"아이코시안 (icosian)"의 두 판 사이의 차이

수학노트
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* $\mathscr{I}\cup \sigma \mathscr{I}$의 240개 원소와 [[E8 루트 시스템]] 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다
 
* $\mathscr{I}\cup \sigma \mathscr{I}$의 240개 원소와 [[E8 루트 시스템]] 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다
 
===테이블===
 
===테이블===
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$$
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[[파일:아이코시안 (icosian)1.png]]
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\begin{array}{c|c|c|c}
 
−
  & \text{icosian} & \text{vector in }E_8 & \text{Dynkin label} \\
 
−
\hline
 
−
1 & \{1,0,0,0\} & \{2,0,0,0,2,0,0,0\} & \{0,0,1,1,1,0,0,1\} \\
 
−
2 & \{0,1,0,0\} & \{0,2,0,0,0,2,0,0\} & \{0,1,1,1,1,1,0,1\} \\
 
−
3 & \{0,0,1,0\} & \{0,0,2,0,0,0,2,0\} & \{0,1,2,1,1,1,1,1\} \\
 
−
4 & \{0,0,0,1\} & \{0,0,0,2,0,0,0,2\} & \{2,4,6,5,3,2,1,3\} \\
 
−
5 & \{-1,0,0,0\} & \{-2,0,0,0,-2,0,0,0\} & \{0,0,-1,-1,-1,0,0,-1\} \\
 
−
6 & \{0,-1,0,0\} & \{0,-2,0,0,0,-2,0,0\} & \{0,-1,-1,-1,-1,-1,0,-1\} \\
 
−
7 & \{0,0,-1,0\} & \{0,0,-2,0,0,0,-2,0\} & \{0,-1,-2,-1,-1,-1,-1,-1\} \\
 
−
8 & \{0,0,0,-1\} & \{0,0,0,-2,0,0,0,-2\} & \{-2,-4,-6,-5,-3,-2,-1,-3\} \\
 
−
9 & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,1,1,1,1,1,1,1\} & \{1,3,5,4,3,2,1,3\} \\
 
−
10 & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-1,-1,-1,-1,-1,-1\} & \{-1,-3,-5,-4,-3,-2,-1,-3\} \\
 
−
11 & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{1,1,1,-1,1,1,1,-1\} & \{-1,-1,-1,-1,0,0,0,0\} \\
 
−
12 & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,-1,1,1,1,-1,1,1\} & \{1,2,4,3,2,1,1,2\} \\
 
−
13 & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,1,-1,1,1,1,-1,1\} & \{1,2,3,3,2,1,0,2\} \\
 
−
14 & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{-1,1,1,1,-1,1,1,1\} & \{1,3,4,3,2,2,1,2\} \\
 
−
15 & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{1,1,-1,-1,1,1,-1,-1\} & \{-1,-2,-3,-2,-1,-1,-1,-1\} \\
 
−
16 & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{1,-1,1,-1,1,-1,1,-1\} & \{-1,-2,-2,-2,-1,-1,0,-1\} \\
 
−
17 & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,-1,-1,1,1,-1,-1,1\} & \{1,1,2,2,1,0,0,1\} \\
 
−
18 & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,1,1,-1,-1,1,1,-1\} & \{-1,-1,-2,-2,-1,0,0,-1\} \\
 
−
19 & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{-1,1,-1,1,-1,1,-1,1\} & \{1,2,2,2,1,1,0,1\} \\
 
−
20 & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{-1,-1,1,1,-1,-1,1,1\} & \{1,2,3,2,1,1,1,1\} \\
 
−
21 & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{1,-1,-1,-1,1,-1,-1,-1\} & \{-1,-3,-4,-3,-2,-2,-1,-2\} \\
 
−
22 & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,1,-1,-1,-1,1,-1,-1\} & \{-1,-2,-4,-3,-2,-1,-1,-2\} \\
 
−
23 & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,1,-1,-1,-1,1,-1\} & \{-1,-2,-3,-3,-2,-1,0,-2\} \\
 
−
24 & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{-1,-1,-1,1,-1,-1,-1,1\} & \{1,1,1,1,0,0,0,0\} \\
 
−
25 & \left\{0,\frac{1}{2},\frac{\sigma }{2},\frac{\tau }{2}\right\} & \{0,0,0,2,0,2,0,0\} & \{0,1,2,2,1,1,0,1\} \\
 
−
26 & \left\{0,\frac{\tau }{2},\frac{1}{2},\frac{\sigma }{2}\right\} & \{0,2,0,0,0,0,2,0\} & \{0,1,1,1,1,1,1,1\} \\
 
−
27 & \left\{0,\frac{\sigma }{2},\frac{\tau }{2},\frac{1}{2}\right\} & \{0,0,2,0,0,0,0,2\} & \{2,4,6,4,3,2,1,3\} \\
 
