Real representation

In mathematics , real representations are a concept of representation theory with numerous applications in physics and mathematics. It denotes representations on a complex vector space that have arisen by tensing with the complex numbers from a representation on a real vector space.

Real representations

If a group operates on a real vector space , then the corresponding representation on the vector space is called real. The vector space is a complex vector space , also called the complexification of This corresponding representation is given by for all ${\displaystyle G}$ ${\displaystyle V_{0}}$ ${\displaystyle V=V_{0}\otimes _{\mathbb {R} }\mathbb {C} }$
${\displaystyle V=V_{0}\otimes _{\mathbb {R} }\mathbb {C} }$ ${\displaystyle V_{0}.}$ ${\displaystyle g(v_{0}\otimes z)=(g(v_{0}))\otimes z}$ ${\displaystyle g\in G,v_{0}\in V_{0},z\in \mathbb {C} .}$

Real characters

Every real representation assigns a real linear mapping to each element of a group . Hence the character of any real representation is real. But conversely, not every representation with a real character is real. So is the trace of each element of the group ${\displaystyle \rho }$ ${\displaystyle g}$ ${\displaystyle G}$ ${\displaystyle \rho (g)}$

${\displaystyle {\text{SU}}(2)=\{{\begin{pmatrix}a&b\\-{\overline {b}}&{\overline {a}}\end{pmatrix}}:\,\,|a|^{2}+|b|^{2}=1\}}$

real. So the character of self-expression is real. But in no basis are all elements of real matrices. ${\displaystyle \rho (g)=g}$ ${\displaystyle {\text{SU}}(2)}$ ${\displaystyle 2\times 2}$

Characterization of real representations

An irreducible representation of on a vector space over can become reducible when the base body is expanded to . An example is the irreducible representation of the cyclic group at given by ${\displaystyle G}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$ ${\displaystyle \mathbb {R} ^{2},}$

${\displaystyle k\mapsto {\begin{pmatrix}{\text{cos}}({\frac {2\pi ik}{m}})&{\text{sin}}({\frac {2\pi ik}{m}})\\-{\text{sin}}({\frac {2\pi ik}{m}})&{\text{cos}}({\frac {2\pi ik}{m}})\end{pmatrix}},}$

the above is reducible. This means that by classifying all irreducible representations that are real, one has not yet classified all irreducible real representations. However, the following is obtained: Let a real vector space on which irreducible operates be the corresponding real representation of If the representation space is not irreducible, it has exactly two irreducible factors and these are conjugate complex representations of ${\displaystyle \mathbb {C} }$
${\displaystyle \mathbb {C} ,}$

${\displaystyle V_{0}}$ ${\displaystyle G}$ ${\displaystyle V=V_{0}\otimes \mathbb {C} }$ ${\displaystyle G.}$ ${\displaystyle V}$ ${\displaystyle G.}$

For evidence and more information on representations about general sub-bodies of see. ${\displaystyle \mathbb {C} }$

example

Let be the cyclic group , i.e. the set with the addition modulo as a group link. ${\displaystyle G}$ ${\displaystyle \mathbb {Z} /3\mathbb {Z} }$ ${\displaystyle \{0,1,2\}}$ ${\displaystyle 3}$

A real representation of this group is obtained by assigning the rotation of the real plane by 120 degrees, i.e. ${\displaystyle 1}$

${\displaystyle \rho (1)=\left({\begin{array}{cc}-{\frac {1}{2}}&{\frac {1}{2}}{\sqrt {3}}\\-{\frac {1}{2}}{\sqrt {3}}&-{\frac {1}{2}}\end{array}}\right).}$

The corresponding representation on is reducible , because complex irreducible representations of Abelian groups are always one-dimensional. The representation on , on the other hand, is irreducible , because one-dimensional subspaces of , i.e. the straight lines through the zero point, cannot be transformed into themselves when rotated by 120 degrees. ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{2}}$

See also

• Quaternionic representation

literature

• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8 ISBN 978-0-387-97527-6

Web links

• Ganev: Real and quaternionic representations of finite groups

Individual evidence

1. ↑ Fulton-Harris, op. Cit.