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imported>Pythagoras0 잔글 (Pythagoras0 사용자가 Degrees and exponent 문서를 Degrees and exponents 문서로 옮겼습니다.) |
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* eigenvalues of Cartan matrices | * eigenvalues of Cartan matrices | ||
* eigenvalues of incidence matrices of Dynkin diagram | * eigenvalues of incidence matrices of Dynkin diagram | ||
| − | * | + | * {{수학노트|url=유한반사군과_콕세터군(finite_reflection_groups_and_Coxeter_groups)}} |
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===example=== | ===example=== | ||
* A4 Cartan matrix has the Coxeter number 5 | * A4 Cartan matrix has the Coxeter number 5 | ||
| − | :<math>\left( \begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array} \right)</math> * incidence matrix:<math>\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right)</math> | + | :<math>\left( \begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array} \right)</math> |
| + | * incidence matrix:<math>\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right)</math> | ||
* eigenvalues of the incidence matrix:<math>\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1+\sqrt{5}\right)\right\}</math> | * eigenvalues of the incidence matrix:<math>\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1+\sqrt{5}\right)\right\}</math> | ||
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* $H^{\bullet}(L)$ is a free super-commutative algebra with homogeneous generator in degrees $2m_1+1,\cdots,2m_l+1$ | * $H^{\bullet}(L)$ is a free super-commutative algebra with homogeneous generator in degrees $2m_1+1,\cdots,2m_l+1$ | ||
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==related items== | ==related items== | ||
| − | + | * [[Coxeter groups and reflection groups]] | |
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2013년 7월 5일 (금) 05:28 판
introduction
- eigenvalues of Cartan matrices
- eigenvalues of incidence matrices of Dynkin diagram
- 틀:수학노트
Cartan matrix
- h : Coxeter number
- eigenvalue \(4\sin^2(m_{i}\pi/2h)\)
- \(m_{i}\) is called the exponents
- \(d_{i}=m_{i}+1\) is called a degree
adjacency matrix
- h : Coxeter number
- eigenvalue \(2\cos(\pi l_n/h)\)
degree and exponent of simple Lie algebra
- appears in invariant theory
- can also be seen as eigenvalues of Cartan matrix or incidence matrix of the Dynkin diagram
- for incidence matrix, the eigenvalues are given by\[2\cos(\pi l_n/h)\] where h is the Coxeter number and \(l_i\)'s are the exponents
example
- A4 Cartan matrix has the Coxeter number 5
\[\left( \begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array} \right)\]
- incidence matrix\[\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right)\]
- eigenvalues of the incidence matrix\[\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1+\sqrt{5}\right)\right\}\]
homological algebraic characterization
- For a semisimple. Lie algebra L
- $H^{\bullet}(L)$ is a free super-commutative algebra with homogeneous generator in degrees $2m_1+1,\cdots,2m_l+1$
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