Project:
Find the composition factors of the representations of the groups N_G(H)/H (where G=S_n is a symmetric group and H is a subgroup) given by the fixed points V^H where V is a simple (=irreducible) representation of G, over the field of 2 elements, H is 2-subgroup of S_n. Find the composition factors that are projective modules.
Reference for group representations: my book pages 1-7.
See http://www.math.umn.edu/~webb/RepBook/index.html
What is a group?
A simple=irreducible representation is one with no non-trivial invariant subspaces. Example: the 2-dimensional representation of S_3.
Fact: The simple representations of S_n over GF(2) are indexed by partitions of n, all of whose parts have distinct sizes.
There are five partitions of 4: 4=4, 4=3+1, 4=2+2, 4=2+1+1, 4=1+1+1+1
Other notation for the same thing:
[4], [3,1], [2,2]= [2^2], [2,1,1]=[2.1^2], [1,1,1,1]=[1^4]
and
XXXX, XXX, XX. These are called Ferrers diagrams.
X XX
Simple=irreducible representations of S_4 over GF(2) are indexed by 4, 3+1. Only two simples.
For S_3: 3= the trivial representation, 2+1= the 2-dimensional rep.
Subgroup of S_n: subset of S_n that is a group in its own right, i.e. is closed under multiplication and taking inverses.
The fixed points V^H of the subgroup H acting on a representation space V is defined to be V^H={v in V | hv = v}.
Here hv is the group element h applied to the vector v via the matrix of the representation of H (and G) on V.
Projective modules?
1. Set up the representations of the symmetric groups using the matrices etc in the files provided.
2. Apply the command FixedPointRep for each 'weight' subgroup and each representation.
3. Find the composition factors using CompositionFactorsRep. The computer should have the actual representations, but it will be too much to display all the matrices etc.
4. Test whether any of them are projective as representations of N_G(H)/H (not of N_G(H)) using IsProjectiveRep. This will require first constructing the factor group N_G(H)/H, and then making the representations of N_G(H) into representations of N_G(H)/H.
A main difficulty in all this is organizing a complicated calculation. It would be nice to have the results presented as a matrix. In the first instance I want to know a) the dimension of the simple composition factors and b) whether any of them are projective.
It would be helpful to have a GAP function written called
Deflate(rep, N)
where N is a normal subgroup of rep.group, and which returns a representation drep with drep.group=rep.group / N, with generators the images of the generators of rep.group, and where the representation space of drep is the quotient representation of rep by the subspace spanned by the vectors in FixedQuotient(RestrictedRep(rep.group, N,rep)). This quotient is the largest quotient of rep on which N acts trivially (=as the identity). This command could then be used in 4. above.
Motivation:
A lot of this is explained in Luis Valero's thesis, see
http://www.math.umn.edu/~webb/PhDStudents/index.html
especially the tables on pages 18, 19, 20. I particularly want to know about the 0 in position (3,3) in table 3.8 for S_9. It is obtained as the Brauer quotient of the weight group on the simple module (there is a command BrauerRep that produces this). I want to know what the fixed point rep is for this representation at the same weight group, and I hope it has a simple projective composition factor. The representation is indexed by the partition [7,2], whose conjugate is [1,1,1,1,1,2,2]. The subgroup is wgtgrp[9][7] in the notation of the online file. I also want to know what the Brauer quotients are of other 2-subgroups of S_9 on this representation (not just the weight groups). There is a list of all 2-subgroups of S_9 in the same place as the other code; however, there are too many subgroups listed, because many of them are conjugate within S_9, and we only need test one subgroup from each conjugacy class.
I may also want to know the fixed point representations of all of these 2-subgroups on this representation.
There is some rather complicated theory behind all this, to do with Mackey functors, that I am not explaining. It is motivated by one of the outstanding conjectures in modular representation theory of finite groups: Alperin's weight conjecture.
To get going: you need to have the code for the set of routines called "reps" (from my GAP page) and also the code in 'SymmetricGroupData' stored in a place where GAP can read it (e.g. your home directory). After starting up GAP do
gap> Read("reps");
gap> Read("SymmetricGroupData");
I have then written a function
FPresults:=function(n)
local output, p, row, rep, parent, g;
output:=[];
for p in regpart[2][n] do
row:=[];
rep:=ModularIrreducible(p,2);
parent:=rep.group;
for g in wgtgrp[n] do
Add(row,FixedPointRep(rep,g));
od;
Add(output,row);
od;
return output;
end;
which can be entered.
gap> output:=FPresults(5);
produces a lot of output which I suppress. Now
gap> List(output,x->List(x,y->y.dimension));
[ [ 1, 1, 1 ], [ 3, 1, 1 ], [ 2, 2, 1 ] ]
gap> CompositionFactorsRep(output[2][1]);
[ rec( dimension := 2, field := GF(2),
genimages := [ ,
,
],
group := Group([ (1,2), (4,5), (3,4) ]), isRepresentation := true ),
rec( dimension := 1, field := GF(2),
genimages := [ ,
,
],
group := Group([ (1,2), (4,5), (3,4) ]), isRepresentation := true ) ]
shows that the fixed point representation in position [2][1] has two composition factors.
I am not sure what is the best way to go about doing all this. I realize there is a steep learning curve, and potentially a daunting amount of material. I suppose I am somehow hoping that a table of composition factors of fixed point representations will magically be constructed.
Peter