Linear representation theory of general linear group:GL(2,3)

This article describes the linear representation theory (in characteristic zero and other characteristics excluding 2,3) of general linear group:GL(2,3), which is the general linear group of degree two over field:F3.

Summary
Note that general linear group:GL(2,3) and the binary octahedral group are isoclinic groups of the same order. We know that isoclinic groups have same proportions of degrees of irreducible representations, therefore, in this case, the degrees of irreducible representations are the same for both groups. However, the character tables themselves are not identical. In fact, the fields generated by character values also differ from one another.

Interpretation as general linear group of degree two
The group is a general linear group of degree two over field:F3. Compare with linear representation theory of general linear group of degree two over a finite field.

Character table
In the table below, we denote by $$\sqrt{-2}$$ a fixed square root of -2.

Degrees of irreducible representations
The degrees of irreducible representations can be computed using GAP's CharacterDegrees function, as follows:

gap> CharacterDegrees(GL(2,3)); [ [ 1, 2 ], [ 2, 3 ], [ 3, 2 ], [ 4, 1 ] ]

Character table
The character table can be computed using GAP's CharacterTable function, as follows:

gap> Irr(CharacterTable(GL(2,3))); [ Character( CharacterTable( GL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( GL(2,3) ), [ 1, 1, 1, 1, 1, -1, -1, -1 ] ), Character( CharacterTable( GL(2,3) ), [ 2, -1, 2, -1, 2, 0, 0, 0 ] ), Character( CharacterTable( GL(2,3) ), [ 2, 1, -2, -1, 0, -E(8)-E(8)^3,     E(8)+E(8)^3, 0 ] ), Character( CharacterTable( GL(2,3) ),    [ 2, 1, -2, -1, 0, E(8)+E(8)^3, -E(8)-E(8)^3, 0 ] ), Character( CharacterTable( GL(2,3) ), [ 3, 0, 3, 0, -1, 1, 1, -1 ] ), Character( CharacterTable( GL(2,3) ), [ 3, 0, 3, 0, -1, -1, -1, 1 ] ), Character( CharacterTable( GL(2,3) ), [ 4, -1, -4, 1, 0, 0, 0, 0 ] ) ]

A visual display of the character table can be achieved as follows:

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

2 4  1  4  1  3  3  3  2     3  1  1  1  1  .  .  ..

1a 6a 2a 3a 4a 8a 8b 2b

X.1    1  1  1  1  1  1  1  1 X.2    1  1  1  1  1 -1 -1 -1 X.3    2 -1  2 -1  2. . . X.4     2  1 -2 -1. A -A. X.5    2  1 -2 -1. -A A. X.6    3. 3 . -1  1  1 -1 X.7     3. 3 . -1 -1 -1  1 X.8     4 -1 -4  1. . ..

A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2

Irreducible representations
The irreducible representations of $$GL(2,3)$$ can be computed using GAP's GAP:IrreducibleRepresentations function, as follows:

gap> IrreducibleRepresentations(GL(2,3)); [ CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8)    ] -> [ [ [ -1 ] ], [ [ 1 ] ] ], ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, E(3) ], [ E(3)^2, 0 ] ],      [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] ->    [ [ [ -1/2*E(24)^11-1/2*E(24)^17, -1/2*E(24)-E(24)^11-E(24)^17-1/2*E(24)^19 ], [ -1/2*E(8)-1/2*E(8)^3, 1/2*E(24)^11+1/2*E(24)^17 ] ],      [ [ E(3), E(3) ], [ 0, E(3)^2 ] ] ], ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] ->    [ [ [ E(24)+E(24)^19, -1 ], [ -E(3)+E(3)^2, -E(24)-E(24)^19 ] ], [ [ E(3)+2*E(3)^2, E(8)+E(8)^3 ], [ -E(24)-E(24)^19, -E(3)^2 ] ] ], ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, 0, 1 ], [ 0, -1, 0 ], [ 1, 0, 0 ] ], [ [ 1, 0, 0 ], [ 0, 0, 1 ], [ 1, -1, -1 ] ] ],    ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ],      [ [ 1, 0, 0 ], [ -1, -1, -1 ], [ 0, 1, 0 ] ] ], ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] ->    [ [ [ 0, 0, -E(3), 0 ], [ 0, 0, -E(3), -E(3) ], [ -E(3)^2, 0, 0, 0 ], [ E(3)^2, -E(3)^2, 0, 0 ] ],      [ [ -E(3)^2, E(3)^2, 0, 0 ], [ -E(3)^2, 0, 0, 0 ], [ 0, 0, E(3)^2, 0 ], [ 0, 0, 1, 1 ] ] ], ) ]