Background Theory on Forms

In this section, we give a very brief overview on the theory of sesquilinear and quadratic forms. The reader can find more in the texts: Cameron , Taylor , Aschbacher , or Kleidman and Liebeck .

Sesquilinear forms, dualities, and polarities
A sesquilinear form Form on a vector space V over a field F, is a map f from V\times V to F which is linear in the first coordinate, but semilinearSemilinear in the second coordinate; that is, there is a field automorphism \alpha (the companion automorphism Companion Automorphism of f) such that f(v,\lambda w)=\lambda^\alpha f(v,w) for all v,w\in V and \lambda\in F. If \alpha is the identity, then f is bilinear Form. Two vectors v and w are orthogonal Orthogonal (w.r.t. f) if f(v,w) = 0. The radical Radical of f is the subspace consisting of vectors which are orthogonal to every vector, and we say that f is non-degenerate Form if its radical is trivial (and degenerate otherwise). A duality Duality \delta of a projective space \mathcal{P} is an incidence reversing permutation of the subspaces of \mathcal{P}, and a polarity Polarity of \mathcal{P} is a duality of order 2. An example of such arises from a non-degenerate sesquilinear form; given a subspace W, we let W^\perp be the set of points which are orthogonal with every element of W. We say that a subspace W is totally isotropic Totally Isotropic with respect to a polarity if W contains or is contained in W^\perp.

The Birkhoff-von Neumann TheoremBirkhoff-von Neumann Theorem states that every duality of the projective space PG(n,q) arises from a non-degenerate sesquilinear form (up to a scalar). Such a duality is a polarity if it is reflexive Form, i.e., f(v,w)=0 implies f(w,v)=0.

Now a sesquilinear form f is hermitian Form if f(v,w)=f(w,v)^\alpha holds where \alpha is the companion automorphism of f and \alpha has order 2. But if \alpha is trivial then f is symmetric Form. If f inturn satisfies f(v,v)=0 (for all v) then f is alternating Form. It is a well-known theorem of polar geometry that a non-degenerate reflexive sesquilinear form is either alternating, symmetric, or similar to an hermitian form. The associated polarity is called symplectic Polarity, orthogonal Polarity, and unitary Polarity respectively (though there are some other conventions for the characteristic 2 case).

Example
Let M be an invertible 4-dimensional square matrix over F and consider the following map on pairs of elements of the 4-dimensional vector space V over F: f(v, w) = v M w^T. Then f is a sesquilinear form of V and M is the Gram matrix of f.
Quadratic forms
We have seen that a polar space can arise from a reflexive sesquilinear form, but there are other polar spaces which do not arise this way, but instead have an associated quadratic form. A map Q from a vector space V to a field F is a quadratic form Form if it satisfies Q(\lambda v) = \lambda^2Q(v) for all v\in V and \lambda\in F. We say that a subspace W is totally singular Totally Singular if the restriction of Q to W is trivial. Note that a subspace is totally isotropic (with respect to the associated polarity) if it is totally singular, but the converse is not always true.

Example
Let M be an invertible 4-dimensional square matrix over F and consider the following map on elements of the 4-dimensional vector space V over F: f(v) = v M v^T. Then f is a quadratic form of V and M is the Gram matrix of f.

Given a quadratic form Q, there is an associated sesquilinear form f (which may not be reflexive) defined as follows f(v,w)=Q(v+w)-Q(v)-Q(w). For characteristic not 2, the quadratic form and its associated sesquilinear form f determine one another, as 2 Q(v)= f(v,v) (for all v).
Morphisms of forms
An isometry Isometry from a formed space (V,f) to a formed space (W,f') is a bijection \phi such that for all v,w in V we have f(v,w) = f'(\phi(v), \phi(w)). The weaker notions of similarity Similarity and semi-similarity Semi-similarity are also important in polar geometry. If there exists a scalar \lambda such that for all v,w in V we have f(v,w) = \lambda f'(\phi(v), \phi(w)) then we say that \phi is a similarity. If we also have a fixed field automorphism \alpha such that f(v,w)=\lambda f'(\phi(v), \phi(w))^\alpha, then \phi is a semi-similarity. Naturally, we say that the formed spaces (V,f) and (W,f') are isometric Isometric (resp. similar Similar) if there exists an isometry (resp. similarity) between them. Every non-degenerate reflexive sesquilinear form is alternating, symmetric, or similar to an hermitian form. Thus, up to similarity, the non-degenerate polar spaces come in five flavours: symplectic, unitary, orthogonal-ellipticPolarity, orthogonal-hyperbolicPolarity, and orthogonal-parabolicPolarity. In the case of the orthogonal spaces, they are distinguished by their Witt IndexWitt Index (the common dimension of their maximal totally singular/isotropic subspaces).
An important convention
In `Forms`, we have stipulated a convention on the creation of forms so as to cause as little confusion as possible. The hermitian forms will simply be those with the Frobenius Automorphism is the companion automorphism. We should also caution the user on what information is "enough" to specify a form as problems can arise in even characteristic.

Example
Let F be a finite field of square order and let M be the following 4\times 4 matrix over F:

0
1
0
0
1
0
0
0
0
0
0
1
0
0
1
0

Let \alpha be the unique automorphism of F of order 2. Then the form (u, v) := uM(v^T)^\alpha defines a non-degenerate hermitian sesquilinear form. If F has odd characteristic, then the form (u, v) := uMv^T defines a non-degenerate orthogonal form, but if F has even characteristic, then this form is both: a symplectic bilinear form, and the associated bilinear form arising from a quadratic form. In the latter, case we see that the bilinear form does not define the quadratic form, but rather that the quadratic form is necessary in order to define the polar geometry.
Canonical forms
Every nondegenerate polar space has a direct decomposition into a sum L_1\perp L_2\perp\cdots L_n\perp U where each of the L_i are hyperbolic lines and U is an anisotropic subspace of dimension at most 2. Thus if the given polar space is defined by a sesquilinear form f, then there is an isometric polar space defined by a Gram Matrix of the form

U
0
1
\epsilon
0
0
1
\epsilon
0
*
*
0
1
\epsilon
0

were the top left hand corner represents the anisotropic part, and there are zeros everywhere else. The value of \epsilon is -1 if the form is alternating, otherwise it is 1.