In mathematics, a complex representation is a representation of a group (or

that ''That'' is an English language word used for several grammatical purposes. These include use as an adjective, conjunction, pronoun, adverb, and intensifier; it has distance from the speaker, as opposed to words like ''this''. The word did not ori ...

of Lie algebra) 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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

nor pseudoreal (quaternionic). In other words, the

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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 In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...

of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the

antifundamental representation In mathematics differential geometry, an antifundamental representation of a Lie group is the complex conjugate of the fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreduci ...

References

*{{Fulton-Harris Representation theory of groups