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### 3. 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 [C00], Taylor [T92], Aschbacher [A00], or Kleidman and Liebeck [KL90].

#### 3.1 Sesquilinear forms, dualities, and polarities

A sesquilinear form on a vector space V over a field F, is a map f from Vtimes V to F which is linear in the first coordinate, but semilinear in the second coordinate; that is, there is a field automorphism alpha (the companion automorphism of f) such that f(v,lambda w)=lambda^alpha f(v,w) for all v,win V and lambdain F. If alpha is the identity, then f is bilinear. Two vectors v and w are orthogonal (w.r.t. f) if f(v,w) = 0. The radical of f is the subspace consisting of vectors which are orthogonal to every vector, and we say that f is non-degenerate if its radical is trivial (and degenerate otherwise). A duality delta of a projective space mathcalP is an incidence reversing permutation of the subspaces of mathcalP, and a polarity of mathcalP 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 with respect to a polarity if W contains or is contained in W^perp. The Birkhoff-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, i.e., f(v,w)=0 implies f(w,v)=0. Now a sesquilinear form f is hermitian 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. If f inturn satisfies f(v,v)=0 (for all v) then f is alternating. 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, orthogonal, and unitary respectively (though there are some other conventions for the characteristic 2 case).

##### 3.1-1 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.

#### 3.2 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 if it satisfies Q(lambda v) = lambda^2Q(v) for all vin V and lambdain F. We say that a subspace W is 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.

##### 3.2-1 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).

#### 3.3 Morphisms of forms

An 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 and 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 (resp. 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-elliptic, orthogonal-hyperbolic, and orthogonal-parabolic. In the case of the orthogonal spaces, they are distinguished by their Witt Index (the common dimension of their maximal totally singular/isotropic subspaces).

#### 3.4 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.

##### 3.4-1 Example

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

1. a symplectic bilinear form, and

2. 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.

#### 3.5 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.

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