"Degrees and exponents"의 두 판 사이의 차이

2020년 11월 13일 (금) 11:11 기준 최신판

introduction

틀:수학노트

related items

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 ==introduction==
+* {{수학노트|url=콕세터_군의_차수와_지수_(degrees_and_exponents)}}
-*  eigenvalues of Cartan matrices
-*  eigenvalues of incidence matrices of Dynkin diagram
-* http://pythagoras0.springnote.com/pages/1938682/attachments/3170605
-==Cartan matrix==
-*  h : [[Coxeter number and dual Coxeter number|Coxeter number]]
-*  eigenvalue <math>4\sin^2(m_{i}\pi/2h)</math>
-* <math>m_{i}</math> is called the exponents
-* <math>d_{i}=m_{i}+1</math> is called a degree
-==adjacency matrix==
-*  h : Coxeter number
-*  eigenvalue <math>2\cos(\pi l_n/h)</math>
-==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:<math>2\cos(\pi l_n/h)</math> where h is the Coxeter number and <math>l_i</math>'s are the exponents
-===example===
-* A4 Cartan matrix has the Coxeter number 5
-:<math>\left( \begin{array}{cccc}  2 & -1 & 0 & 0 \\  -1 & 2 & -1 & 0 \\  0 & -1 & 2 & -1 \\  0 & 0 & -1 & 2 \end{array} \right)</math> * incidence matrix:<math>\left( \begin{array}{cccc}  0 & 1 & 0 & 0 \\  1 & 0 & 1 & 0 \\  0 & 1 & 0 & 1 \\  0 & 0 & 1 & 0 \end{array} \right)</math>
-* eigenvalues of the incidence matrix:<math>\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\}</math>
-==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]]
+* [[Coxeter number and dual Coxeter number]]
+* [[Macdonald constant term conjecture]]
-==encyclopedia==
-* http://en.wikipedia.org/wiki/Coxeter_number
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 [[분류:math and physics]]
 [[분류:math]]
+[[분류:migrate]]

"Degrees and exponents"의 두 판 사이의 차이

2020년 11월 13일 (금) 11:11 기준 최신판

introduction

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