−
28 & \left\{\frac{1}{2},0,\frac{\tau }{2},\frac{\sigma }{2}\right\} & \{1,1,1,-1,1,-1,1,1\} & \{1,2,3,2,2,1,1,2\} \\
 
−
29 & \left\{\frac{\sigma }{2},0,\frac{1}{2},\frac{\tau }{2}\right\} & \{0,0,2,2,0,0,0,0\} & \{0,1,2,1,0,0,0,1\} \\
 
−
30 & \left\{\frac{\tau }{2},0,\frac{\sigma }{2},\frac{1}{2}\right\} & \{1,-1,-1,1,1,1,1,1\} & \{1,2,4,4,3,2,1,2\} \\
 
−
31 & \left\{\frac{1}{2},\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \{1,-1,1,1,1,1,-1,1\} & \{1,2,4,3,2,1,0,2\} \\
 
−
32 & \left\{\frac{\tau }{2},\frac{1}{2},0,\frac{\sigma }{2}\right\} & \{1,1,-1,-1,1,1,1,1\} & \{1,2,3,3,3,2,1,2\} \\
 
−
33 & \left\{\frac{\sigma }{2},\frac{\tau }{2},0,\frac{1}{2}\right\} & \{0,2,0,2,0,0,0,0\} & \{0,1,1,1,0,0,0,1\} \\
 
−
34 & \left\{\frac{1}{2},\frac{\tau }{2},\frac{\sigma }{2},0\right\} & \{1,1,-1,1,1,1,1,-1\} & \{-1,-1,-1,0,0,0,0,0\} \\
 
−
35 & \left\{\frac{\sigma }{2},\frac{1}{2},\frac{\tau }{2},0\right\} & \{0,2,2,0,0,0,0,0\} & \{0,1,1,0,0,0,0,1\} \\
 
−
36 & \left\{\frac{\tau }{2},\frac{\sigma }{2},\frac{1}{2},0\right\} & \{1,-1,1,-1,1,1,1,1\} & \{1,2,4,3,3,2,1,2\} \\
 
−
37 & \left\{0,-\frac{1}{2},\frac{\sigma }{2},\frac{\tau }{2}\right\} & \{0,-2,0,2,0,0,0,0\} & \{0,0,1,1,0,0,0,0\} \\
 
−
38 & \left\{0,\frac{\tau }{2},-\frac{1}{2},\frac{\sigma }{2}\right\} & \{0,2,-2,0,0,0,0,0\} & \{0,0,-1,0,0,0,0,0\} \\
 
−
39 & \left\{0,\frac{\sigma }{2},\frac{\tau }{2},-\frac{1}{2}\right\} & \{0,0,2,-2,0,0,0,0\} & \{0,0,0,-1,0,0,0,0\} \\
 
−
40 & \left\{-\frac{1}{2},0,\frac{\tau }{2},\frac{\sigma }{2}\right\} & \{-1,1,1,-1,-1,-1,1,1\} & \{1,2,2,1,1,1,1,1\} \\
 
−
41 & \left\{\frac{\sigma }{2},0,-\frac{1}{2},\frac{\tau }{2}\right\} & \{0,0,0,2,0,0,-2,0\} & \{0,0,0,0,-1,-1,-1,0\} \\
 
−
42 & \left\{\frac{\tau }{2},0,\frac{\sigma }{2},-\frac{1}{2}\right\} & \{1,-1,-1,-1,1,1,1,-1\} & \{-1,-2,-2,-1,0,0,0,-1\} \\
 
−
43 & \left\{-\frac{1}{2},\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \{-1,-1,1,1,-1,1,-1,1\} & \{1,2,3,2,1,1,0,1\} \\
 
−
44 & \left\{\frac{\tau }{2},-\frac{1}{2},0,\frac{\sigma }{2}\right\} & \{1,-1,-1,-1,1,-1,1,1\} & \{1,1,2,2,2,1,1,1\} \\
 
−
45 & \left\{\frac{\sigma }{2},\frac{\tau }{2},0,-\frac{1}{2}\right\} & \{0,2,0,0,0,0,0,-2\} & \{-2,-3,-5,-4,-3,-2,-1,-2\} \\
 
−
46 & \left\{-\frac{1}{2},\frac{\tau }{2},\frac{\sigma }{2},0\right\} & \{-1,1,-1,1,-1,1,1,-1\} & \{-1,-1,-2,-1,-1,0,0,-1\} \\
 
−
47 & \left\{\frac{\sigma }{2},-\frac{1}{2},\frac{\tau }{2},0\right\} & \{0,0,2,0,0,-2,0,0\} & \{0,0,0,-1,-1,-1,0,0\} \\
 
−
48 & \left\{\frac{\tau }{2},\frac{\sigma }{2},-\frac{1}{2},0\right\} & \{1,-1,-1,-1,1,1,-1,1\} & \{1,1,2,2,2,1,0,1\} \\
 
