}y_{m+1}=
 \begin{cases}
-  y_m^b+1, & \text{if $m$ is odd}\\
+  y_m^b+1, & \text{if </math>m<math> is odd}\\
-  y_m^c+1, & \text{if $m$ is even} \\
+  y_m^c+1, & \text{if </math>m<math> is even} \\
 \end{cases}
-$$
+</math>
 * 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)
+* Note : we can use the exchange relation any <math>y_m</math> in terms of arbitrary cluster <math>\{y_i,y_{i+1}\}</math> (rational expression)
@@ 34번째 줄: / 34번째 줄: @@
 :<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_1x_1'=x_2^c+1</math> call <math>x_1'=x_3</math>
-* <math>x_2x_2'=x_1^b+1</math> call $x_2'=x_4$
+* <math>x_2x_2'=x_1^b+1</math> call <math>x_2'=x_4</math>
@@ 41번째 줄: / 41번째 줄: @@
 ==observations==
-;theorem (FZ) : For any $b,c$, $y_m$ 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 positive (numerator and denomonator always have )
 * In this example, <math>bc\leq 3</math> iff the recurrence is periodic
@@ 54번째 줄: / 54번째 줄: @@
 * 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> 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>bc=1, h=2</math>
-* $bc=2, h=4$
+* <math>bc=2, h=4</math>
-* $bc=3, h=6$
+* <math>bc=3, h=6</math>
-* $bc\geq 4, h=\infty$
+* <math>bc\geq 4, h=\infty</math>
-* If $bc\geq 4$, all $y_m$ are distinct
+* If <math>bc\geq 4</math>, all <math>y_m</math> 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 $m$
+* By "Laurent phenomenon" each element in <math>A(b,c)</math> can be uniquely expressed as Laurent polynomial in <math>y_m</math> and <math>y_{m+1}</math> for any <math>m</math>
 ;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>