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For the symmetric group of degree n the irreducible representations
can be indexed by partitions of weight n. For more information on
partitions see Section Partitions.
It is possible to define representing matrices of the symmetric group
over the integers.
Al: MonStgElt Default: "JamesKerber"
Given a partition pa of weight n and a permutation pe in a symmetric
group of degree n, return an
irreducible representing matrix for pe, indexed by pa, over the integers.
If Al is set to the default "JamesKerber" then
the method described in [JK81] is used.
If Al is set to "Boerner"
the method described in the book of Boerner [Boe67] is used.
If Al is set to "Specht"
then the method used is a direct implementation of that used by Specht
in his paper from 1935 [Spe35].
We compute a representing matrix of a permutation using two different
algorithms and check whether the results have the same character.
> a:=SymmetricRepresentation([3,2],Sym(5)!(3,4,5) : Al := "Boerner");a;
[ 0 0 1 -1 0]
[ 1 0 0 -1 0]
[ 0 1 0 -1 0]
[ 0 0 0 -1 1]
[ 0 0 0 -1 0]
> b:=SymmetricRepresentation([3,2],Sym(5)!(3,4,5) : Al := "Specht");b;
[ 0 1 0 -1 0]
[ 0 0 1 0 -1]
[ 1 0 0 0 0]
[ 0 0 0 0 -1]
[ 0 0 0 1 -1]
> IsSimilar(Matrix(Rationals(), a), Matrix(Rationals(), b));
true
The matrices are similar as they should be.
The seminormal and orthogonal representations involve matrices
which are not necessarily integral.
The method Magma uses to construct these matrices is described
in [JK81, Section 3.3];
Given a partition pa of weight n and a permutation pe in a symmetric
group of degree n, return the
matrix of the seminormal representation for pe, indexed by pa,
over the rationals.
Given a partition pa of weight n and a permutation pe in a symmetric
group of degree n, return the
matrix of the orthogonal representation for pe, indexed by pa.
An orthogonal basis is used to compute the matrix which may have entries
in a cyclotomic field.
We compare the seminormal and orthogonal representations of a permutation
and note that they are similar.
> g:=Sym(5)!(3,4,5);
> a:=SymmetricRepresentationSeminormal([3,2],g);a;
[-1/2 0 -3/4 0 0]
[ 0 1/2 0 3/4 0]
[ 1 0 -1/2 0 0]
[ 0 1/3 0 -1/6 8/9]
[ 0 1 0 -1/2 -1/3]
> b:=SymmetricRepresentationOrthogonal([3,2],g);b;
[-1/2 0 zeta(24)_8^2*zeta(24)_3 + 1/2*zeta(24)_8^2 0 0]
[0 1/2 0 -zeta(24)_8^2*zeta(24)_3 - 1/2*zeta(24)_8^2 0]
[-zeta(24)_8^2*zeta(24)_3 - 1/2*zeta(24)_8^2 0 -1/2 0 0]
[0 -1/3*zeta(24)_8^2*zeta(24)_3 - 1/6*zeta(24)_8^2 0
-1/6 2/3*zeta(24)_8^3 - 2/3*zeta(24)_8]
[0 2/3*zeta(24)_8^3*zeta(24)_3 + 1/3*zeta(24)_8^3
+ 2/3*zeta(24)_8*zeta(24)_3 + 1/3*zeta(24)_8 0
-1/3*zeta(24)_8^3 + 1/3*zeta(24)_8 -1/3]
> IsSimilar(a,b);
true
They should both be of finite order, 3.
> IsOne(a^Order(g));
true
> IsOne(b^Order(g));
true
>
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