−
49 & \left\{0,\frac{1}{2},-\frac{\sigma }{2},\frac{\tau }{2}\right\} & \{1,1,1,1,-1,1,-1,1\} & \{1,2,3,2,1,1,0,2\} \\
 
−
50 & \left\{0,\frac{\tau }{2},\frac{1}{2},-\frac{\sigma }{2}\right\} & \{1,1,1,1,-1,1,1,-1\} & \{-1,-1,-1,-1,-1,0,0,0\} \\
 
−
51 & \left\{0,-\frac{\sigma }{2},\frac{\tau }{2},\frac{1}{2}\right\} & \{1,1,1,1,-1,-1,1,1\} & \{1,2,3,2,1,1,1,2\} \\
 
−
52 & \left\{\frac{1}{2},0,\frac{\tau }{2},-\frac{\sigma }{2}\right\} & \{2,0,2,0,0,0,0,0\} & \{0,0,1,0,0,0,0,1\} \\
 
−
53 & \left\{-\frac{\sigma }{2},0,\frac{1}{2},\frac{\tau }{2}\right\} & \{1,-1,1,1,-1,1,1,1\} & \{1,2,4,3,2,2,1,2\} \\
 
−
54 & \left\{\frac{\tau }{2},0,-\frac{\sigma }{2},\frac{1}{2}\right\} & \{2,0,0,0,0,0,0,2\} & \{2,3,5,4,3,2,1,3\} \\
 
−
55 & \left\{\frac{1}{2},-\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \{2,0,0,2,0,0,0,0\} & \{0,0,1,1,0,0,0,1\} \\
 
−
56 & \left\{\frac{\tau }{2},\frac{1}{2},0,-\frac{\sigma }{2}\right\} & \{2,0,0,0,0,2,0,0\} & \{0,0,1,1,1,1,0,1\} \\
 
−
57 & \left\{-\frac{\sigma }{2},\frac{\tau }{2},0,\frac{1}{2}\right\} & \{1,1,-1,1,-1,1,1,1\} & \{1,2,3,3,2,2,1,2\} \\
 
−
58 & \left\{\frac{1}{2},\frac{\tau }{2},-\frac{\sigma }{2},0\right\} & \{2,2,0,0,0,0,0,0\} & \{0,0,0,0,0,0,0,1\} \\
 
−
59 & \left\{-\frac{\sigma }{2},\frac{1}{2},\frac{\tau }{2},0\right\} & \{1,1,1,-1,-1,1,1,1\} & \{1,2,3,2,2,2,1,2\} \\
 
−
60 & \left\{\frac{\tau }{2},-\frac{\sigma }{2},\frac{1}{2},0\right\} & \{2,0,0,0,0,0,2,0\} & \{0,0,1,1,1,1,1,1\}
 
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\end{array}
 
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$$
 
  
 
==메모==
 
==메모==

2014년 7월 8일 (화) 05:41 판

개요

  • 아이코시안 군과 아이코시안 환


아이코시안 군

  • 사원수의 부분군으로 크기는 120이며 다음과 같은 벡터의 좌표에 짝치환을 적용하여 얻어지는 원소들로 구성
  • 8개 $\frac{1}{2}(\pm 2,0,0,0)$
  • 16개 $\frac{1}{2}(\pm 1,\pm 1,\pm 1,\pm 1)$
  • 96개 $\frac{1}{2}(0,\pm 1,\pm \sigma,\pm \tau)$, 여기서 $\sigma=\frac{1-\sqrt{5}}{2},\tau=\frac{1+\sqrt{5}}{2}$


아이코시안 환

  • 계수를 $\mathbb{Z}[\sqrt{5}]$에서 갖는 사원수들이 이루는 환
  • E8 격자에 isometric


아이코시안과 $E_8$

  • 아이코시안 군 $\mathscr{I}$의 원소 120개를 갖는다
  • $\sigma \mathscr{I}:=\{\sigma s|s\in \mathscr{I}\}$라 두자. 여기서 $\sigma=\frac{1-\sqrt{5}}{2}$
  • $\mathscr{I}\cup \sigma \mathscr{I}$의 240개 원소와 E8 루트 시스템 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다

테이블

아이코시안 (icosian)1.png

메모


관련된 항목들


매스매티카 파일 및 계산 리소스


사전 형태의 자료


리뷰, 에세이, 강의노트

  • Biggs, Norman. “The Icosian Calculus of Today.” Proceedings of the Royal Irish Academy. Section A. Mathematical and Physical Sciences 95, no. suppl. (1995): 23–34.


관련논문

  • Moody, R. V., and J. Patera. “Quasicrystals and Icosians.” Journal of Physics. A. Mathematical and General 26, no. 12 (1993): 2829–53.
  • Wilson, Robert A. "The geometry of the Hall-Janko group as a quaternionic reflection group." Geometriae Dedicata 20.2 (1986): 157-173.
원본 주소 "https://wiki.mathnt.net/index.php?title=아이코시안_(icosian)&oldid=30038"