Unitary group

Definition

Suppose $L$ is a separable quadratic extension of a field $K$ and $\sigma:L \to L$ is the unique automorphism of $L$ that fixes $K$ pointwise. The unitary group of degree $n$ for this quadratic extension, denoted $U(n,L)$ (if the extension being referred to is understood) is defined as the subgroup of the general linear group $GL(n,L)$ comprising those matrices on which the transpose-inverse map gives the same result as the entry-wise application of $\sigma$.

$U(n,L) = \{ A \in GL(n,L) \mid \sigma(A) = (A^t)^{-1} \}$

Here, $\sigma(A)$ is the matrix obtained by applying $\sigma$ to each of the entries of $A$.

For the real and complex numbers

The most typical usage of the term unitary group is in the context where $K$ is the field of real numbers, $L$ is the field of complex numbers, and the automorphism $\sigma$ is complex conjugation. In this case, the group $U(n,\mathbb{C})$ is the subgroup of the general linear group $GL(n,\mathbb{C})$ comprising those matrices whose complex conjugate equals the transpose-inverse. When it's understood that we are working over the complex numbers, this group is sometimes just denoted $U(n)$.

For a finite field

If $K$ is the (unique up to isomorphism) finite field of size a prime power $q$, there is a unique quadratic extension $L$ of $K$, and this extension is separable. The extension field is the finite field (unique up to isomorphism) of order $q^2$. The automorphism $\sigma$ is the map $x \mapsto x^q$. The unitary group for this extension may be denoted $U(n,q)$ (the more standard choice) or $U(n,q^2)$ (a less standard choice). Note that due to the multiplicity of notation, it is important to understand from context what exactly is meant.