Given a group G which acts on a vector space V over a finite field F, the space of all G-invariant bilinear forms is isomorphic to HomG(V, V * ), where V * is the dual space of V. The isomorphism associates the form β to θ∈HomG(V, V * ), where β(u, v) = < v, uθ > and where < v, varphi > denotes the action of varphi on v. If v1, v2, ..., vn is a basis for V with dual basis ω1, ω2, ..., ωn, the matrix of θ with respect to these bases is J = (β(ei, ej)).
A linear transformation with matrix A preserves the form if and only if AJA^(tr) = J.
If the characteristic of the field is not 2, then J = (1/2)(J + J^(tr)) + (1/2)(J - J^(tr)). Therefore, in this case, every G-invariant form is the sum of a G-invariant symmetric form and a G-invariant alternating form. If the characteristic of the field is 2, every alternating form is symmetric. Thus in this case the space of G-invariant alternating forms is a subspace of the space of G-invariant symmetric forms. If the G-module is irreducible these two spaces coincide.
If G acts irreducibly on a vector space V of dimension n over a (finite) field F and if θ0 : V to V * is a G-invariant isomorphism, then D to HomG(V, V * ) : θ |-> θθ0 is an isomorphism of vector spaces, where D = EndG(V). The algebra D is a division ring and hence a field (since F is finite). Thus V becomes a vector space of dimension m over D, where n = m|D:F| and G is isomorphic to a subgroup of GL(m, D).
If the characteristic of the field is not two and if J is a symmetric bilinear form there is a unique upper triangular matrix Q such that J = Q + Q^(tr).
On the other hand, if the characteristic is two and J is alternating, the upper triangular matrices Q such that J = Q + Q^(tr) form an affine space of dimension dim V.
Suppose that the characteristic is two. If G preserves a symmetric bilinear form which is not alternating, then G is reducible. Conversely, if G is irreducible and if J is the matrix of a symmetric form preserved by G, then the form must be alternating and there is a unique G-invariant quadratic form Q such that J = Q + Q^(tr).
Given a group G which acts on a vector space V over a finite field F with an automorphism F to F :a |-> /line(a) of order 2, the space of G-invariant sesquilinear forms is isomorphic to the space of G-invariant semilinear maps from V to V * ; equivalently it is isomorphic to HomG(V, /line(V) * ), where /line(V) * is the semilinear dual of V, namely the space of all semilinear maps from V to F.
If θ∈HomG(V, /line(V) * ), the corresponding sesquilinear form β is defined by β(u, v) = < v, uθ > where, as before, < v, varphi > denotes the action of varphi on v.
Given a matrix group G this function returns two sequences: a basis for the space of G-invariant symmetric forms and a basis for the space of G-invariant alternating forms.
> F<x> := GF(25);
> G := MatrixGroup< 4, F |
> [ 1, 0, 0, 0, 0, 1, 0, 0, 0, x^14, 1, 0, 0, 0, 0, 1 ],
> [ 3, x^23, x^20, x^10, 2, 3, 0, x^13, 4, x^10, x^13, x^23,
> x^5, x^11, x, x^17 ] >;
> IsIrreducible(G);
false
> InvariantBilinearForms(G);
[]
[
[ 0 0 0 1]
[ 0 0 1 0]
[ 0 4 0 0]
[ 4 0 0 0]
]
> F<a> := GF(25); > G := MatrixGroup< 4, F | > [ a^10, a^21, a^4, 4, > a^16, 4, a^9, a^8, > a^20, 4, 4, a^13, > 0, a^2, a^11, a ] >; > IsIrreducible(G), #G; true 626 > sym, alt := InvariantBilinearForms(G); > #sym,#alt; 2 2
A basis for the space of quadratic forms preserved by the irreducible matrix group G.
> F<z> := GF(4);
> H := MatrixGroup<6,F |
> [ z, 0, z^2, z, z, 1,
> 1, z, 0, z, z, z,
> 0, z^2, z, 1, z^2, z^2,
> z, 1, z, 1, 1, 0,
> 1, z^2, z, z, 0, 1,
> 1, 0, 1, 0, z^2, 1 ] >;
>
> InvariantQuadraticForms(H);
[
[ 1 1 0 1 0 0]
[ 0 0 1 z^2 z^2 z]
[ 0 0 1 0 z^2 z]
[ 0 0 0 0 0 z]
[ 0 0 0 0 z^2 z]
[ 0 0 0 0 0 z^2],
medbreak
[ 1 0 1 z^2 1 0]
[ 0 z 1 1 1 z]
[ 0 0 z 0 1 0]
[ 0 0 0 z^2 z^2 z^2]
[ 0 0 0 0 z^2 1]
[ 0 0 0 0 0 1],
medbreak
[ 0 0 0 0 0 1]
[ 0 0 0 0 1 0]
[ 0 0 1 1 0 0]
[ 0 0 0 z 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
]
The semilinear dual of the G-module M with respect to the field automorphism mu.
