permutation representation

Let $C$ be a groupoid.

A *permutation representation* of $C$ is a representation of $C$ on Set, i.e. a functor $C \to \Set$.

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

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

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

The category

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

is the classifying topos for the group $G$.

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

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

$G
\;\simeq\;
Aut_{G Set}(G,G)$

(See also at *torsor*.)

Consider an function

$f \;\colon\; G \longrightarrow G
\,.$

By $G$-equivariance, its value on any $g \in G$ is fixed by its value on the neutral element $1$

$f(g) = f(g \cdot 1) = g \cdot f(1)
\,.$

Moreover, if $f_1$ is a $G$-invariant function given by $f_1(1) = g_1$ and $f_2$ is given by $f_2(1) = g_2$, then their composite is given by

$f_2 \circ f_1(1)
=
f_2( g_1 ) = g_1 \cdot f_2(1) = g_1 \cdot g_2
\,.$

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