[GAP Forum] character^automorphism

[Hulpke,Alexander]

Dear Forum,

On Mar 5, 2017, at 11:24 AM, Benjamin Sambale <benjamin.sambale at gmail.com<mailto:benjamin.sambale at gmail.com>> wrote:
the class function manual says: "The power of a class function by an element that normalizes the underlying group or by a Galois automorphism is the conjugate class function.”

G:=Group((1,2),(3,4));
a:=Elements(AutomorphismGroup(G))[2];
Irr(G)[1]^a;
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 3rd choice method found for `^' on 2 arguments

Well, `a` is not an element (of a larger group) that normalizes the underlying group (in fact this automorphism cannot be realized by conjugation), nor a Galois automorphism. (The identity *is* a Galois automorphism, albeit a boring one.)

If you take the regular representation, you can take equivalent conjugating elements of S_4, and it works as expected.

gap> G:=Group((1,2)(3,4),(1,3)(2,4));;
gap> c:=Irr(G)[1];
Character( CharacterTable( Group([ (1,2)(3,4), (1,3)(2,4) ]) ),
 [ 1, 1, 1, 1 ] )
gap> a:=Elements(AutomorphismGroup(G));;Length(a);
6
gap> List(a,IsConjugatorAutomorphism);
[ true, true, true, true, true, true ]
gap> a:=List(a,ConjugatorOfConjugatorIsomorphism);
[ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4) ]
gap> List(a,x->c^x);
[ Character( CharacterTable( Group([ (1,2)(3,4), (1,3)(2,4) ]) ),
  [ 1, 1, 1, 1 ] ), Character( CharacterTable( Group([ (1,2)(3,4), (1,3)
  (2,4) ]) ), [ 1, 1, 1, 1 ] ), Character( CharacterTable( Group([ (1,2)
  (3,4), (1,3)(2,4) ]) ), [ 1, 1, 1, 1 ] ),
  Character( CharacterTable( Group([ (1,2)(3,4), (1,3)(2,4) ]) ),
  [ 1, 1, 1, 1 ] ), Character( CharacterTable( Group([ (1,2)(3,4), (1,3)
  (2,4) ]) ), [ 1, 1, 1, 1 ] ), Character( CharacterTable( Group([ (1,2)
  (3,4), (1,3)(2,4) ]) ), [ 1, 1, 1, 1 ] ) ]

Regards,

   Alexander Hulpke


-- Colorado State University, Department of Mathematics,
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email: hulpke at colostate.edu<mailto:hulpke at colostate.edu>, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke