Degrees and exponents

introduction

• eigenvalues of Cartan matrices
• eigenvalues of incidence matrices of Dynkin diagram
• http://pythagoras0.springnote.com/pages/1938682/attachments/3170605

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

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

encyclopedia

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