"Degrees and exponents"의 두 판 사이의 차이

2013년 2월 8일 (금) 15:09 판

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
\(\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1+\sqrt{5}\right)\right\}\)
you can evalutate the following in mathematica to get the same set
1. Table[2 Cos[Pi*l/5], {l, 1, 4}]

Table[Simplify[2 Cos[Pi*l/5]], {l, 1, 4}]
Table[Simplify[4 Sin[Pi*l/10]^2], {l, 1, 4}]

For a s.s. Lie algebra L

(a)H'(L) is a free super- commutative algebra with homogeneous generator in degrees 2m_1+1,\cdots,2m_l+1

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 [[분류:Lie theory]]
 [[분류:math and physics]]
+[[분류:math]]