24차원 짝수 자기쌍대 격자

개요

• $\Gamma\subset \mathbb{R}^{24}$가 짝수 unimodular 격자라 하자
• 루트로 생성되는 격자 $(\Gamma_2)_{\mathbb{Z}}$는 다음과 같은 24가지 경우만이 가능하다

$$ \begin{aligned} \emptyset &, & A_1^{24} &, & A_2^{12}&,& A_3^8&,& A_4^6&,& A_5^4D_4&,& D_4^6&,&A_6^4\\ A_7^2D_5^2&,&A_8^3&,&A_9^2D_6&,& D_6^4 &,& E_6^4&,&A_{11}D_7E_6&,&A_{12}^2&,&D_8^3 \\ A_{15}D_9 &,& A_{17}E_7&,&D_{10}E_7^2&,&D_{12}^2&,& A_{24}&,&D_{16}E_8&,& E_8^3&,&D_{24} \end{aligned} $$

테이블

\begin{array}{ccccc} \text{root system} & \text{Coxeter} & \text{num. of roots} & \text{order of Aut} & \text{factorization} \\ \hline \emptyset & \cdot & 0 & 8315553613086720000 & 2^{22}\cdot 3^9\cdot 5^4\cdot 7^2\cdot 11^1\cdot 13^1\cdot 23^1 \\ A_1^{24} & 2 & 48 & 4107449023856640 & 2^{34}\cdot 3^3\cdot 5^1\cdot 7^1\cdot 11^1\cdot 23^1 \\ A_2^{12} & 3 & 72 & 413762786426880 & 2^{19}\cdot 3^{15}\cdot 5^1\cdot 11^1 \\ A_3^8 & 4 & 96 & 295882444505088 & 2^{31}\cdot 3^9\cdot 7^1 \\ A_4^6 & 5 & 120 & 716636160000000 & 2^{22}\cdot 3^7\cdot 5^7 \\ A_5^4 D_4 & 6 & 144 & 2476694568960000 & 2^{26}\cdot 3^{10}\cdot 5^4 \\ D_4^6 & 6 & 144 & 108208436847575040 & 2^{40}\cdot 3^9\cdot 5^1 \\ A_6^4 & 7 & 168 & 15485790781440000 & 2^{19}\cdot 3^9\cdot 5^4\cdot 7^4 \\ A_7^2 D_5^2 & 8 & 192 & 47943914618880000 & 2^{31}\cdot 3^6\cdot 5^4\cdot 7^2 \\ A_8^3 & 9 & 216 & 573416710078464000 & 2^{23}\cdot 3^{13}\cdot 5^3\cdot 7^3 \\ A_9^2 D_6 & 10 & 240 & 1213580338790400000 & 2^{27}\cdot 3^{10}\cdot 5^5\cdot 7^2 \\ D_6^4 & 10 & 240 & 6763027302973440000 & 2^{39}\cdot 3^9\cdot 5^4 \\ E_6^4 & 12 & 288 & 346657985428193280000 & 2^{32}\cdot 3^{17}\cdot 5^4 \\ A_{11} D_7 E_6 & 12 & 288 & 16019260472033280000 & 2^{28}\cdot 3^{11}\cdot 5^4\cdot 7^2\cdot 11^1 \\ A_{12}^2 & 13 & 312 & 155103152174530560000 & 2^{22}\cdot 3^{10}\cdot 5^4\cdot 7^2\cdot 11^2\cdot 13^2 \\ D_8^3 & 14 & 336 & 824788751971516416000 & 2^{43}\cdot 3^7\cdot 5^3\cdot 7^3 \\ A_{15} D_9 & 16 & 384 & 3887340541213409280000 & 2^{31}\cdot 3^{10}\cdot 5^4\cdot 7^3\cdot 11^1\cdot 13^1 \\ A_{17} E_7 & 18 & 432 & 37172693925353226240000 & 2^{27}\cdot 3^{12}\cdot 5^4\cdot 7^3\cdot 11^1\cdot 13^1\cdot 17^1 \\ D_{10} E_7^2 & 18 & 432 & 31316197926418513920000 & 2^{38}\cdot 3^{12}\cdot 5^4\cdot 7^3 \\ D_{12}^2 & 22 & 528 & 1924703466207817236480000 & 2^{43}\cdot 3^{10}\cdot 5^4\cdot 7^2\cdot 11^2 \\ A_{24} & 25 & 600 & 31022420086661971968000000 & 2^{23}\cdot 3^{10}\cdot 5^6\cdot 7^3\cdot 11^2\cdot 13^1\cdot 17^1\cdot 19^1\cdot 23^1 \\ D_{16} E_8 & 30 & 720 & 477676405704303732326400000 & 2^{44}\cdot 3^{11}\cdot 5^5\cdot 7^3\cdot 11^1\cdot 13^1 \\ E_8^3 & 30 & 720 & 2029289625631919702016000000 & 2^{43}\cdot 3^{16}\cdot 5^6\cdot 7^3 \\ D_{24} & 46 & 1104 & 5204698426366666226930810880000 & 2^{45}\cdot 3^{10}\cdot 5^4\cdot 7^3\cdot 11^2\cdot 13^1\cdot 17^1\cdot 19^1\cdot 23^1 \end{array}

메모

• Kneser p-neighbors of Niemeier lattices
• Chenevier, Gaëtan, and Jean Lannes. “Formes Automorphes et Voisins de Kneser Des R’eseaux de Niemeier.” arXiv:1409.7616 [math], September 26, 2014. http://arxiv.org/abs/1409.7616.
• http://larmor.nuigalway.ie/~detinko/Gabi2.pdf
• http://www.math.rwth-aachen.de/~Gabriele.Nebe/talks/lat3op.pdf

관련된 항목들

• 리치 격자(Leech lattice)

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxSXJwV21kaV9XSEk/edit
• http://oeis.org/search?q=Niemeier+lattice&sort=&language=&go=Search

사전 형태의 자료

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

관련논문

• Nebe, Gabriele, and Boris Venkov. On Siegel Modular Forms of Weight 12. http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/Siemod12.pdf
• Borcherds, Richard E., Eberhard Freitag, and Rainer Weissauer. "A Siegel cusp form of degree 12 and weight 12." Journal fur die Reine und Angewandte Mathematik (1998): 141-153.
• Erokhin, V. A. 1984. “Automorphism Groups of 24-Dimensional Even Unimodular Lattices.” Journal of Soviet Mathematics 26 (3): 1876–79. doi:10.1007/BF01670573.
• Erokhin, V. A. “Theta-Series of Even Unimodular Lattices.” Journal of Soviet Mathematics 25, no. 2 (April 1, 1984): 1012–20. doi:10.1007/BF01680824.
• Erokhin, V. A. “Theta-Series of Even Unimodular 24-Dimensional Lattices.” Journal of Soviet Mathematics 17, no. 4 (November 1, 1981): 1999–2008. doi:10.1007/BF01465457.
• Niemeier, Hans-Volker. 1973. “Definite Quadratische Formen Der Dimension 24 Und Diskriminante 1.” Journal of Number Theory 5 (2): 142–78. doi:10.1016/0022-314X(73)90068-1.