Operations on Elements

The operations in this section can be applied to elements of either a group algebra or of a group algebra subalgebra of type AlgGrpSub. Only those operations are listed, which are additional to those available for general algebras.

a + r: AlgGrpElt, RngElt -> AlgGrpElt

r + a: RngElt, AlgGrpElt -> AlgGrpElt

The sum of the group algebra element \(a \in R[G]\) and the scalar \(r \in R\).

a + g: AlgGrpElt, GrpElt -> AlgGrpElt

g + a: GrpElt, AlgGrpElt -> AlgGrpElt

The sum of the group algebra element \(a \in R[G]\) and the group element \(g \in G\).

a - r: AlgGrpElt, RngElt -> AlgGrpElt

r - a: RngElt, AlgGrpElt -> AlgGrpElt

The difference of the group algebra element \(a \in R[G]\) and the scalar \(r \in R\).

a - g: AlgGrpElt, GrpElt -> AlgGrpElt

g - a: GrpElt, AlgGrpElt -> AlgGrpElt

The difference of the group algebra element \(a \in R[G]\) and the group element \(g \in G\).

a * r: AlgGrpElt, RngElt -> AlgGrpElt

r * a: RngElt, AlgGrpElt -> AlgGrpElt

The product of the group algebra element \(a \in R[G]\) and the scalar \(r \in R\).

g * a: GrpElt, AlgGrpElt -> AlgGrpElt

a * g: AlgGrpElt, GrpElt -> AlgGrpElt

The product of the group algebra element \(a \in R[G]\) and the group element \(g \in G\).

Support(a): AlgGrpElt -> SeqEnum

The support of \(a\); that is, the sequence of group elements whose coefficients in \(a\) are non-zero.

Trace(a): AlgGrpElt -> RngElt

The trace of \(a\); that is, the coefficient of \(1_G\) in \(a\).

Augmentation(a): AlgGrpElt -> RngElt

The augmentation of the group algebra element \(a\); that is, \(\sum_{g \in G} r_g\) where \(a = \sum_{g \in G} r_g * g\).

Involution(a): AlgGrpElt -> AlgGrpElt

If \(a = \sum_{g \in G} r_g * g\), returns \(\sum_{g \in G} r_g * g^{-1}\).

Coefficient(a, g): AlgGrpElt, GrpElt -> RngElt

a[g]: AlgGrpElt, GrpElt -> RngElt

The coefficient of \(g \in G\) in \(a \in R[G]\).

ElementToSequence(a): AlgGrpElt -> SeqEnum

Eltseq(a): AlgGrpElt -> SeqEnum

If \(a\) is an element from a group algebra \(A\) given in vector representation, this returns the sequence of coefficients with respect to the fixed basis of \(A\). If \(A\) is given in terms representation, this returns a sequence of tuples, where the second entry is a group element and the first is the coefficient of that group element in \(a\).

Coefficients(a): AlgGrpElt -> SeqEnum

For an element \(a\) from a group algebra \(A\) given in vector representation, this returns the sequence of coefficients with respect to the fixed basis of \(A\).

Centraliser(a): AlgGrpElt -> AlgGrpSub

Centralizer(a): AlgGrpElt -> AlgGrpSub

The centralizer in the group algebra \(A\) of the element \(a\) of \(A\).

Centraliser(S, a): AlgGrpSub, AlgGrpElt -> AlgGrpSub

Centralizer(S, a): AlgGrpSub, AlgGrpElt -> AlgGrpSub

The centralizer of the element \(a\) (of a group algebra \(A\)) in the subalgebra \(S\) of \(A\).

Example: powering (ex-76435e)

We use the group algebra to determine the diameter of the Cayley graph of a group.

> G := Alt(6);
> QG := GroupAlgebra( Rationals(), G );
> e := QG!1 + &+[ QG!g : g in Generators(G) ];
> e;
Id(G) + (1, 2)(3, 4, 5, 6) + (1, 2, 3)

Run in calculatorCopy input

The group elements that can be expressed as words of length at most \(n\) in the generators of \(G\) have non-zero coefficient in \(e^n\). The following function returns for a group algebra element \(e\) a sequence with the cardinalities of the supports of \(e^n\) and breaks when the group order is reached.

> wordcount := function(e)
>     f := e;
>     count := [ #Support(f) ];
>     while count[#count] lt #Group(Parent(e)) do
>         f *:= e;
>         Append(~count, #Support(f));
>     end while;
>     return count;
> end function;

Run in calculatorCopy input

Now apply this function to the above defined element:

> wordcount( e );
[ 3, 7, 14, 26, 47, 83, 140, 219, 293, 345, 360 ]

Run in calculatorCopy input

Thus, every element in \(A_6\) can be expressed as a word of length at most 11 in the generators \((1,2)(3,4,5,6)\) and \((1,2,3)\). A better 2-generator set is for example \((1,2,3,4,5)\) and \((1,5,3,6,4)\), where all elements can be expressed as words of length at most 10 and this is in fact optimal. A worst 2-generator set is given by \((1,2)(3,4)\) and \((1,5,3,2)(4,6)\).

> wordcount( QG!1 + G!(1,2,3,4,5) + G!(1,5,3,6,4) );
[ 3, 7, 15, 31, 60, 109, 183, 274, 350, 360 ]
> wordcount( QG!1 + G!(1,2)(3,4) + G!(1,5,3,2)(4,6) );
[ 3, 6, 11, 18, 28, 43, 63, 88, 119, 158, 206, 255, 297, 329, 352, 360 ]

Run in calculatorCopy input

Example: average (ex-1a2ff9)

The group algebra can also be used to investigate the random distribution of words of a certain length in the generators of the group.

> M11 := sub< Sym(11) | (1,11,9,10,4,3,7,2,6,5,8), (1,5,6,3,4,2,7,11,9,10,8) >;
> A := GroupAlgebra(RealField(16), M11 : Rep := "Vector");
> A;
Group algebra with vector representation
Coefficient ring: Real Field of precision 16
Group: Permutation group M11 acting on a set of cardinality 11
    Order = 7920 = 2^4 * 3^2 * 5 * 11
        (1, 11, 9, 10, 4, 3, 7, 2, 6, 5, 8)
        (1, 5, 6, 3, 4, 2, 7, 11, 9, 10, 8)
> e := (A!M11.1 + A!M11.2) / 2.0;
> eta := Eta(A) / #M11;

Run in calculatorCopy input

For growing \(n\), the words of length \(n\) in the generators of M11 converge towards a random distribution iff e\(^n\) converges towards eta. We look at the quadratic differences of the coefficients of e^n-eta for \(n = 10,20,30,40,50\).

> e10 := e^10;
> f := A!1;
> for i in [1..5] do
>     f *:= e10;
>     print &+[ c^2 : c in Eltseq(f - eta) ];
> end for;
0.0012050667195213
1.289719354694155e-5
5.9390965208879e-7
3.394099291966e-8
2.19432454574986e-9

Run in calculatorCopy input