Complex representation
Encyclopedia
The term complex representation has slightly different meanings in mathematics and physics.
In mathematics
, a complex representation is a group representation
of a group (or Lie algebra) on a complex vector space.
In physics
, a complex representation is a group representation
of a group (or Lie algebra) on a complex vector space that is neither real
nor pseudoreal. 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
.
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a complex representation is a group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of a group (or Lie algebra) on a complex vector space.
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a complex representation is a group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of a group (or Lie algebra) on a complex vector space that is neither real
Real representation
In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant mapj\colon V\to V\,which...
nor pseudoreal. In other words, the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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
Frobenius-Schur indicator
In mathematics the Schur indicator, named after Issai Schur, or Frobenius–Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has...
can be used to tell whether a representation is real, complex, or pseudo-real.
For example, the N-dimensional fundamental representation
Fundamental representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...
of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the antifundamental representation
Antifundamental representation
In mathematics, an antifundamental representation is the complex conjugate of the fundamental representation, although the distinction between the fundamental and the antifundamental representation is a matter of convention. However, these two are often non-equivalent, because each of them is a...
.