# Changes

## Linear representation theory of symmetric group:S3

, 04:55, 17 June 2012
Verification of the McKay conjecture
Hence, the McKay conjecture is true for this group.

==GAP implementation==

===Degrees of irreducible representations===

The [[degrees of irreducible representations]] can be computed using the [[GAP:CharacterDegrees|CharacterDegrees]] function:

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

===Character table===

The character table can be computed using the [[GAP:Irr|Irr]] and [[GAP:CharacterTable|CharacterTable]] functions:

<pre>gap> Irr(CharacterTable(SymmetricGroup(3)));
[ Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, -1, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, -1 ] ),
Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] ) ]</pre>

A nicer display can be achieved using the Display function:

<pre>gap> Display(CharacterTable(SymmetricGroup(3)));
CT1

2 1 1 .
3 1 . 1

1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a

X.1 1 -1 1
X.2 2 . -1
X.3 1 1 1</pre>

===Irreducible representations===

The irreducible representations can be computed using the [[GAP:IrreducibleRepresentations|IrreducibleRepresentations]] function:

<pre>gap> IrreducibleRepresentations(SymmetricGroup(3));
[ Pcgs([ (2,3), (1,2,3) ]) -> [ [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ (2,3), (1,2,3) ]) -> [ [ [ -1 ] ], [ [ 1 ] ] ],
Pcgs([ (2,3), (1,2,3) ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ] ]</pre>