[GAP Forum] GAP4 question: character table as a matrix

[Alexander Hulpke]

Dear Roman Schmied,

> I am trying to use a character table of a permutation group "g" as a 
> matrix, for further matrix multiplications. In order to extract this 
> matrix, my code is
> 
>        TransposedMat(TransposedMat(Irr(CharacterTable(g))));
> 
> which is very roundabout. Would anyone know of a nice way of doing this?

The following is probably the shortest
  List(Irr(g),ShallowCopy);

though

n:=Length(Irr(g));Irr(g){[1..n]}{[1..n]};

is a close contender.

> My goal is the decomposition of a vector "c" of characters into a direct 
> sum of irreducible representations, which I then do with
> 
>        TransposedMat(TransposedMat(Irr(CharacterTable(g))))
>        * DiagonalMat(SizesConjugacyClasses(CharacterTable(g)))
>        * c
>        / Length(Elements(g));

As you have irreducible characters calculating scalar products will be
easier and more efficient:

  MatScalarProducts(Irr(g),[c]);

or List(Irr(g),i->ScalarProduct(i,c));  

(If you prefer to solve a system of equations, 
   SolutionMat(Irr(g),c);
will work.)

Best wishes,

  Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke