Given a linear representation $\rho$ of $SL_n(\mathbb C)$ of finite dimension $m$, the image $\rho(U)$ of a maximal unipotent Jordan block $U\in SL_n$ decomposes into generally several Jordan blocks of size $m_1,\dots,m_k$.

Is it possible to describe the partition $m=m_1+m_2+\dots+m_k$, say in terms of the highest weight vector associated to an irreducible representation $\rho$?