= One-dimensional representation
{title2=$\rho:G\to\mathbb C^\times$}
A one-dimensional complex <group representation> acts on a one-dimensional complex <vector space>, and is equivalently a <group homomorphism> into the multiplicative <group> $\mathbb C^\times$. It is automatically an <irreducible representation>. Its <character of a representation> is its scalar value, a <one-dimensional character>. A change of <basis> cannot change those scalar values, so two such <group representations> are equivalent exactly when their homomorphisms agree. For a <symmetric group> $S_n$ with $n\geq2$, all <transpositions> are conjugate, square to the identity and generate the group. Their common scalar image is therefore $1$ or $-1$, giving exactly the <trivial representation> and the <sign representation>.
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