Real representation
Encyclopedia
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 map
which satisfies
The two viewpoints are equivalent because if U is a real vector space acted on by a group G (say), then V = U⊗C is a representation on a complex vector space with an antilinear equivariant map given by complex conjugation. Conversely, if V is such a complex representation, then U can be recovered as the fixed point set of j (the eigenspace with eigenvalue 1).
In physics
, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors.
A real representation on a complex vector space is isomorphic to its complex conjugate representation
, but the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a pseudoreal representation. An irreducible pseudoreal representation V is necessarily a quaternionic representation: it admits an invariant quaternionic structure, i.e., an antilinear equivariant map
which satisfies
A direct sum of real and quaternionic representations is neither real nor quaternionic in general.
A representation on a complex vector space can also be isomorphic to the dual representation
of its complex conjugate. This happens precisely when the representation admits a nondegenerate invariant sesquilinear form
, e.g. a hermitian form. Such representations are sometimes said to be complex or (pseudo-)hermitian.
s G) for reality of irreducible representations in terms of character theory
is based on the Frobenius-Schur indicator
defined by
where χ is the character of the representation and μ is the Haar measure
with μ(G) = 1. For a finite group, this is given by
The indicator may take the values 1, 0 or −1. If the indicator is 1, then the representation is real. If the indicator is zero, the representation is complex (hermitian), and if the indicator is −1, the representation is quaternionic.
are real (and in fact rational), since we can build a complete set of irreducible representations using Young tableaux.
All representations of the rotation groups
are real, since they all appear as subrepresentations of tensor products of copies of the fundamental representation, which is real.
Further examples of real representations are the spinor
representations of the spin groups in 8k−1, 8k, and 1 + 8k dimensions for k = 1, 2, 3 ... . This periodicity modulo
8 is known in mathematics not only in the theory of Clifford algebra
s, but also in algebraic topology
, in KO-theory; see spin representation.
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...
field of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
a real representation is usually a 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...
on a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
U, but it can also mean a representation on a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
vector space V with an invariant real structure
Real structure
In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces...
, i.e., an antilinear equivariant map
which satisfies
The two viewpoints are equivalent because if U is a real vector space acted on by a group G (say), then V = U⊗C is a representation on a complex vector space with an antilinear equivariant map given by complex conjugation. Conversely, if V is such a complex representation, then U can be recovered as the fixed point set of j (the eigenspace with eigenvalue 1).
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...
, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors.
A real representation on a complex vector space is isomorphic to its complex conjugate representation
Complex conjugate representation
In mathematics, if G is a group and ρ is a representation of it over the complex vector space V, then the complex conjugate representation ρ* is defined over the conjugate vector space V* as follows:...
, but the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a pseudoreal representation. An irreducible pseudoreal representation V is necessarily a quaternionic representation: it admits an invariant quaternionic structure, i.e., an antilinear equivariant map
which satisfies
A direct sum of real and quaternionic representations is neither real nor quaternionic in general.
A representation on a complex vector space can also be isomorphic to the dual representation
Dual representation
In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation is defined over the dual vector space as follows:...
of its complex conjugate. This happens precisely when the representation admits a nondegenerate invariant sesquilinear form
Sesquilinear form
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...
, e.g. a hermitian form. Such representations are sometimes said to be complex or (pseudo-)hermitian.
Frobenius-Schur indicator
A criterion (for compact groupCompact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
s G) for reality of irreducible representations in terms of character theory
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
is based on 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...
defined by
where χ is the character of the representation and μ is the Haar measure
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
with μ(G) = 1. For a finite group, this is given by
The indicator may take the values 1, 0 or −1. If the indicator is 1, then the representation is real. If the indicator is zero, the representation is complex (hermitian), and if the indicator is −1, the representation is quaternionic.
Examples
All representation of the symmetric groupsSymmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
are real (and in fact rational), since we can build a complete set of irreducible representations using Young tableaux.
All representations of the rotation groups
Rotation group
In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
are real, since they all appear as subrepresentations of tensor products of copies of the fundamental representation, which is real.
Further examples of real representations are the spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
representations of the spin groups in 8k−1, 8k, and 1 + 8k dimensions for k = 1, 2, 3 ... . This periodicity modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
8 is known in mathematics not only in the theory of Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
s, but also in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, in KO-theory; see spin representation.