[GAP Forum] Does Irr(g) return the principle character as its first entry?

[Thomas Breuer]

Dear Sashar,

you asked the GAP Forum

> I am working with SmallGroups Library.I have used the function
> Irr(SmallGroup(n,i))  for some ns and is. I have noticed that the pincipal
> character 1_G is the first entry for these groups. Is this always true for
> all the groups in SmallGroups Library?

No.

The method that is chosen by GAP for computing
the irreducible characters of a group
depends on properties of this group
and not on the library from which it is constructed.

For example,
there is a method for computing the irreducible characters
of a symmetric group in its natural representation;
this method uses the fact that the conjugacy classes of the group
are parametrized by the cycle structures of the group elements.
This method is applicable only if the group has this property
and if this property is stored in the group
before the call to the function `Irr',
and currently this method can be expected to be used in this situation.

Here is a sample GAP session.

    gap> g:= SmallGroup( 120, 34 );     # the symmetric group on five points
    Group([ (1,2,3,4,5), (1,2) ])
    gap> Position( Irr( g ), TrivialCharacter( g ) );
    1
    gap> g:= SmallGroup( 120, 34 );     # fetch the group again
    Group([ (1,2,3,4,5), (1,2) ])
    gap> IsNaturalSymmetricGroup( g );  # now the group stores the property
    true
    gap> Position( Irr( g ), TrivialCharacter( g ) );
    7

Note that the GAP manual states explicitly that the trivial character
need not be the first entry in the list of irreducible characters.

All the best,
Thomas