[Next][Prev] [Right] [Left] [Up] [Index] [Root]
Given a K[G]-module M, return the action of G on M as homomorphism
f of G into the matrix group GLn(K).
Given a K[G]-module M, return the action of G on M as homomorphism
f of G into the matrix algebra Mn(K).
The function Representation allows the easy calculation of
group characters. We illustrate this with the 6-dimension module
for the group A7 constructed above.
> A7 := AlternatingGroup(7);
> M := PermutationModule(A7, Vector(GF(11), [1,0,1,0,1,0,1]));
> phi := Representation(M);
> [ Trace(phi(c[3])) : c in Classes(A7) ];
[ 7, 3, 4, 1, 1, 2, 0, 0, 0 ]
We present a procedure which, given a K[G]-module M, constructs
its dual D.
> DualModule := function(M)
> G := Group(M);
> f := Representation(M);
> return GModule(G, [ Transpose(f(G.i))^-1 : i in [1 .. Ngens(G)] ]);
> end function;
RightActionGenerator(M, i) : ModGrp, RngIntElt -> AlgMatElt
The i-th generator of the (right) acting matrix algebra for the module M.
That is, the image of the i-th group generator in the corresponding
representation.
Return the matrices giving the action on the module M as a sequence. These
are the images of the generators of the group in the corresponding representation.
Nagens(M) : ModGrp -> RngIntElt
The number of action generators (the number of generators of the algebra)
for the R[G]-module M.
The matrix group generated by the action generators of M.
Given a matrix group G defined over a finite field K, return the
action of G on each composition factor of the natural K[G]-module
for G.
We construct the tensor square T of the natural module M of the
matrix group G = SL(3, 5) and then determine the action of G
on each composition factor of T.
> G := SL(3, 5);
> M := GModule(G);
> T := TensorProduct(M, M);
> A := ActionGroup(T);
> S := Sections(A);
> #S;
2
There are just two composition factors of T, the symmetric square
and the exterior square of M.
> S[2];
MatrixGroup(3, GF(5))
Generators:
[1 0 0]
[0 2 0]
[0 0 3]
[0 1 0]
[1 0 1]
[1 0 0]
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|
