Degrees and exponents - 편집 역사

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imported>Pythagoras0: /* articles */

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articles

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← 이전 판		2020년 11월 13일 (금) 18:11 판
16번째 줄:		16번째 줄:
	[[분류:math and physics]]		[[분류:math and physics]]
	[[분류:math]]		[[분류:math]]
		+	[[분류:migrate]]

@@ 8번째 줄: / 8번째 줄: @@
 * [[Coxeter number and dual Coxeter number]]
 * [[Macdonald constant term conjecture]]
-* [[Poincare Series of Coxeter Groups]]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* appears in invariant theory
+* {{수학노트|url=콕세터_군의_차수와_지수_(degrees_and_exponents)}}
 ==related items==
@@ 73번째 줄: / 11번째 줄: @@
 [[분류:개인노트]]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
+* appears in invariant theory
-*  eigenvalues of Cartan matrices
+* eigenvalues of Coxeter element
-*  eigenvalues of incidence matrices of Dynkin diagram
+* eigenvalues of Cartan matrices
 * {{수학노트|url=유한반사군과_콕세터군(finite_reflection_groups_and_Coxeter_groups)}}
@@ 8번째 줄: / 9번째 줄: @@
 ==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
@@ 33번째 줄: / 39번째 줄: @@
 * conjectured by Coxeter
@@ 61번째 줄: / 44번째 줄: @@
 * 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>
+* adjacency 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>
+* eigenvalues of the adjacency 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>
@@ 85번째 줄: / 68번째 줄: @@
 ==related items==
 * [[Coxeter groups and reflection groups]]
 * [[Macdonald constant term conjecture]]
 * [[Poincare Series of Coxeter Groups]]

@@ 4번째 줄: / 4번째 줄: @@
 *  eigenvalues of incidence matrices of Dynkin diagram
 * {{수학노트|url=유한반사군과_콕세터군(finite_reflection_groups_and_Coxeter_groups)}}
-http://mathoverflow.net/questions/143548/uniform-proof-that-a-finite-reflection-group-is-determined-by-its-degrees
 ==exponent==
@@ 54번째 줄: / 54번째 줄: @@
 *  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
+*  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 $m_i$, $1\le m_i < h$, then $m_i+m_{r-i+1}=h$ where $r$ is the rank
+* 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
@@ 71번째 줄: / 71번째 줄: @@
+==memo==
 ==history==

← 이전 판		2014년 6월 26일 (목) 07:26 판
4번째 줄:		4번째 줄:
	* eigenvalues of incidence matrices of Dynkin diagram		* eigenvalues of incidence matrices of Dynkin diagram
	* {{수학노트\|url=유한반사군과_콕세터군(finite_reflection_groups_and_Coxeter_groups)}}		* {{수학노트\|url=유한반사군과_콕세터군(finite_reflection_groups_and_Coxeter_groups)}}
		+
		+	;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

@@ 67번째 줄: / 67번째 줄: @@
 ==articles==
 * 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.

@@ 17번째 줄: / 17번째 줄: @@
 *  h : Coxeter number
-*  eigenvalue <math>2\cos(\pi l_n/h)</math>
+*  eigenvalue <math>2\cos(\pi m_i/h)</math>
@@ 25번째 줄: / 25번째 줄: @@
 *  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
+*  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 $a_i$, $1\le a_i < h$, then $a_i+a_{h-i+1}=h$.
+* 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===