EXERCISE:

Decompose the module of the regular representation of S_3 into irreducible submodules

> S3:=Sn(3);

> regS3:=regmodule(S3);

> charS3:=module2char(regS3);

> map(charS3(classfunc),permute(3));

Step 1: The sum of all of the elements is a one dimensional module

> triv(group):=S3; triv(classfunc):=g->1;

> reynoldsop(triv,regS3,v[[1,2,3]]);

> blinform(charS3,triv);

Step 2: The sign representation is also a one dimensional submodule

> sgn(group):=S3; sgn(classfunc):=g->(-1)^lengthperm(g);

> blinform(charS3,sgn);

> reynoldsop(sgn,regS3,v[[1,2,3]]);

Step 3: The remaining 4 dimensional module will not be irreducible

> rest(group):=S3; rest(classfunc):=g->charS3(classfunc)(g)-triv(classfunc)(g)-sgn(classfunc)(g);

> blinform(rest,rest);

There are only 3 conjugacy classes and so only 3 irreducible characters, therefore
the multiplicity of the last

> nops(conjclass(S3));

> map(rest(classfunc),permute(3));

> permute(3);

Find an irreducible submodule of dimension 2, show that it is invariant by acting
on the basis elements and show that they are still in the submodule.

> tdim(group):=S3;
tdim(classfunc)([1,2,3]):=2;
tdim(classfunc)([1,3,2]):=0;
tdim(classfunc)([2,1,3]):=0;
tdim(classfunc)([2,3,1]):=-1;
tdim(classfunc)([3,1,2]):=-1;
tdim(classfunc)([3,2,1]):=0;

> reynoldsop(tdim,regS3,v[[1,2,3]]);

>

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Exercise: To continue, decompose the regular representation of S4