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

Ravi Kulkarni ravi.kulk at gmail.com
Sun Oct 18 07:19:52 BST 2009

```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

```