Determinant representation ($\det V={\Lambda }^{\dim V}V$)

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.