nLab
permutation representation

Contents

Definition

Let CC be a groupoid.

A permutation representation of CC is a representation of CC on Set, i.e. a functor CSetC \to \Set.

A linear permutation representation is a functor CC \to Vect that factors through a permutation representation via the free functor k ||:SetVectk^{|-|}\colon Set \to Vect which sends a set to the vector space for which this set is a basis.

Warning

In the usual literature of representation theory, “linear permutation representations” are just called “permutation representations”.

Examples

GG b Notably for C=BGC = \mathbf{B}G the delooping groupoid of a group GG, a permutation representation BGSet\mathbf{B}G \to Set is a set equipped with a GG-action.

The category

Rep(G,Set)PSh(BG) Rep(G, Set) \simeq PSh(\mathbf{B} G)

is the classifying topos for the group GG.

For other general perspectives on this see also at infinity-action the section Examples – Discrete group actions on sets.

Example

Let GG be a group. The automorphism group of GG regarded as a permutation representation of itself via left multiplication, is isomorphic to GG:

GAut GSet(G,G) G \;\simeq\; Aut_{G Set}(G,G)

(See also at torsor.)

Proof

Consider an function

f:GG. f \;\colon\; G \longrightarrow G \,.

By GG-equivariance, its value on any gGg \in G is fixed by its value on the neutral element 11

f(g)=f(g1)=gf(1). f(g) = f(g \cdot 1) = g \cdot f(1) \,.

Moreover, if f 1f_1 is a GG-invariant function given by f 1(1)=g 1f_1(1) = g_1 and f 2f_2 is given by f 2(1)=g 2f_2(1) = g_2, then their composite is given by

f 2f 1(1)=f 2(g 1)=g 1f 2(1)=g 1g 2. f_2 \circ f_1(1) = f_2( g_1 ) = g_1 \cdot f_2(1) = g_1 \cdot g_2 \,.

Revised on July 3, 2017 10:40:55 by Urs Schreiber (88.77.226.246)