[GAP Forum] Co0 generators in SO24

[mim_ at op.pl]

The conjecture is that whether we can decompose Leech lattice into 273 of E8 tripples. 273*720=4095*48=196560.

Maybe you know the answer already.

Regards,
Marek
 
mim_ at op.pl napisał(a): 
 > Thank you for your answers. Sorry, I am interested in Co0 - automorphism group of Leech lattice. My goal is to find the decomposition of Leech lattice into 4095 "crosses" i.e. orthonormal frames of 48 vectors. I have heard that such decomposition exists, but I want to have it explicite.
 > 
 > My plan is following. Take simple frame built with 4^2,0^22 vectors. Calculate image of it by random element from Co0. See if new frame is received. If yes then add it to the set. Continue until all is done.
 > 
 > What I suspect is that maybe E8 sublattices are distinct generated by those crosses, but I am not sure.
 > 
 > I would be grateful if you can provide me with generators of such group for my Leech lattice mentioned below.
 > 
 > Regards,
 > Marek
 > 
 >  
 > mim_ at op.pl napisał(a): 
 >  > Hello,
 >  > 
 >  > I have received following email from one matematician. I have asked him for the matrix generators of Conway group Co1 in SO(24). Do you know how to obtain such generators in GAP ? 
 >  > 
 >  > <quote>
 >  > The following Magma code should work:
 >  > 
 >  > L := Lattice("Lambda",24);
 >  > G := AutomorphismGroup(L);
 >  > B := BasisMatrix(L);
 >  > S := ShortestVectors(L);
 >  > S := S cat [-S[i] : i in [1..#S]];
 >  > M := MatrixRing(Rationals(),24);
 >  > G := MatrixGroup<24, Rationals() | [B^(-1) * M!G.i * B : i in [1..Ngens(G)]]>;
 >  > 
 >  > Then S will be the list of minimal vectors and G will be the
 >  > automorphism group, as a subgroup of SO(24). The code for G is
 >  > a little ugly, because by default Magma will express it as a
 >  > subgroup of GL_24(Z) instead.
 >  > < end of quote>
 >  > 
 >  > Here is the base matrix of my leech lattice. The determinant is 8^12.
 >  > B:=[[4,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [2,2,2,2,2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [2,2,2,2,0,0,0,0,2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0],
 >  > [4,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],
 >  > [2,2,0,0,2,2,0,0,2,2,0,0,2,2,0,0,0,0,0,0,0,0,0,0],
 >  > [2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,0,0,0,0,0,0,0,0],
 >  > [2,0,0,2,2,0,0,2,2,0,0,2,2,0,0,2,0,0,0,0,0,0,0,0],
 >  > [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0],
 >  > [2,0,2,0,2,0,0,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0,2,2],
 >  > [2,0,0,2,2,2,0,0,2,0,2,0,0,0,0,0,0,0,0,0,0,2,0,2],
 >  > [2,2,0,0,2,0,2,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2],
 >  > [0,2,2,2,2,0,0,0,2,0,0,0,2,0,0,0,0,0,0,2,0,0,0,2],
 >  > [0,0,0,0,0,0,0,0,2,2,0,0,2,2,0,0,2,2,0,0,2,2,0,0],
 >  > [0,0,0,0,0,0,0,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0],
 >  > [-3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]];
 >  > 
 >  > Regards,
 >  > Marek
 >  > 
 >  > 
 >