Complex representation

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title: "Complex representation" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["representation-theory-of-groups"] topic_path: "general/representation-theory-of-groups" source: "https://en.wikipedia.org/wiki/Complex_representation" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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

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