Projective special 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 projective special unitary group of degree $n$ for this quadratic extension, denoted $PSU(n,L)$ (if the extension being referred to is understood), is defined as the quotient of the special unitary group $SU(n,L)$ by its center, which is precisely the set of scalar matrices in $SU(n,L)$.

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 $PSU(n,\mathbb {C} )$ is the quotient of $SU(n,\mathbb {C} )$ by its center, which is a cyclic group of order $n$ comprising those scalar matrices whose scalar entry is a $n^{th}$ root of unity. The group $PSU(n,\mathbb {C} )$ is sometimes also denoted $PSU(n)$.

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 $PSU(n,q)$ (the more standard choice) or $PSU(n,q^{2})$ (a less standard choice).