[GAP Forum] Re: Working with symmetric powers of a linear transformation

[Laurent Bartholdi]

Dear Ravi:
What you're looking for is the coefficients of GSym2[i][j] in the
basis GSym2. Do the following:

Sym2V := VectorSpace(F,Sym2);
m1_6 := List(GSym2[1],x->Coefficients(Basis(Sym2V),x));
m2_6 := List(GSym2[2],x->Coefficients(Basis(Sym2V),x));

(or shorter: m_6 := List(GSym2,m->List(m,x->Coefficients(Basis(Sym2V),x)));
)
Note that the answer will not be represented using a, but using E(12),
a twelvth root of unity.

Cheers, Laurent

On Sun, Oct 18, 2009 at 8:19 AM, Ravi Kulkarni <ravi.kulk at gmail.com> wrote:
> Dear GAP Forum,
>
> Given a linear transformation on a vector space V with basis {x,y,z},
> I need to calculate the matrix of the induced linear transformation on
> the symmetric powers of V. In the code below, GSym1 gives the action
> of the (generators of the) group G on {x,y,z}. GSym2 represents the
> induced action of G on the second symmetric power of V.
>
> a := Sqrt(3);
> F := Field(a);
>
> m1 := [ [-1/2,-a/2,0],[a/2,-1/2,0],[0,0,1] ];;
> m2 := [ [-1,0,0],[0,1,0],[0,0,1] ];;
> G := Group(m1,m2);;
>
> R := PolynomialRing(Rationals,3);;
> inds := IndeterminatesOfPolynomialRing(R);;
> x := inds[1];; y := inds[2];; z := inds[3];;
>
> Sym1 := [x,y,z];
> Sym2 := [x^2,x*y,x*z,y^2,y*z,z^2];
>
> GSym1 := List([m1,m2], g -> g*Sym1);
> GSym2 := List(GSym1, r -> List(Sym2, a -> Value(a,Sym1,r)));
>
> What I would now like to have is the 6x6 matrix that represents the
> action of G with respect to the basis Sym2. (i.e. the coefficients of
> the elements of Sym2 in GSym2). By hand, this is:
>
> m1_6 := [ [1/4, -a/4,0,3/4,0,0],
>          [a/2,-1/2,0,-a/2,0,0],
>          [0,0,-1/2,0,a/2,0],
>          [3/4,a/4,0,1/4,0,0],
>          [0,0,-a/2,0,-1/2,0],
>          [0,0,0,0,0,1] ];
>
> m2_6 := [ [1,0,0,0,0,0],
>          [0,-1,0,0,0,0],
>          [0,0,-1,0,0,0],
>          [0,0,0,1,0,0],
>          [0,0,0,0,1,0],
>          [0,0,0,0,0,1] ];
>
> One would obviously not want to do this by hand for larger symmetric
> powers. Is there a GAP function that will extract the matrix of the
> induced transformation as above?
>
> Thanks,
>  Ravi
>



-- 
Prof. Dr. Laurent Bartholdi   \ laurent.bartholdi<at>gmail<dot>com
G.-A. Universität zu Göttingen \ Phone: +49 551 39 7826
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Sent from Gottingen, NI, Germany