# Invariant Forms 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 ${\operatorname{Hom}}_G(V,V^*)$, where $V^*$ is the dual space of $V$. The isomorphism associates the form $\beta$ to $\theta\in{\operatorname{Hom}}_G(V,V^*)$, where $\beta(u,v) = \langle v, u\theta \rangle$ and where $\langle v, \varphi \rangle$ denotes the action of $\varphi$ on $v$. If $v_1$, $v_2$, …, $v_n$ is a basis for $V$ with dual basis $\omega_1$, $\omega_2$, …, $\omega_n$, the matrix of $\theta$ with respect to these bases is $J = (\beta(e_i,e_j))$. A linear transformation with matrix $A$ preserves the form if and only if $AJA^{\hbox{tr}} = J$. If the characteristic of the field is not 2, then $J = {1\over2}(J+J^{\hbox{tr}}) + {1\over2}(J-J^{\hbox{tr}})$. 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. ## `InvariantBilinearForms(G): GrpMat -> SeqEnum[AlgMatElt], SeqEnum[AlgMatElt]` 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. ## `Example: reducible (ex-246b09)` In this example the group $G$ is reducible but (up to a scalar multiple) there is a unique $G$-invariant bilinear form. ```magma > F := 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] ] ``` If $G$ acts irreducibly on a vector space $V$ of dimension $n$ over a (finite) field $F$ and if $\theta_0 : V\to V^*$ is a $G$-invariant isomorphism, then $D \to {\operatorname{Hom}}_G(V,V^*) : \theta \mapsto \theta\theta_0$ is an isomorphism of vector spaces, where $D = {\operatorname{End}}_G(V)$. 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 ${\operatorname{GL}}(m,D)$. ## `Example: nonabs (ex-8559f1)` If $G$ acts irreducibly on $V$, the spaces of symmetric and alternating $G$-invariant forms are isomorphic (as vector spaces) to subfields of ${\operatorname{End}}_G(V)$ and therefore their dimensions are either 0 or divide $\dim_F(V)$. ```magma > F := 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 ``` 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 $J = Q + Q^{\hbox{tr}}$. On the other hand, if the characteristic is two and $J$ is alternating, the upper triangular matrices $Q$ such that $J = Q + Q^{\hbox{tr}}$ 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 $J = Q + Q^{\hbox{tr}}$. ## `InvariantQuadraticForms(G): GrpMat -> SeqEnum[AlgMatElt]` A basis for the space of quadratic forms preserved by the irreducible matrix group $G$. ## `Example: invquadform (ex-e24e49)` In the following example the quadratic forms which are invariant under the action of a cyclic group $H$ of order 13 form a vector space of dimension 3 over ${\bf F}_{4}$. ```magma > F := 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], [ 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], [ 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] ] ``` Given a group $G$ which acts on a vector space $V$ over a finite field $F$ with an automorphism $F \to F :a \mapsto \overline{a}$ 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 ${\operatorname{Hom}}_G(V, \overline{V}^*)$, where $\overline{V}^*$ is the semilinear dual of $V$, namely the space of all semilinear maps from $V$ to $F$. If $\theta\in {\operatorname{Hom}}_G(V,\overline{V}^*)$, the corresponding sesquilinear form $\beta$ is defined by $\beta(u,v) = \langle v,u\theta\rangle$ where, as before, $\langle v, \varphi \rangle$ denotes the action of $\varphi$ on $v$. ## `SemilinearDual(M, mu): ModGrp, Map -> ModGrp` The semilinear dual of the $G$-module $M$ with respect to the field automorphism `mu`. ## `InvariantSesquilinearForms(G): GrpMat -> SeqEnum[AlgMatElt]` A basis for the space of *hermitian* forms preserved by the