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
m...@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
>
>
> 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
> >
> >
>
_______________________________________________
Forum mailing list
Forum@mail.gap-system.org
http://mail.gap-system.org/mailman/listinfo/forum
