Sign representation of symmetric group

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Definition

Let $S_{n}$ be the symmetric group on $n$ letters.

Then the map $\varepsilon :S_{n}\to GL(\mathbb {C} )$ 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.