matrix group $G$. ## `Example: sesquiforms (ex-17b876)` Let $F_0$ be the fixed field of the involution. The set $\cal H$ of $G$-invariant hermitian forms is a vector space over $F_0$ and if the characteristic of $F$ is not 2, then ${\operatorname{Hom}}_G(V,\overline{V}^*) \simeq {\cal H}\otimes_{F_0} F$. ```magma > F := 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 ``` ## `Example: hermandalt (ex-9dd62c)` If an irreducible group preserves both a bilinear and a sesquilinear form then it is realisable over a subfield of its base field. Conversely, this observation can be used to construct an example: ```magma > F := 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] ] ``` ## `InvariantFormBases(G): GrpMat -> SeqEnum[AlgMatElt], SeqEnum[AlgMatElt], SeqEnum[AlgMatElt], SeqEnum[AlgMatElt]` This function returns four sequences: bases for the spaces of symmetric, alternating, hermitian and quadratic forms preserved by the matrix group $G$. ## Semi-invariant Forms Given a vector space $V$ over a finite field $F$ and a group $G$ which acts on $V$, a bilinear form $\beta : V\times V \to F$ is *semi-invariant* if for all $g\in G$ there is a scalar $\lambda(g)$ such that $\beta(ug,vg) = \lambda(g)\beta(u,v)$ for all $u,v\in V$. The function $\lambda : G \to F^\times$ is a homomorphism and its kernel contains the derived group of $G$. The *twisted dual* $V^*_\lambda$ of the $G$-module $V$ is the dual space of $V$ with $G$-action given by $\langle v, \varphi g\rangle = \lambda(g)\langle vg^{-1},\varphi\rangle$; thus if $A$ is the matrix of $g$ acting on $V$ the matrix of the action on $V_\lambda^*$ with respect to the dual basis is $\lambda(g)A^{-\hbox{tr}}$. The space of all semi-invariant bilinear forms is isomorphic to ${\operatorname{Hom}}_G(V,V_\lambda^*)$. The isomorphism associates the form $\beta$ to $\theta\in{\operatorname{Hom}}_G(V,V_\lambda^*)$, where $\beta(u,v) = \langle v, u\theta \rangle$. If $\beta$ is a bilinear form with matrix $J$, then the linear transformation $g$ with matrix $A$ preserves the form up to multiplication by $\lambda(g)$ if and only if $AJA^{\hbox{tr}} = \lambda(g)J$. ### `TwistedDual(M, lambda): ModGrp, Map -> ModGrp` The twisted dual of the $G$-module M with respect to the linear character `lambda`. ### `SemiInvariantBilinearForms(G): GrpMat -> SeqEnum` A sequence of triples $\langle L, S, A\rangle$ 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). ### `SemiInvariantQuadraticForms(G): GrpMat -> SeqEnum` A sequence of pairs $\langle L, Q\rangle$ 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). ### `TwistedSemilinearDual(M, lambda, mu): ModGrp, Map, Map -> ModGrp` The twisted semilinear dual of the $G$-module $M$ with respect to the linear character $\lambda$ and the field automorphism $\mu$. ### `SemiInvariantSesquilinearForms(G): GrpMat -> SeqEnum` A sequence of pairs $\langle L, H\rangle$ where $L$ is a sequence of field elements (one for each generator) which define a homomorphism from the matrix group $G$ to the field $F_0$, where the base field of $G$ is a quadratic extension of $F_0$, and $H$ is a basis for the space of hermitian forms preserved by $G$ (up to multiplication by scalars). ### `Example: semiinv (ex-f6ba6b)` In this example $H$ is a normal subgroup of the absolutely irreducible group $N$ an $H$ is irreducible but not absolutely irreducible. ```magma > F := 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], [ 0 x^3 x] [ x 0 x^5] [x^3 x^7 0], [ 0 0 1] [ 0 1 0] [ 1 0 0] ]> ] > SemiInvariantSesquilinearForms(N); [ <[ 1, 2, 1 ], [ [ 0 0 1] [ 0 1 0] [ 1 0 0] ]> ] ```