In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a complex representation is a
representation of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
(or
that
''That'' is an English language word used for several grammar, grammatical purposes. These include use as an adjective, conjunction (grammar), conjunction, pronoun, adverb and intensifier; it has distance from the speaker, as opposed to words li ...
of
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
) on a complex vector space. Sometimes (for example in physics), the term complex representation is reserved for a representation on a complex vector space that is neither
real nor
pseudoreal (quaternionic). In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation. For compact groups, the
Frobenius-Schur indicator can be used to tell whether a representation is real, complex, or pseudo-real.
For example, the N-dimensional
fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the
antifundamental representation.
References
*{{Fulton-Harris
Representation theory of groups