A basis for the space of hermitian forms preserved by the matrix group G.
> F<x> := GF(5,2); > mu := hom< F->F | x :-> x^5 >; > H := MatrixGroup< 5, F | > [ 0, x^3, 0, 1, x^9, x^8, 1, 0, x^11, x^7, x^20, x^16, 1, > x^11, x^3, x^21, 4, 1, x^3, x^23, x^4, x^3, x, x^3, 2 ] >; > M := GModule(H); > D := SemilinearDual(M,mu); > E := AHom(M,D); > Dimension(E); 5 > herm := InvariantSesquilinearForms(H); > #herm; 5
> F<x> := GF(81);
> H := MatrixGroup< 4, F | [ChangeRing(g,F) : g in Generators(Sp(4,9))]>;
> InvariantBilinearForms(H);
[]
[
[ 0 0 0 1]
[ 0 0 1 0]
[ 0 2 0 0]
[ 2 0 0 0]
]
> InvariantSesquilinearForms(H);
[
[ 0 0 0 x^45]
[ 0 0 x^45 0]
[ 0 x^5 0 0]
[ x^5 0 0 0]
]
This function returns four sequences: bases for the spaces of symmetric, alternating, hermitian and quadratic forms preserved by the matrix group G.
Given a vector space V over a finite field F and a group G which acts on V, a bilinear form β : V x V to F is semi-invariant if for all g∈G there is a scalar λ(g) such that β(ug, vg) = λ(g)β(u, v) for all u, v∈V. The function λ : G to F x is a homomorphism and its kernel contains the derived group of G. The twisted dual V * λ of the G-module V is the dual space of V with G-action given by < v, varphi g > = λ(g)< vg - 1, varphi >; thus if A is the matrix of g acting on V the matrix of the action on Vλ * with respect to the dual basis is λ(g)A^(-tr).
The space of all semi-invariant bilinear forms is isomorphic to HomG(V, Vλ * ). The isomorphism associates the form β to θ∈HomG(V, Vλ * ), where β(u, v) = < v, uθ >.
If β is a bilinear form with matrix J, then the linear transformation g with matrix A preserves the form up to multiplication by λ(g) if and only if AJA^(tr) = λ(g)J.
The twisted dual of the G-module M with respect to the linear character lambda.
A sequence of triples < L, S, A > where L is a sequence of field elements (one for each generator) which define a homomorphism from the matrix group G to its base field and S and A are bases for the spaces of symmetric and alternating forms preserved by G (up to multiplication by scalars).
A sequence of pairs < L, Q > where L is a sequence of field elements (one for each generator) which define a homomorphism from the matrix group G to its base field and Q is a basis for the space of quadratic forms preserved by G (up to multiplication by scalars).
The twisted semilinear dual of the G-module M with respect to the linear character λ and the field automorphism μ.
A sequence of pairs < L, H > where L is a sequence of field elements (one for each generator) which define a homomorphism from the matrix group G to the field F0, where the base field of G is a quadratic extension of F0, and H is a basis for the space of hermitian forms preserved by G (up to multiplication by scalars).
> F<x> := GF(3,2);
> H := MatrixGroup<3,F|
> [x^2,x^7,x^3, x,0,1, x^3,x^6,2],
> [x^3, 0, 0, 0, x^3, 0, 0, 0, x^3 ] >;
> N := MatrixGroup<3,F|H.1,H.2,[x^5,x^5,2, 0,x^2,x^6, x^7,x^7,2]>;
> IsNormal(N,H);
true
> IsIrreducible(H), IsAbsolutelyIrreducible(H);
true false
> IsIrreducible(N), IsAbsolutelyIrreducible(N);
true true
> SemiInvariantSesquilinearForms(H);
[
<[ 1, 2 ],
[
[ 1 x x]
[x^3 0 x^3]
[x^3 x 1],
medbreak
[ 0 x^3 x]
[ x 0 x^5]
[x^3 x^7 0],
medbreak
[ 0 0 1]
[ 0 1 0]
[ 1 0 0]
]>
]
> SemiInvariantSesquilinearForms(N);
[
<[ 1, 2, 1 ],
[
[ 0 0 1]
[ 0 1 0]
[ 1 0 0]
]>
]