# Brauer Character in GAP.

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.

• @Shaun thanks for edit and answer ... – neelkanth Dec 23 '18 at 4:32

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" ]

• Thanks ... I will try this way and will tell.... – neelkanth Dec 23 '18 at 4:31
• Yes sir it is fine and working for me ... you explained exactly that I need... thank you very much ... – neelkanth Dec 23 '18 at 7:24
• Sir can i ask one more question? If i want to learn about GAP from where to start ? I mean is there elementary level book of GAP? Please suggest me. Thanks. – neelkanth Dec 23 '18 at 7:36
• I suggest to start with the GAP Software Carpentry lesson. It is aimed at beginners, and was taught at workshops, but should also be suitable for self-study. Then look also at my answer to the question What is the best way to self-study GAP?. – Alexander Konovalov Dec 23 '18 at 21:50
• Thanks .................... – neelkanth Dec 24 '18 at 4:23

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.

• If want to learn about GAP from where to start ? I mean is there elementary level book of GAP? Please suggest me. Thanks – neelkanth Dec 23 '18 at 10:12
• @neelkanth Simply search online for the GAP tutorial and the GAP manual. They're free PDFs with all the information you'll ever need. – Shaun Dec 23 '18 at 11:46
• There's also this, @neelkanth :) – Shaun Dec 23 '18 at 11:48
• @ Shaun Thank you very much ... – neelkanth Dec 23 '18 at 15:26
• please look at my problem math.stackexchange.com/questions/3051924/… – neelkanth Dec 25 '18 at 8:39