Let A = R[G] be a group algebra defined in Magma, where R is a unital ring with a matrix echelon algorithm and G is a group. If A was constructed to use the vector representation for its elements, then subalgebras and left, right and two-sided ideals of A can be constructed. If R is a field, then quotient algebras of A are also possible.
A subalgebra or ideal of a group algebra A is not in general a group algebra; hence, a different type is needed for these objects. Two possibilities are provided in Magma. The first is to construct the subalgebra or ideal to have type AlgGrpSub: a subalgebra (which may be an ideal) of a group algebra. Its elements are elements of the group algebra and it remains closely connected to its defining group algebra. By default, ideals of group algebras will be of this type.
The alternative is to create the subalgebra or ideal as an associative algebra given by structure constants (type AlgAss). The required structure constants are easily computable from the group algebra. This is the default Magma uses when creating subalgebras of a group algebra.
Regardless of the type given to a subalgebra or ideal, the inclusion map back into the group algebra is also provided.
cat: Cat Default: {hbox AlgGrpSub}
Create the subalgebra S of the group algebra A generated by the elements defined by L, where L is a list of one or more items of the following types:The constructor returns the subalgebra as an element of the category cat, where cat is either AlgGrpSub (the default) or AlgAss. As well as the subalgebra itself, the constructor returns the inclusion homomorphism f : S -> A.
- (a)
- an element of A;
- (b)
- a set or sequence of elements of A;
- (c)
- a subalgebra of A;
- (d)
- a set or sequence of subalgebras of A;
- (e)
- a sequence of ring elements specifying an element of A, as returned when the ElementToSequence (or Eltseq) function is applied to the element.
cat: Cat Default: {hbox AlgGrpSub}
Create the left ideal of the group algebra A, generated by the elements defined by L, where L is a list as for the sub constructor above.The constructor returns the ideal as an element of the category cat, where cat is either AlgGrpSub (the default) or AlgAss. As well as the ideal itself, the constructor returns the inclusion homomorphism f : I -> A.
cat: Cat Default: {hbox AlgGrpSub}
Create the right ideal of the group algebra A generated by the elements defined by L, where L is a list as for the sub constructor above.The constructor returns the ideal as an element of the category cat, where cat is either AlgGrpSub (the default) or AlgAss. As well as the ideal itself, the constructor returns the inclusion homomorphism f : I -> A.
cat: Cat Default: {hbox AlgGrpSub}
Create the (two-sided) ideal I of the group algebra A generated by the elements defined by L, where L is a list as for the sub constructor above.The constructor returns the ideal as an element of the category cat, where cat is either AlgGrpSub (the default) or AlgAss. As well as the ideal itself, the constructor returns the inclusion homomorphism f : I -> A.
The above operations can also be applied with a subalgebra (type AlgGrpSub) S in place of the group algebra A. In this case the subalgebra and ideal closures are taken in S.
For a right ideal I of the group algebra A construct the right ideal {a * b: b ∈I}.
For a left ideal I of the group algebra A construct the left ideal {b * a: b ∈I}.
Create the quotient of the group algebra A by the ideal of A generated by the elements defined by L, where L is a list as for the sub constructor above.The constructor returns the quotient as an associative algebra. As well as the quotient Q itself, the constructor returns the natural homomorphism f : A -> Q.
The quotient of the algebra A by the ideal closure of its subalgebra S.
We demonstrate how the degrees of the absolutely irreducible characters of a group can be found by computing the Wedderburn decomposition of the group algebra over a suitable finite field. Of course, this is neither the smartest nor the quickest way to get this information. In the example we deal with an extraspecial group of order 33.
> G := ExtraSpecialGroup(3, 1); > Exponent(G); 3We have to choose a finite field that contains the roots of unity that may be required by the absolutely irreducible characters. This is the case for GF(q) if the exponent of G divides q - 1. In our example, q=4 or q=7 are possible choices.
> FG := GroupAlgebra(GF(4), G);
> FG;
Group algebra with vector representation and stored multiplication table
Coefficient ring: Finite field of size 2^2
Group: Permutation group G acting on a set of cardinality 27
Order = 27 = 3^3
(1, 19, 10)(2, 20, 11)(3, 21, 12)(4, 22, 13)(5, 23, 14)(6, 24, 15)(7,
25, 16)(8, 26, 17)(9, 27, 18)
(1, 7, 4)(2, 8, 5)(3, 9, 6)(10, 18, 14)(11, 16, 15)(12, 17, 13)(19, 26,
24)(20, 27, 22)(21, 25, 23)
(1, 3, 2)(4, 6, 5)(7, 9, 8)(10, 12, 11)(13, 15, 14)(16, 18, 17)(19, 21,
20)(22, 24, 23)(25, 27, 26)
> MI := MinimalIdeals(FG);
> #MI;
11
The minimal ideals are the simple Wedderburn components of the group algebra,
corresponding to the absolutely irreducible characters of the group, and
their dimensions are the squares of the character degrees.
> [ Isqrt(Dimension(I)) : I in MI ]; [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3 ]Hence, there are 9 linear characters and two of degree 3 (which we might have concluded immediately from the fact that the derived quotient of G has order 9).