nLab
induced representation of the trivial 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
Statement
Let $G$ be a finite group and $H \overset{\iota}{\hookrightarrow} G$ a subgroup -inclusion. Then the induced representation in Rep(G) of the 1-dimensional trivial representation $\mathbf{1} \in Rep(H)$ is the permutation representation $k[G/H]$ of the coset G-set $G/H$ :

$\mathrm{ind}_H^G\big( \mathbf{1}\big)
\;\simeq\;
k[G/H]
\,.$

This follows directly as a special case of the general formula for induced representations of finite groups (this Example ).

It follows that every virtual permutation representation (hence every element of the representation ring $R_k(G)$ in the image of the canonical morphism $A(G) \overset{\beta}{\to} R_k(G)$ from the Burnside ring ) is a virtual combination of induced representations of trivial representations.

A generalization of this statement including non-permutation representations is the Brauer induction theorem .

References
See also

Created on January 28, 2019 at 04:45:40.
See the history of this page for a list of all contributions to it.