Dear GAP Forum,
I am interested in decomposing the symmetric powers of a
representation. To give an example: take the group G :=
AllSmallGroups(6)[1] acting on V := C^3 (with coordinates x,y,z) as
the sum of its 2 dim irrep (X.3) and trivial rep (X.1).
When I look at Sym^2(V), I can see using characters that Sym^2(V)
decomposes into two 2-dim reps and 2 1-dim reps. A little manipulation
by hand shows that the bases are {x^2-y^2, xy}, {xz, yz}, x^2+y^2 and
z^2. I thought I should be able to do the same with primitive central
idempotents using the Wedderga package. But I find the output
confusing:
GR := GroupRing(Rationals, G);
gap> idempots := PrimitiveCentralIdempotentsByCharacterTable(GR);
[ (1/6)*<identity> of ...+(1/6)*f1+(1/6)*f2+(1/6)*f1*f2+(1/6)*f2^2+(1/
6)*f1*f2^2, (1/6)*<identity> of ...+(-1/6)*f1+(1/6)*f2+(-1/6)*f1*f2+(1/
6)*f2^2+(-1/6)*f1*f2^2, (2/3)*<identity> of ...+(-1/3)*f2+(-1/3)*f2^2 ]
gap> idempots[1];
(1/6)*<identity> of ...+(1/6)*f1+(1/6)*f2+(1/6)*f1*f2+(1/6)*f2^2+(1/6)*f1*f2^2
gap> Length(idempots);
3
How do I use this information to get a decomposition like the one I
can obviously see above? Any help will be welcome...
Ravi
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