"Degrees and exponents"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 26번째 줄: | 26번째 줄: | ||
* 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 l_n/h)</math> where h is the Coxeter number and <math>l_i</math>'s are the exponents | * for incidence matrix, the eigenvalues are given by:<math>2\cos(\pi l_n/h)</math> where h is the Coxeter number and <math>l_i</math>'s are the exponents | ||
| + | * if we denote the exponents by $a_i$, $1\le a_i < h$, then $a_i+a_{h-i+1}=h$. | ||
| + | |||
===example=== | ===example=== | ||
* A4 Cartan matrix has the Coxeter number 5 | * A4 Cartan matrix has the Coxeter number 5 | ||
2013년 12월 7일 (토) 12:40 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
- if we denote the exponents by $a_i$, $1\le a_i < h$, then $a_i+a_{h-i+1}=h$.
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