Linear representation theory of special linear group:SL(2,3)

Summary information
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.

Character table
In the table below, we denote by $$\omega$$ a primitive cube root of unity.

Isoclinism and projective representations
Please compare this with projective representation theory of alternating group:A4.

Degrees of irreducible representations
The degrees of irreducible representations can be computed using the CharacterDegrees function:

gap> CharacterDegrees(SL(2,3)); [ [ 1, 3 ], [ 2, 3 ], [ 3, 1 ] ]

This says that there are three irreducible representations of degree one, three irreducible representations of degree two, and one irreducible representation of degree three.

Character table
The characters of irreducible representations can be computed using Irr and CharacterTable as follows:

gap> Irr(CharacterTable(SL(2,3))); [ Character( CharacterTable( SL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( SL(2,3) ), [ 1, E(3)^2, E(3), 1, E(3), E(3)^2, 1    ] ), Character( CharacterTable( SL(2,3) ),    [ 1, E(3), E(3)^2, 1, E(3)^2, E(3), 1 ] ), Character( CharacterTable( SL(2,3) ), [ 2, 1, 1, -2, -1, -1, 0 ] ), Character( CharacterTable( SL(2,3) ), [ 2, E(3), E(3)^2, -2, -E(3)^2,     -E(3), 0 ] ), Character( CharacterTable( SL(2,3) ),    [ 2, E(3)^2, E(3), -2, -E(3), -E(3)^2, 0 ] ), Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]

A nicer display of the character table can be achieved with the Display function:

gap> Display(CharacterTable(SL(2,3))); CT1

2 3  1  1  3   1   1  2     3  1  1  1  1   1   1.

1a 6a 6b 2a 3a  3b 4a

X.1    1  1  1  1   1   1  1 X.2    1  A /A  1  /A   A  1 X.3    1 /A  A  1   A  /A  1 X.4    2  1  1 -2  -1  -1. X.5    2 /A  A -2  -A -/A. X.6    2  A /A -2 -/A  -A. X.7    3. . 3   .   . -1

A = E(3)^2 = (-1-ER(-3))/2 = -1-b3

Irreducible representations
The irreducible representations can be computed using IrreducibleRepresentations as follows:

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