Unitary group

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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 ambiguity of notation, it is important to understand from context what exactly is meant.