Degrees and exponents
introduction
- appears in invariant theory
- eigenvalues of Coxeter element
- eigenvalues of Cartan matrices
- eigenvalues of adjacency matrices of Dynkin diagram
- 틀:수학노트
exponent
- An eigenvalue of a Coxeter element is always of the form $\zeta^{m_i}$ for some integer $m_i$ where $\zeta$ is a primitive $h$-th root of unity. We call the integers $m_i$ such that $1\leq m_i\leq h-1$ the exponents.
- if we denote the exponents by $m_i$, $1\le m_i \le h$, then $m_i+m_{r-i+1}=h$ where $r$ is the rank
- eigenvalue of the Cartan matrix \(4\sin^2(m_{i}\pi/2h)\)
- eigenvalue of the adjacency matrix \(2\cos(\pi m_i/h)\)
example
The characteristic polynomial of a Coxeter element for $E_8$ acting on $\mathbb{R}^8$ is given by $$ x^8+x^7-x^5-x^4-x^3+x+1 $$ Actually, this is the 30-th cyclotomic polynomial $\Phi_{30}(x)$ where $$ \Phi_n(x) =\prod_{1\le k\le n,\gcd(k,n)=1}\left(x-e^{2i\pi\frac{k}{n}}\right). $$
Its roots are $\left\{\zeta ,\zeta ^7,\zeta ^{11},\zeta ^{13},\zeta ^{17},\zeta ^{19},\zeta ^{23},\zeta ^{29}\right\}$ where $\zeta$ is a primitive 30-th root of unity.
property
- thm (Kostant, 1959)
Let $m_1,\cdots,m_n$ be the exponents and $k_1,\cdots,k_n$ be the degrees arranged in an increasing order. Then $$ m_i=k_i-1. $$ Thus $$ \sum_{}m_i=\sum_{}(k_i-1)=\frac{nh}{2}=|\Phi^{+}| $$
- conjectured by Coxeter
example
- A4 Cartan matrix has the Coxeter number 5
\[\left( \begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array} \right)\]
- adjacency 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 adjacency 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 $2m_1+1,\cdots,2m_l+1$
memo
history
- Coxeter groups and reflection groups
- Coxeter number and dual Coxeter number
- Macdonald constant term conjecture
- Poincare Series of Coxeter Groups
computational resource
encyclopedia
articles
- Burns, John M., and Ruedi Suter. 2012. “Power Sums of Coxeter Exponents.” Advances in Mathematics 231 (3-4): 1291–1307. doi:10.1016/j.aim.2012.06.020.
- 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.