Consider the determinant representation of $U(N)$ defined by $\det:U(N)\ni U\mapsto\det U\in U(1)$. If I'm not mistaken, $\det^{n}$ for $n\geq 1$ are all irreducible representations. When classifying irreducible representations with Young tableux, they correspond to tableux with $n$ columns, each of height $N$.
My question is: are negative powers of the determinant representation, $\det^n$ for $n\leq -1$, also irreducible representations? If not, how do they decompose into irreducible representations? Can we classify them using Young tableux?