Degrees and exponents

introduction

• eigenvalues of Cartan matrices
• eigenvalues of incidence matrices of Dynkin diagram
• 틀:수학노트

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

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

• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• Coxeter groups and reflection groups
• Macdonald constant term conjecture
• Poincare Series of Coxeter Groups

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxcjRTSkZXbWxwOFE/edit

encyclopedia

• http://en.wikipedia.org/wiki/Coxeter_number

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.