"Degrees and exponents"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 17번째 줄: | 17번째 줄: | ||
* h : Coxeter number | * h : Coxeter number | ||
| − | * eigenvalue <math>2\cos(\pi | + | * eigenvalue <math>2\cos(\pi m_i/h)</math> |
| 25번째 줄: | 25번째 줄: | ||
* appears in invariant theory | * appears in invariant theory | ||
* can also be seen as eigenvalues of Cartan matrix or incidence matrix of the Dynkin diagram | * can also be seen as eigenvalues of Cartan matrix or incidence matrix of the Dynkin diagram | ||
| − | * for incidence matrix, the eigenvalues are given by:<math>2\cos(\pi | + | * for incidence matrix, the eigenvalues are given by:<math>2\cos(\pi m_i/h)</math> where h is the Coxeter number and <math>m_i</math>'s are the exponents |
| − | * if we denote the exponents by $ | + | * if we denote the exponents by $m_i$, $1\le m_i < h$, then $m_i+m_{r-i+1}=h$ where $r$ is the rank |
| + | |||
===example=== | ===example=== | ||
2014년 6월 21일 (토) 20:58 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 \(2\cos(\pi m_i/h)\)
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\[2\cos(\pi m_i/h)\] where h is the Coxeter number and \(m_i\)'s are the exponents
- if we denote the exponents by $m_i$, $1\le m_i < h$, then $m_i+m_{r-i+1}=h$ where $r$ is the rank
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 $2m_1+1,\cdots,2m_l+1$
history
- Coxeter groups and reflection groups
- Macdonald constant term conjecture
- Poincare Series of Coxeter Groups
computational resource
encyclopedia
articles
- Kostant, Bertram. “The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group.” American Journal of Mathematics 81 (1959): 973–1032.
- Coleman, A. J. “The Betti Numbers of the Simple Lie Groups.” Canadian Journal of Mathematics. Journal Canadien de Mathématiques 10 (1958): 349–56.