"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
-* {{수학노트|url=유한반사군과_콕세터군(finite_reflection_groups_and_Coxeter_groups)}}
-==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.
-===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
-==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 m_i/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 m_i/h)</math> where $h$ is the Coxeter number and <math>m_i</math>'s are 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
-===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 $2m_1+1,\cdots,2m_l+1$
-==memo==
-* http://mathoverflow.net/questions/143548/uniform-proof-that-a-finite-reflection-group-is-determined-by-its-degrees
-==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]]
-* [[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.
 [[분류:개인노트]]
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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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