nLab
virtual representation
Contents
Context
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
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.
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