How to read GAP's output on "IrreducibleRepresentations"?

Question

For example for the group $SL_2(\mathbb{F}_3)$ I get the following,

gap> IrreducibleRepresentations(SL(2,3));
[ CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] ->
[ [ [ 1 ] ], [ [ 1 ] ] ], <action isomorphism> ),
  CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] ->
[ [ [ E(3)^2 ] ], [ [ 1 ] ] ], <action isomorphism> ),
  CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] ->
[ [ [ E(3) ] ], [ [ 1 ] ] ], <action isomorphism> ),
  CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] ->
[ [ [ E(3), -E(3) ], [ 0, E(3)^2 ] ], [ [ E(3), 1 ], [ E(3), -E(3) ] ]
 ], <action isomorphism> ),
  CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] ->
[ [ [ E(3)^2, E(3) ], [ 0, 1 ] ],
  [ [ -E(3)^2, -E(3) ], [ -E(3), E(3)^2 ] ] ], <action isomorphism> ),
  CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] ->
[ [ [ E(3), E(3)^2 ], [ 0, 1 ] ], [ [ 0, 1 ], [ -1, 0 ] ]
 ], <action isomorphism> ),
  CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] ->
[ [ [ 0, 0, 1 ], [ 0, 1, 0 ], [ -1, -1, -1 ] ],
  [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ -1, -1, -1 ] ] ], <action isomorphism> ) ]

It seems to me that it is specifying the two matrices in in each irrep of $SL_2(\mathbb{F}_3)$ which correspond to the two elements which it denotes as $(4,5,6)(7,9,8)$ and $(2,7,3,4)(5,8,9,6)$. But I don't know what is its scheme of denoting of denoting the group elements are? (what group elements do these two pairs of tuples correspond to?) And how to get the matrix for every group element from this?

Answer 1

What GAP returns is a list of homomorphisms, i.e. the actual irreducible representations. Each of these homomorphisms is a bit more complicated, as it actually factors from SL(2,3) through an isomorphic permutation group. What you can do is to take any such homomorphism, and evaluate it at arbitrary elements of SL(2,3).

gap> G:=SL(2,3);;
gap> irr:=IrreducibleRepresentations(G);
[ CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> 
    [ [ [ 1 ] ], [ [ 1 ] ] ], <action isomorphism> ), [...]

gap> rep:=irr[6];
CompositionMapping( [ (4,5,6)(7,9,8), (2,7,3,4)(5,8,9,6) ] -> 
 [ [ [ -E(3)^2, -E(3) ], [ 1, 0 ] ], [ [ 0, 1 ], [ -1, 0 ] ] ],
 <action isomorphism> )
gap> mat:=[[1,0],[1,1]]*One(GF(3));
[ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ] ]
gap> Image(rep,mat);
[ [ -E(3), E(3)^2 ], [ -1, 0 ] ]

i.e. this group element is mapped to $\left(\begin{array}{cc}-\zeta&\zeta^2\\ -1&0\end{array}\right)$ where $\zeta=e^{\frac{2\pi i}{3}}$.

If you want to see what happens with generators under all representations, you could use:

gap> gens:=GeneratorsOfGroup(G);
[ [ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), Z(3)^0 ] ], 
  [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ] ]
gap> List(irr,r->List(gens,g->Image(r,g)));
[ [ [ [ 1 ] ], [ [ 1 ] ] ], [ [ [ E(3)^2 ] ], [ [ 1 ] ] ], 
  [ [ [ E(3) ] ], [ [ 1 ] ] ], 
  [ [ [ E(3), -E(3)^2 ], [ 0, E(3)^2 ] ], 
      [ [ -E(3)^2, 1 ], [ E(3)^2, E(3)^2 ] ] ], 
  [ [ [ E(3)^2, -E(3) ], [ 0, 1 ] ], [ [ E(3), E(3)^2 ], [ E(3)^2, -E(3) ] ] ]
    , [ [ [ -E(3)^2, -E(3) ], [ 1, 0 ] ], [ [ 0, 1 ], [ -1, 0 ] ] ], 
  [ [ [ -1, -1, -1 ], [ 0, 1, 0 ], [ 1, 0, 0 ] ], 
      [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ -1, -1, -1 ] ] ] ]