Say that $W$ is a $SL_4(\mathbb{Q})$ representation that decomposes into irreducibles (using the notation of Fulton-Harris) $$W = \Gamma_{a,b,c} \oplus \Gamma_{d,e,f}$$
Now say that I have a vector $v \in W$, and I know something about it (maybe I can write it as a wedge product or tensor product of the basis vectors for the standard representation). How can I know if $v$ belongs to $\Gamma_{a,b,c}$ or $\Gamma_{d,e,f}$ or if it is a sum of elements that belong to both irreducible subrepresentations?
I've tried to pin down some very easy (but nontrivial - so $W$ itself is not irreducible) cases. For example, if $$W = V \otimes V^* = \Gamma_{1,0,1} \otimes \Gamma_{0,0,0}$$ what would vectors belonging to $\Gamma_{1,0,1}$ or $\Gamma_{0,0,0}$ or neither look like? What do the highest weight vectors look like in the form $\Sigma w_i \otimes w_j^*$?
(I also don't mean for my question to be specific to $SL_4(\mathbb{Q})$...it would be great to eventually do this for any semisimple Lie group! At the very least for $SL_n(\mathbb{Q})$ for any $n$.)