"Degrees and exponents"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 51번째 줄: | 51번째 줄: | ||
* [[Coxeter groups and reflection groups]] | * [[Coxeter groups and reflection groups]] | ||
* [[Macdonald constant term conjecture]] | * [[Macdonald constant term conjecture]] | ||
| + | * [[Poincare Series of Coxeter Groups]] | ||
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| 61번째 줄: | 63번째 줄: | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
2013년 7월 5일 (금) 06:23 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 $2m_1+1,\cdots,2m_l+1$
history
- Coxeter groups and reflection groups
- Macdonald constant term conjecture
- Poincare Series of Coxeter Groups
computational resource