Sign representation of symmetric group

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Definition

Let $S_n$ be the symmetric group on $n$ letters.

Then the map $\epsilon :S_n\to GL(\mathbb{C})$ sending each element of $S_n$ to its sign is a linear representation. It is called the sign representation.

Properties

The sign representation is one-dimensional, hence it is irreducible.

Character

The character of this representation is $1$ on all even elements (that is, elements in the alternating group $A_n$) and $-1$ on the odd elements.