To find Brauer Character for the special linear group SL(2,5) in GAP.

Question

As suggest by Alexander Konovalov I load LoadPackage("ctbllib") and them worked for symmetric group $S5$. It worked for me as

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 ] ) ]

But i tried same for Special linear group $SL(2,5)$, it does not works

gap> LoadPackage("ctbllib");

true

gap> m:=SL(2,5);

SL(2,5)

gap> t:=CharacterTable("m") mod 2;

fail

gap>

Where is problem. Please help. Thanks.

Answer 1

You are mixing up the (string) names of groups in the library and variables you choose to assign a group to. The group "M" in the libary is the monster simple group, and for it the Brauer table is not known.

The ATLAS name of $SL(2,5)$ is 2.L2(5), indeed we can get the table for it as

gap> t:=CharacterTable("2.L2(5)") mod 3;
BrauerTable( "2.A5", 3 )

Note that all of this is about pre-computed Brauer tables from the character table library. If you wanted to compute the Brauer table for an arbitrary finite group, you would have to find all irreducible modules (e.g. by splitting up the regular module), and then lift the Brauer character values appropriately (there currently is no pre-defined function which does so automatically).