Composition Series of $sl_3(\Bbb C)$

Question

Let $Z(\mathbb \lambda)$ denote the standard cyclic(or highest weight) $L$-module of highest weight $\lambda$ and $V(\lambda)$ denote the finite dimensional $L$-module of highest weight $\lambda$. Now for any $\lambda$ $\in$ $H^*$, we know that $ch_{V(\lambda)}$ = $\sum_{w.\lambda \leq \lambda}$ $b(\lambda,w)ch_{Z(w.\lambda)}$ with $b(\lambda,w)$ $\in$ $\mathbb{Z}$ and $b(\lambda,1)$ = $1$. For $L$ = $sl_2(\mathbb{C})$, we get $b(\lambda,w)$ = $\pm 1$ where $\lambda$ is a dominant integral weight. I want to find the coefficients $b(\lambda,w)$ for $L$ = $sl_3(C)$ where $\lambda$ is a dominant integral weight. I was trying to find it using the fact that the unique proper submodule of $Z(\lambda)$ for $\lambda$ integral dominant is given by $\bigoplus_{\alpha \in \Delta}$ $Z(s_{\alpha}. \lambda)$. Can anyone please help me with this. Thanks for any help. $Notations$ - $L$ is finite dimensional semisimple Lie algebra, $\Delta$ is the set of simple roots and $s_{\alpha}$ denotes the simple reflections for $\alpha$ $\in$ $\Delta$,$w. \lambda$ = $w(\lambda$ + $\delta)$ - $\delta$ $\forall$ $w$ $\in$ $W$ where $W$ is the Weyl group and $\delta$ is the half sum of the positive roots. In general, is it possible to find out how $b(\lambda,w)$ depends on $\lambda$ and $w$ for a dominant integral weight $\lambda$?