Orthogonalizable linear representation

This article describes a property to be evaluated for a linear representation of a group, i.e. a homomorphism from the group to the general linear group of a vector space over a field

This article provides a semi-basic definition in the following area: linear representation theory

Definition

Symbol-free definition

A linear representation of a group over a field is termed orthogonalizable if it saisfies the following equivalent conditions:

• It is equivalent, as a linear representation, to a linear representation whose image lies completely within the orthogonal group (i.e. it is equivalent to an orthogonal representation)
• There is a nondegenerate, symmetric, bilinear form that is equivalent to the diagonal form, and which is invariant under the action of every element of the group representation

Definition with symbols

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Facts

• Linear representation of finite group over reals has invariant dot product: It turns out that over the reals, every linear representation of a finite group is orthogonalizable. The proof again relies on an averaging technique. Further, the orthogonalizability can also be used to prove Maschke's averaging lemma for the case of real representations.