Let \(V\) be an \(n\)-dimensional vector space over a finite field \(K=\mathbb{F}_q\). A classical form on \(V\) is either a \(\sigma\)-sesquilinear form \(\beta: V \times V \to K\) (for some field automorphism \(\sigma \in {\rm Aut}(K)\)) or a quadratic form \(Q: V \to K\).
A map \(\beta: V \times V \to K\) is a \(\sigma\)-sesquilinear form if it is additive in both arguments and satisfies
\[\beta(\lambda u, \mu v) = \lambda \mu^\sigma \beta(u, v)\]
for all \(u, v \in V\) and \(\lambda, \mu \in K\). When \(\sigma = 1\) (the identity automorphism), \(\beta\) is simply called bilinear. The form is symmetric if \(\beta(u, v) = \beta(v, u)\) for all \(u, v\).
A map \(Q: V \to K\) is a quadratic form if \(Q(\lambda v) = \lambda^2 Q(v)\) for all \(v \in V\) and \(\lambda \in K\), and the associated polar form
\[\beta(u, v) := Q(u + v) - Q(u) - Q(v)\]
is a symmetric bilinear form. Note that \(Q\) and \(\beta\) uniquely determine each other if \({\rm char}(K) \neq 2\).
In this package, classical forms on \(V\) are realized as (Gram) matrices.
For a \(\sigma\)-sesquilinear form \(\beta\), the Gram matrix \(B\) satisfies
\[\beta(u, v) = u B v^{\sigma {\rm T}}\]
for all column vectors \(u, v \in V\). For a quadratic form \(Q\), the Gram matrix \(A\) satisfies
\[Q(v) = v A v^{\rm T}\]
for all column vectors \(v \in V\).
For relevant classical forms, the package fixes a standard choice of Gram matrices in Subsection 2.2-6.
A sesquilinear form \(\beta\) is non-degenerate if its radical
\[{\rm Rad}(\beta) = \{v \in V \mid \forall u \in V: \beta(u, v) = 0 \}\]
is trivial. A quadratic form is non-degenerate if its polar form is non-degenerate.
Let \(g \in {\rm GL}(V)\). Then \(g\) is an isometry of \(\beta\) if
\[\forall u, v \in V: \beta(ug, vg) = \beta(u, v)\]
and an isometry of \(Q\) if \(Q(vg) = Q(v)\) for all \(v\). If instead equality holds up to a non-zero scalar \(\lambda \in K^\times\), \(g\) is a similarity.
Let \(\beta\) be a \(\sigma\)-sesquilinear form on \(V\). Then following [BHR13, Theorem 1.5.13], we focus on the following four classes of such forms, and the corresponding isometry groups:
\(\beta = 0\).
\(\sigma = 1\), \(\beta(u, v) = -\beta(v, u)\) for all \(u, v\), and \(\beta(v, v) = 0\) for all \(v\). Then \(\beta\) is called alternating or symplectic.
\(\sigma^2 = 1 \neq \sigma\) and \(\beta(v, u) = \beta(u, v)^\sigma\) for all \(u, v\). Then \(\beta\) is called \(\sigma\)-Hermitian or unitary.
\(\sigma = 1\) and \(\beta(v, u) = \beta(u, v)\) for all \(u, v\). Then \(\beta\) is symmetric bilinear.
In characteristic 2 the symplectic and symmetric cases overlap. Otherwise all cases are mutually exclusive.
Classical groups in this package are realized as matrix groups over finite fields that preserve a specific form on a vector space. These forms are represented explicitly by their Gram matrices (with respect to a standard basis), and all constructions in the package are carried out relative to fixed choices of such matrices.
Linear groups preserve the zero form. We will denote the isometry group by \({\rm GL}_n(q)\).
Symplectic groups preserve non-degenerate symplectic forms. We will use \({\rm antidiag}(1,\dots,1,-1,\dots,-1)\) as our standard symplectic form matrix and denote the isometry group by \({\rm Sp}_n(q)\). Note that \(n\) must be even.
Unitary groups preserve non-degenerate unitary forms, where \(\sigma\) is a field automorphism of order 2. So they are only defined over \(\mathbb{F}_{q^2}\) with \(\sigma: x\mapsto x^q\). We will use \({\rm antidiag}(1,\dots,1)\) as our standard unitary form matrix and denote the isometry group by \({\rm GU}_n(q)\).
If \(Q\) is a quadratic form on \(\mathbb{F}_q^n\) with \(q\) even and \(n\) odd, then the isometry group of \(Q\) is isomorphic to a symplectic group of dimension \(n-1\).
For this reason, we restrict attention to orthogonal groups of odd dimension over fields of odd characteristic. It is sufficient to define the polar form of \(Q\). We will use \({\rm antidiag}(1,\dots,1,\frac{1}{2},1,\dots,1)\) as our standard non-degenerate symmetric bilinear form matrix and denote the isometry group by \({\rm GO}_n(q)\). In odd dimension, there are two isometry classes of non-degenerate symmetric bilinear forms, distinguished by whether the determinant of the form matrix is a square or a non-square in \(\mathbb{F}_q^\times\). These two isometry classes, however, belong to the same similarity class.
In even dimension, non-degenerate quadratic forms fall into two distinct isometry classes, which also correspond to two different similarity classes. Let \(\beta\) denote the polar form associated with a quadratic form \(Q\). Over a field of odd characteristic, \(\beta\) and \(Q\) uniquely determine each other.
