Brauer Character in GAP.

Question

What is command to obtain Brauer Character in GAP? In magma, it is like

> G := Sym(5);
> I := AbsolutelyIrreducibleModules(G, GF(3));
> [BrauerCharacter(i): i in I];
[
    ( 1, 1, 1, 0, 1, 1, 0 ),
    ( 1, -1, 1, 0, -1, 1, 0 ),
    ( 4, 2, 0, 0, 0, -1, 0 ),
    ( 4, -2, 0, 0, 0, -1, 0 ),
    ( 6, 0, -2, 0, 0, 1, 0 )
] 

Please tell me the same for GAP.

Thanks.

Answer 1

Indeed, as in the quote from the GAP manual given by @Shaun, trying to do this immediately in GAP fails:

gap> G:=SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap> t:=CharacterTable(G);
CharacterTable( Sym( [ 1 .. 5 ] ) )
gap> t mod 3;
fail

The GAP Character Table Library is provided by the CTblLib package (in most installations, it is loaded by default, otherwise one should load it first with LoadPackage("ctbllib");. Then you can retrieve the character table from the library. In your example, do the following:

gap> t:=CharacterTable("S5") mod 3;
BrauerTable( "A5.2", 3 )
gap> Irr(t);
[ Character( BrauerTable( "A5.2", 3 ), [ 1, 1, 1, 1, 1 ] ), 
  Character( BrauerTable( "A5.2", 3 ), [ 1, 1, 1, -1, -1 ] ), 
  Character( BrauerTable( "A5.2", 3 ), [ 6, -2, 1, 0, 0 ] ), 
  Character( BrauerTable( "A5.2", 3 ), [ 4, 0, -1, 2, 0 ] ), 
  Character( BrauerTable( "A5.2", 3 ), [ 4, 0, -1, -2, 0 ] ) ]

See ?AllCharacterTableNames in GAP to search in identifiers of library tables. It could narrow the search if you are not sure which identifier to use, For example,

gap> AllCharacterTableNames(Size,120);
[ "2.A5", "2.A6M2", "2.Alt(5)", "2xA5", "A5.2", "A6.2_1M3", "D120", "L2(25)M3", "Sym(5)" ]
gap> AllCharacterTableNames(Size,7920,IsSimple,true);
[ "HSM9", "M11", "M12M2", "ONM11" ]

Answer 2

Currently GAP does not provide special functionality for computing Brauer character tables, but there is an interface to the MOC system (see [HJLP]), and the GAP Character Table Library contains many known Brauer character tables.

Found here.

See "71.3-1 CharacterTable" onwards ibid. for the capabilities.