nLab
alternating representation
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
Definition
For $S_n$ a symmetric group , its alternating representation is the 1-dimensional linear representation

$alt \;\; S_n \longrightarrow GL(1)$

which sends even permutations to $+1$ and odd permutations to $-1$ .

More generally, for $G$ any finite group and $H \subset G$ a subgroup of index 2, then the corresponding alternating representation of $G$ sends elements of $H$ to $+1$ and all other elements to -1.

This reduces to the previous special case by setting $G \coloneqq S_n$ and $H \coloneqq A_n \subset S_n$ the alternating group .

References
For instance

João Pedro Martins dos Santos, Def. 3 in Representation Theory of Symmetric Groups , 2012 (pdf )