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Linear representation theory of symmetric group:S3

1,354 bytes added, 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>
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