# Reflexive Forms

Let $V$ be a vector space of dimension $n$ over a field $K$. If $\sigma$ is an automorphism of $K$, a $\sigma$-sesquilinear form on the vector space $V$ over $K$ is a map $\beta : V\times V\to K$ such that

$$
\begin{aligned}\beta(u_1+u_2,v) &= \beta(u_1,v)+\beta(u_2,v),\\
  \beta(u,v_1+v_2) &= \beta(u,v_1)+\beta(u,v_2)\\ \text{and} &\\
  \beta(au,bv) &= a\sigma(b)\beta(u,v).\end{aligned}
$$

for all $u,u_1,u_2,v,v_1,v_2\in V$ and all $a,b\in K$. If $\sigma$ is the identity, the form is said to be *bilinear*.

A linear transformation $g$ of $V$ is an *isometry* if $g$ preserves $\beta$; it is a *similarity* if it preserves $\beta$ up to a non-zero scalar multiple.

A $\sigma$-sesquilinear form $\beta$ is *reflexive* if for all $u,v\in V$, $\beta(u,v) = 0$ implies $\beta(v,u) = 0$. Any non-zero multiple of a reflexive form is again reflexive with the same group of isometries. By a theorem of Brauer [[Brauer, 1936](../../references.md#cite-brauer-1936)] (but sometimes referred to as the Birkhoff–von Neumann theorem), up to a non-zero scalar multiple, there are three types of *non-degenerate* reflexive forms:

**Alternating.** In this case $\sigma$ is the identity, $\beta(u,u) = 0$ for all $u\in V$ and consequently $\beta(u,v) = -\beta(v,u)$ for all $u,v \in V$. The group of isometries is a *symplectic* group.

**Symmetric.** In this case $\sigma$ is the identity and $\beta(u,v) = \beta(v,u)$ for all $u,v \in V$. If the characteristic of $K$ is not two, the group of isometries is an *orthogonal* group. If the characteristic is two, the form is either alternating or pseudo-alternating (see below).

**Hermitian.** In this case $\sigma$ is an automorphism of order two and $\beta(u,v) = \sigma\beta(v,u)$ for all $u,v \in V$. The group of isometries is a *unitary* group.

If $V$ is a vector space $V$ and if $\beta$ is a reflexive form defined on $V$, the partially ordered set of totally isotropic subspaces with respect to $\beta$ is often referred to as a . Similarly, there are polar spaces associated with quadratic forms (see Section [Quadratic Forms](#subsectquadratic)). But throughout this chapter by polar space we shall simply mean a vector space furnished with either a reflexive $\sigma$-sesquilinear form or a quadratic form. See [[Buekenhout, 1995](../../references.md#cite-buekenhout95), Chap. 2] for an account of polar spaces in a more general context.

Let $K_0$ be the fixed field of $\sigma$. Multiplying an alternating, symmetric or hermitian form by a non-zero element of $K_0$ leaves the type of the form unchanged.

However, multiplying an hermitian form by a non-zero element of $K$ produces a sesquilinear form $\xi$ and an element $\varepsilon\in K$ such that for all $u,v \in V$, $\xi(v,u) = \varepsilon\sigma\xi(u,v)$, where $\varepsilon\sigma(\varepsilon) = 1$. In this case $\xi$ is said to be $\varepsilon$-hermitian.

**Skew-hermitian.** A reflexive $\sigma$-sesquilinear form is skew-hermitian if the order of $\sigma$ is two and $\xi(v,u) = -\sigma\xi(v,u)$ for all $u,v\in V$. If $\beta$ is hermitian and if $d\in K$ is chosen so that $d\ne\sigma(d)$, then $e = d-\sigma(d)$ satisfies $\sigma(e) = -e$. Thus $\xi(u,v) = e\beta(u,v)$ is skew-hermitian. The group of isometries of $\xi$ coincides with the group of isometries of $\beta$ and it is therefore a *unitary* group.

In the case of fields of characteristic two there is no distinction between hermitian and skew-hermitian forms and moreover, every alternating form is symmetric.

**Pseudo-alternating.** A symmetric form (in characteristic two) which is not alternating is said to be *pseudo-alternating*.

The three types of forms—alternating, symmetric and hermitian—correspond to the three types of classical groups of isometries: symplectic, orthogonal and unitary. But this is not quite the whole story because it does not include orthogonal groups over fields of characteristic two. In order to include these groups it is necessary to consider quadratic forms in addition to symmetric bilinear forms.

## Quadratic Forms

If $\beta$ is a bilinear form, a *quadratic form* with *polar form* $\beta$ is a function $Q : V \to K$ such that

$$
\begin{aligned}Q(av) &= a^2Q(v)\\ \text{and} &\\
  \beta(u,v) &= Q(u+v) - Q(u) - Q(v)\end{aligned}
$$

for all $u,v\in V$ and all $a\in K$. We have $\beta(v,v) = 2Q(v)$ and therefore, if the characteristic of $K$ is not two, $\beta$ determines $Q$.

We extend the notion of polar space to include vector spaces $V$ with an associated quadratic form $Q$. The pair $(V,Q)$ is an orthogonal geometry and $V$ is a *quadratic* space.