# nLab virtual representation

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

A virtual representation of a group is a formal difference with respect to direct sum of two ordinary representations, hence the isomorphism class of a virtual representation is an element of the Grothendieck group of $(G Rep, \oplus)$. Equivalence classes of virtual representations form the elements of the representation ring of the group, see there for more.

If we regard an ordinary representation as an equivariant vector bundle over the point, then a virtual representation is a corresponding equivariant virtual vector bundle. Accordingly the representation ring of a finite group is its equivariant K-theory of the point.

Last revised on December 20, 2018 at 03:41:38. See the history of this page for a list of all contributions to it.