One-dimensional representation ($\rho :G\to {\mathbb{C}}^{\times }$)

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 $n\ge 2$, 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.

The top exterior power of an $n$-dimensional group representation is one-dimensional, and $g$ acts on it by $\det R(g)$. It is trivial for the defining representation of $SU(n)$. This is why a full $n$-box column in an $SU(n)$ Young diagram can be removed without changing the representation.