[GAP Forum] Co1 generators in SO24

[MCKAY john]

Isn't the obvious place the book SPLAG by Conway & Sloane? Chapter 4?

Rob WIlson at QMUL is master of the generators. He has a new book
on the finite simples.


John McKay

==




On Fri, 9 Apr 2010, Mathieu Dutour wrote:

> There are a number of ways to do that.
>
> If you are interested in the automorphism group of the Leech lattice, that
> is the double cover of the group Co1, then you can use my package
> "polyhedral" (from http://www.liga.ens.fr/~dutour/Polyhedral/index.html)
> which is not official or the package "cryst" which is official.
> Both rely on the use of autom by B. Souvignier and W. Plesken and Magma
> rely as well on this program.
>
> But I should point to you that what you are asking is Co1, i.e. the quotient
> of the automorphism group of the Leech lattice by the antipodal involution.
> The atlas of finite groups http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/
> does not list obvious 24-dimensional rational representations of this group.
>
>   Mathieu
>
> >> 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
> y
>
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