Special 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 special unitary group of degree n for this quadratic extension, denoted SU(n,L) (if the extension being referred to is understood) is defined as the subgroup of the special linear group SL(n,L) comprising those matrices on which the transpose-inverse map gives the same result as the entry-wise application of \sigma.

SU(n,L) = \{ A \in SL(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.

Alternatively, we can define it as the intersection of the unitary group and the special linear group, both viewed as subgroups of the general linear group:

SU(n,L) = U(n,L) \cap SL(n,L)

For the real and complex numbers

The most typical usage of the term special 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 SU(n,\mathbb{C}) is the subgroup of the special linear group SL(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 SU(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 special unitary group for this extension may be denoted SU(n,q) (the more standard choice) or SU(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.

Note that, if we denote this group by SU(n,q), then, somewhat confusingly:

SU(n,q) = U(n,q) \cap SL(n,q^2)