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

, 18:29, 31 August 2013
Character table
| four-dimensional, induced from one-dimensional representation of Borel subgroup || 4 || -4 || 0 || 0 || 0 || 1 || -1 || 0
|}

==GAP implementation==

===Degrees of irreducible representations===

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

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

===Character table===

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

<pre>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 ] ) ]</pre>

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

<pre>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</pre>

===Irreducible representations===

The irreducible representations of [itex]GL(2,3)[/itex] can be computed using GAP's [[GAP:IrreducibleRepresentations]] function, as follows:

<pre>gap> IrreducibleRepresentations(GL(2,3));
[ CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8)
] -> [ [ [ -1 ] ], [ [ 1 ] ] ], <action isomorphism> ), 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 ] ] ], <action isomorphism> ), 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 ] ] ], <action isomorphism> ), 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 ] ] ], <action isomorphism> ),
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 ] ] ],
<action isomorphism> ), 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 ] ] ], <action isomorphism> ), 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 ] ] ], <action isomorphism> ) ]</pre>
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