Quaternionic representation

In mathematics , quaternionic representations are a concept of representation theory that is used, among other things, in spin geometry .

definition

A quaternionic representation is a (complex) representation of a group that has an invariant homomorphism that is anti-linear and satisfies. The complex vector space thus has a structure of a quaternionic vector space defined by the complex number and and . So a quaternionic representation defines a group homomorphism . ${\displaystyle V}$ ${\displaystyle G,}$ ${\displaystyle G}$ ${\displaystyle J\colon V\to V}$ ${\displaystyle J^{2}=-{\text{Id}}}$
${\displaystyle i}$ ${\displaystyle j:=J}$ ${\displaystyle k=ij}$ ${\displaystyle \rho \colon G\to GL(V,\mathbb {H} )}$

example

Rotations of the 3-dimensional space can be described by quaternions. That defines a quaternionic representation

${\displaystyle \rho \colon Spin(3)\to GL(1,\mathbb {H} )}$

the spin group . ${\displaystyle Spin(3)}$

In general, the spinor representations of the spin group are quaternionic representations for and with . ${\displaystyle Spin(d)}$ ${\displaystyle d=8k+3,d=8k+4}$ ${\displaystyle d=8k+5}$ ${\displaystyle k\in \mathbb {N} }$

Characterization of quaternionic representations

A skew-symmetrical non-degenerate -invariant bilinear form defines a quaternionic structure ${\displaystyle G}$ ${\displaystyle V.}$

Conversely, for every quaternionic representation there is an invariant skew-symmetric, non-degenerate bilinear form . For irreducible representations , this bilinear form is unique except for scaling. ${\displaystyle G}$ ${\displaystyle V}$

An irreducible representation is exactly one of the following: ${\displaystyle V}$

• complex: the character is not real-valued and has no -invariant, non-degenerate bilinear form ${\displaystyle \chi _{V}}$ ${\displaystyle V}$ ${\displaystyle G}$
• real: a real representation ; has an -invariant symmetric non-degenerate bilinear form ${\displaystyle V=V_{0}\otimes \mathbb {C} ,}$ ${\displaystyle V}$ ${\displaystyle G}$
• quaternionic: the character is real, but is not a real representation ; has an -invariant skew-symmetric, non-degenerate bilinear form. ${\displaystyle \chi _{V}}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle G}$

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
• Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Springer-Verlag, ISBN 0-387-90190-6

Web links

• Ganev: Real and quaternionic representations of finite groups