[GAP Forum] Co1 generators in SO24

[Asst. Prof. Dmitrii (Dima) Pasechnik]

Dear Marek,
you can find matrices for 2.Co1 here:
http://brauer.maths.qmul.ac.uk/Atlas/spor/Co1/gap0/2Co1G1-Zr24B0.g
as a subgroup of SL(24,Z).
To find the form F that is invariant under the group, you can, for instance,
solve an obvious system of linear equations g_i F=F (g_i^T)^-1, with i=1,2.

Note that Co1 itself does not have a 24-dimensional faithful
representation in characteristic 0.
The minimal degree of such a representation is 276.

Best,
Dmitrii


On 8 April 2010 22:27, mim_ at op.pl <mim_ at op.pl> wrote:
> 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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Dmitrii Pasechnik
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