Group Theory/Representations

Definition (representation):

Let $G$ be a group, ${\mathcal {C}}$ a category, $A$ an object of ${\mathcal {C}}$. Then a representation (also called action) of $G$ on $A$ by automorphisms of ${\mathcal {C}}$ is a group homomorphism

$G\to \operatorname {Aut} (A)$.

Example (symmetric group acting on a product set):

Let $S$ be a set and let $S^{n}$ be the product of $n$ copies of $S$. The symmetric group acts on $S^{n}$ via

$\sigma (x_{1},\ldots ,x_{n}):=(x_{\sigma (1)},\ldots ,x_{\sigma (n)})$.

Note that in this notation, we identified the element of $G$ with the automorphism of $A$ to which the element is sent via the homomorphism of the representation. This convention is followed throughout group theory and will be understood by every mathematician.

Definition (equivalence of representations):

Let $G$ be a group, ${\mathcal {C}}$ a category, $A,B$ objects of ${\mathcal {C}}$ and $\rho :G\to A$, $\psi :G\to B$ two representations of $G$ on $A$ resp. $B$. Then an equivalence of representations is an isomorphism $f:A\to B$ such that

$\forall g\in G:g\circ f=f\circ g$.

Proposition (inverse of equivalence of representations is equivalence of representations):

Let $G$ be a group, ${\mathcal {C}}$ a category, $A,B$ objects of ${\mathcal {C}}$ and $\rho :G\to A$, $\psi :G\to B$ two representations of $G$ on $A$ resp. $B$. Let $f:A\to B$ be an equivalence of representations. Then $f^{-1}:B\to A$ is also an equivalence of representations.

Proof: We have

$g\circ f^{-1}=f^{-1}\circ g\Leftrightarrow f\circ g\circ f^{-1}=g\Leftrightarrow g=g$,

since $f$ is an equivalence of representations. $\Box$