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 $\rho$? 



Dimensions of Jordan blocks associated to representations
The answer is yes. The Jordan block decomposition of the generic nilpotent in $SL\_m$ on a representation is the same as the decomposition of any representation under the principal $SL\_2$ (which is a map of $SL\_2$ to $SL\_n$ which sends the generic nilpotent in $SL\_2$ to a generic one in $SL\_n$; people usually have a particular one in mind, but they are all the same up to conjugation by Jacobson-Morozov).


This can be extracted from the formula for the character of the principal $SL\_2$ usually called the "quantum Weyl dimension formula"


$$\chi(V\_\lambda)=\frac{\prod\_{\alpha\in \Delta^+}q^{\langle\rho+\lambda,\alpha\rangle}-q^{-\langle\rho+\lambda,\alpha\rangle}}{\prod\_{\alpha\in \Delta^+}q^{\langle\rho,\alpha\rangle}-q^{-\langle\rho,\alpha\rangle}}$$


One "only" needs to expand this out in terms of the characters of the irreducible $SL\_2$ reps $\frac{q^{n+1}-q^{-n-1}}{q-q^{-1}}$.


