Complex Lorentzian Leech Lattice and the Bimonster - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-18T17:39:05Z http://mathoverflow.net/feeds/question/46696 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46696/complex-lorentzian-leech-lattice-and-the-bimonster Complex Lorentzian Leech Lattice and the Bimonster Paul Hjelmstad 2010-11-20T00:52:17Z 2010-11-20T00:52:17Z <p>I'm reading an excellent paper on the complex Lorentzian Leech Lattice and the bimonster (Tathagata Basak). Instead of using the binary Golay Code, the author uses the ternary Golay code and the complex (lorentzian) leech lattice.</p> <p>Many references in the paper are to SPLAG, however, I am having trouble finding (2) on page 6 of the paper in SPLAG. When I master LaTex I will post it here. Basically, it defines the ternary Golay code over the Eisenstein integers (Z(exp(2pi*i/3), the Complex Lorentzian Leech Lattice as vectors over E^12, and C12, the ternary Golay code in F3^12. I've gotten pretty good with the MOG and the MINIMOG, so I grasp the use of the code, just not in this context. </p> <p>And then, of course, how it all leads to the Bimonster, and the Inc(P^2(F3)) with 26 nodes. Also reading Conway's (26 Implies the Bimonster) which also discusses the 13 line 13 point projective plane of order 3, in the same context. And deflating the 12-gons, etc. </p> <p>But for today, just want to make sense of this use of the TGC in conjunction with the CLLL:</p> <p>The complex Leech Lattice ^ consists of the set of vectors in E^12 of the form</p> <p>{(m + theta*c(sub i) + 3z(sub i))i=1...12 : m = 0, or +/- 1, (c(sub i) member of C12, </p> <p>Sigma z(sub i) =cong= m mod theta} where </p> <p>E = the ring of Z[exp(2 pi*i/3)] of Eisenstein integers C12 = Ternary Golay Code over F(sub 3)^12 and Theta = sqrt(-3) or w - w bar where w = exp(2*pi*i)</p> <p>If someone could help my tie in my (better) understanding of the TGC with this more novel use of a Lattice over the Eisenstein integers, and new concepts of Weyl vectors, Coxeter-Dynkins Diagrams, and the rest, (I've read the whole paper) it would be greatly appreciated. But I would be happy just to understand the formula above, which is in SPLAG, I just cannot find it.</p> <p>PGH</p>