아이코시안 (icosian)

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

$$ \begin{array}{cccc} & \text{icosian} & \text{vector in }E_8 & \text{Dynkin label} \\ \hline 1 & \{1,0,0,0\} & \{1,0,0,0,1,0,0,0\} & \{0,0,1,1,1,0,0,1\} \\ 2 & \{0,1,0,0\} & \{0,1,0,0,0,1,0,0\} & \{0,1,1,1,1,1,0,1\} \\ 3 & \{0,0,1,0\} & \{0,0,1,0,0,0,1,0\} & \{0,1,2,1,1,1,1,1\} \\ 4 & \{0,0,0,1\} & \{0,0,0,1,0,0,0,1\} & \{2,4,6,5,3,2,1,3\} \\ 5 & \{-1,0,0,0\} & \{-1,0,0,0,-1,0,0,0\} & \{0,0,-1,-1,-1,0,0,-1\} \\ 6 & \{0,-1,0,0\} & \{0,-1,0,0,0,-1,0,0\} & \{0,-1,-1,-1,-1,-1,0,-1\} \\ 7 & \{0,0,-1,0\} & \{0,0,-1,0,0,0,-1,0\} & \{0,-1,-2,-1,-1,-1,-1,-1\} \\ 8 & \{0,0,0,-1\} & \{0,0,0,-1,0,0,0,-1\} & \{-2,-4,-6,-5,-3,-2,-1,-3\} \\ 9 & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,3,5,4,3,2,1,3\} \\ 10 & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-3,-5,-4,-3,-2,-1,-3\} \\ 11 & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-1,-1,0,0,0,0\} \\ 12 & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,3,2,1,1,2\} \\ 13 & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,3,2,1,0,2\} \\ 14 & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,3,4,3,2,2,1,2\} \\ 15 & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-3,-2,-1,-1,-1,-1\} \\ 16 & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-2,-2,-1,-1,0,-1\} \\ 17 & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,1,2,2,1,0,0,1\} \\ 18 & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-2,-2,-1,0,0,-1\} \\ 19 & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,2,2,1,1,0,1\} \\ 20 & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,1,1,1,1\} \\ 21 & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-3,-4,-3,-2,-2,-1,-2\} \\ 22 & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-4,-3,-2,-1,-1,-2\} \\ 23 & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-3,-3,-2,-1,0,-2\} \\ 24 & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,1,1,1,0,0,0,0\} \\ 25 & \left\{0,\frac{1}{2},\frac{\sigma }{2},\frac{\tau }{2}\right\} & \{0,0,0,1,0,1,0,0\} & \{0,1,2,2,1,1,0,1\} \\ 26 & \left\{0,\frac{\tau }{2},\frac{1}{2},\frac{\sigma }{2}\right\} & \{0,1,0,0,0,0,1,0\} & \{0,1,1,1,1,1,1,1\} \\ 27 & \left\{0,\frac{\sigma }{2},\frac{\tau }{2},\frac{1}{2}\right\} & \{0,0,1,0,0,0,0,1\} & \{2,4,6,4,3,2,1,3\} \\ 28 & \left\{\frac{1}{2},0,\frac{\tau }{2},\frac{\sigma }{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,2,1,1,2\} \\ 29 & \left\{\frac{\sigma }{2},0,\frac{1}{2},\frac{\tau }{2}\right\} & \{0,0,1,1,0,0,0,0\} & \{0,1,2,1,0,0,0,1\} \\ 30 & \left\{\frac{\tau }{2},0,\frac{\sigma }{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,4,3,2,1,2\} \\ 31 & \left\{\frac{1}{2},\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,3,2,1,0,2\} \\ 32 & \left\{\frac{\tau }{2},\frac{1}{2},0,\frac{\sigma }{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,3,3,2,1,2\} \\ 33 & \left\{\frac{\sigma }{2},\frac{\tau }{2},0,\frac{1}{2}\right\} & \{0,1,0,1,0,0,0,0\} & \{0,1,1,1,0,0,0,1\} \\ 34 & \left\{\frac{1}{2},\frac{\tau }{2},\frac{\sigma }{2},0\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-1,0,0,0,0,0\} \\ 35 & \left\{\frac{\sigma }{2},\frac{1}{2},\frac{\tau }{2},0\right\} & \{0,1,1,0,0,0,0,0\} & \{0,1,1,0,0,0,0,1\} \\ 36 & \left\{\frac{\tau }{2},\frac{\sigma }{2},\frac{1}{2},0\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,3,3,2,1,2\} \\ 37 & \left\{0,-\frac{1}{2},\frac{\sigma }{2},\frac{\tau }{2}\right\} & \{0,-1,0,1,0,0,0,0\} & \{0,0,1,1,0,0,0,0\} \\ 38 & \left\{0,\frac{\tau }{2},-\frac{1}{2},\frac{\sigma }{2}\right\} & \{0,1,-1,0,0,0,0,0\} & \{0,0,-1,0,0,0,0,0\} \\ 39 & \left\{0,\frac{\sigma }{2},\frac{\tau }{2},-\frac{1}{2}\right\} & \{0,0,1,-1,0,0,0,0\} & \{0,0,0,-1,0,0,0,0\} \\ 40 & \left\{-\frac{1}{2},0,\frac{\tau }{2},\frac{\sigma }{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,2,1,1,1,1,1\} \\ 41 & \left\{\frac{\sigma }{2},0,-\frac{1}{2},\frac{\tau }{2}\right\} & \{0,0,0,1,0,0,-1,0\} & \{0,0,0,0,-1,-1,-1,0\} \\ 42 & \left\{\frac{\tau }{2},0,\frac{\sigma }{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-2,-1,0,0,0,-1\} \\ 43 & \left\{-\frac{1}{2},\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,1,1,0,1\} \\ 44 & \left\{\frac{\tau }{2},-\frac{1}{2},0,\frac{\sigma }{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,1,2,2,2,1,1,1\} \\ 45 & \left\{\frac{\sigma }{2},\frac{\tau }{2},0,-\frac{1}{2}\right\} & \{0,1,0,0,0,0,0,-1\} & \{-2,-3,-5,-4,-3,-2,-1,-2\} \\ 46 & \left\{-\frac{1}{2},\frac{\tau }{2},\frac{\sigma }{2},0\right\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-2,-1,-1,0,0,-1\} \\ 47 & \left\{\frac{\sigma }{2},-\frac{1}{2},\frac{\tau }{2},0\right\} & \{0,0,1,0,0,-1,0,0\} & \{0,0,0,-1,-1,-1,0,0\} \\ 48 & \left\{\frac{\tau }{2},\frac{\sigma }{2},-\frac{1}{2},0\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,1,2,2,2,1,0,1\} \\ 49 & \left\{0,\frac{1}{2},-\frac{\sigma }{2},\frac{\tau }{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,1,1,0,2\} \\ 50 & \left\{0,\frac{\tau }{2},\frac{1}{2},-\frac{\sigma }{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-1,-1,-1,0,0,0\} \\ 51 & \left\{0,-\frac{\sigma }{2},\frac{\tau }{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,1,1,1,2\} \\ 52 & \left\{\frac{1}{2},0,\frac{\tau }{2},-\frac{\sigma }{2}\right\} & \{1,0,1,0,0,0,0,0\} & \{0,0,1,0,0,0,0,1\} \\ 53 & \left\{-\frac{\sigma }{2},0,\frac{1}{2},\frac{\tau }{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,3,2,2,1,2\} \\ 54 & \left\{\frac{\tau }{2},0,-\frac{\sigma }{2},\frac{1}{2}\right\} & \{1,0,0,0,0,0,0,1\} & \{2,3,5,4,3,2,1,3\} \\ 55 & \left\{\frac{1}{2},-\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \{1,0,0,1,0,0,0,0\} & \{0,0,1,1,0,0,0,1\} \\ 56 & \left\{\frac{\tau }{2},\frac{1}{2},0,-\frac{\sigma }{2}\right\} & \{1,0,0,0,0,1,0,0\} & \{0,0,1,1,1,1,0,1\} \\ 57 & \left\{-\frac{\sigma }{2},\frac{\tau }{2},0,\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,3,2,2,1,2\} \\ 58 & \left\{\frac{1}{2},\frac{\tau }{2},-\frac{\sigma }{2},0\right\} & \{1,1,0,0,0,0,0,0\} & \{0,0,0,0,0,0,0,1\} \\ 59 & \left\{-\frac{\sigma }{2},\frac{1}{2},\frac{\tau }{2},0\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,2,2,1,2\} \\ 60 & \left\{\frac{\tau }{2},-\frac{\sigma }{2},\frac{1}{2},0\right\} & \{1,0,0,0,0,0,1,0\} & \{0,0,1,1,1,1,1,1\} \end{array} $$

메모

• Wilson, The Finite Simple Groups
• The `Icosian Calculus' By William R. Hamilton
• http://mathoverflow.net/questions/117604/a-non-commutative-ring-from-su2
• http://www.tony5m17h.net/E8H4H4.pdf
• http://cp4space.wordpress.com/2013/09/12/leech-lattice/

관련된 항목들

• 해밀턴의 사원수(quarternions)

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxbnlqcFJRQVE0ZWs/edit

사전 형태의 자료

• http://en.wikipedia.org/wiki/Icosian

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.