Assume that \(q\) is odd, let \(\beta\) be a non-degenerate symmetric bilinear form on \(\mathbb{F}_q^n\) with \(n\) even. Then \(\beta\) is of plus-type if it is isometric to our standard form matrix \({\rm antidiag}(1,\dots,1)\), and otherwise of minus-type.
In even characteristic, a non-degenerate quadratic form \(Q\) in even dimension is of plus-type if it is isometric to our standard form matrix \({\rm antidiag}(1,\dots,1,0,\dots,0)\) and of minus-type otherwise. We denote the isometry group of a standard quadratic or bilinear form of plus-type by \({\rm GO}_n^+(q)\) and the isometry group of a standard form of minus-type (see 2.2-6) by \({\rm GO}_n^-(q)\).
Let \(G\) be the isometry group of one of the following forms on \(\mathbb{F}_q^n\): the zero form, a unitary form, a symplectic form, a symmetric bilinear form, or a quadratic form, as described above.
The following table lists the corresponding form matrices preserved by \(G\), which we adopt as our standard form matrices.
In odd characteristic, we specify quadratic forms via the matrix of their associated polar form rather than the quadratic form itself. We use the notation \(\nu=Z(q^2)\) and \(\xi=Z(q)\).
| Case | conditions | form type | isom. grp. | form |
| \({\bf L}\) | — | zero | \({\rm GL}_n(q)\) | \(0_{n \times n}\) |
| \({\bf U}\) | — | hermitian | \({\rm GU}_n(q)\) | \({\rm antidiag}(1,\dots,1)\) |
| \({\bf S}\) | \(n\) even | alternating | \({\rm Sp}_n(q)\) | \({\rm antidiag}(1,\dots,1,-1,\dots,-1)\) |
| \({\bf O}\) | \(q\) odd, \(n\) odd | symmetric | \({\rm GO}_n(q)\) | \({\rm antidiag}(1,\dots,1,\frac{1}{2},1,\dots,1)\) |
| \({\bf O}^+\) | \(q\) odd, \(n\) even | symmetric | \({\rm GO}^+_n(q)\) | \({\rm antidiag}(1,\dots,1)\) |
| \({\bf O}^-\) | \(q\) odd, \(n\) even \(n>2\) |
symmetric | \({\rm GO}^-_n(q)\) | \({\rm antidiag}(1,\dots,1,-\nu-\nu^q,-\nu-\nu^q,1,\dots,1)\) \(-2E_{m,m}-2\xi E_{m+1,m+1}\) |
| \({\bf O}^+\) | \(q\) even, \(n\) even | quadratic | \({\rm GO}^+_n(q)\) | \({\rm antidiag}(1,\dots,1,0,\dots,0)\) |
| \({\bf O}^-\) | \(q\) even, \(n\) even \(n>2\) |
quadratic | \({\rm GO}^-_n(q)\) | \({\rm antidiag}(1,\dots,1,-\nu-\nu^q,0,0,\dots,0)\) \(-E_{m,m}-\xi E_{m+1,m+1}\) |
These are exactly the forms preserved by the groups returned by GAP's constructors SL, SU, Sp, and Omega.
Aschbacher's theorem is the organising principle behind the entire package. It says that maximal subgroups of finite classical groups fall into nine broad families:
Let \(G\) be a quasisimple classical group acting naturally on a vector space \(V\) of dimension \(n \geq 2\) over \(\mathbb{F}_q\) and one of \({\rm SL}_n(q)\), \({\rm SU}_n(q)\), \({\rm Sp}_n(q)\) or \(\Omega^\varepsilon_n(q)\).
Let \(H\) be a maximal subgroup of \(G\). Then one of the following holds:
\(H\) is a geometric subgroup and belongs to one of the geometric Aschbacher classes \({\cal C}_1,\dots,{\cal C}_8\). These classes consist roughly of groups that preserve some kind of geometric structure on \(V\).
\(H\) belongs to the non-geometric class \({\cal S}\) (sometimes called \({\cal C}_9\)). In this case,
\(H\) is almost simple modulo scalars,
\(H^\infty\) acts absolutely irreducibly,
there does not exist a \(g \in {\rm GL}_n(q)\) such that \((H^\infty)^g\) is defined over a proper subfield of \(\mathbb{F}_q\),
\(H^\infty\) does not preserve a non-zero classical form on \(V\) other than those already preserved by \(G\).
The following table gives a rough description of the eight geometric Aschbacher classes, based on [KL90, Table 1.2.A].
| Class | Type | Rough description |
| \({\cal C}_1\) | reducible groups | stabilisers of totally singular or non-singular subspaces |
| \({\cal C}_2\) | imprimitive groups | stabilisers of decompositions \(V=\bigoplus_{i=1}^t V_i, {\rm dim}(V_i)=n/t\) |
| \({\cal C}_3\) | semilinear groups | stabilisers of extension fields of \(\mathbb{F}_q\) of prime index dividing \(n\) |
| \({\cal C}_4\) | tensor product groups | stabilisers of tensor product decompositions \(V=V_1 \otimes V_2\) |
| \({\cal C}_5\) | subfield groups | stabilisers of subfields of \(\mathbb{F}_q\) of prime index |
| \({\cal C}_6\) | extraspecial normaliser groups |
normalisers of symplectic-type or extraspecial groups in absolutely irreducible representations |
| \({\cal C}_7\) | tensor induced groups | stabilisers of decompositions \(V=\bigotimes_{i=1}^t V_i, {\rm dim}(V_i)=a, n=a^t\) |
| \({\cal C}_8\) | classical normaliser groups | groups of similarities of non-degenerate classical forms |
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