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
m...@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
>
>
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