# Group Theory/Representations

**Definition (representation)**:

Let be a group, a category, an object of . Then a **representation** (also called **action**) of on by automorphisms of is a group homomorphism

- .

**Example (symmetric group acting on a product set)**:

Let be a set and let be the product of copies of . The symmetric group acts on via

- .

Note that in this notation, we identified the element of with the automorphism of 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 be a group, a category, objects of and , two representations of on resp. . Then an **equivalence of representations** is an isomorphism such that

- .

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

Let be a group, a category, objects of and , two representations of on resp. . Let be an equivalence of representations. Then is also an equivalence of representations.

**Proof:** We have

- ,

since is an equivalence of representations.