Given a Lie Algebra $\mathfrak{g}$, its Cartan Matrix $A$ and a finite representation $R$, is there a way of determining its highest weight $\Lambda$ in a simple way?


In my course, we consider $\mathfrak{g}=A\_2= \mathfrak{L}\_{\mathbb{C}}(SU(3))$. It is stated that the highest weight of the fundamental representation has Dynkin labels $\Lambda = (1,0)$ and the highest weight of the adjoint representation has Dynkin labels $\Lambda = (1,1)$. Why is it so? From there, I can work out the other roots by removing weights given by the Cartan Matrix but it is of no use if I can't compute the highest weight in the first place.


Taking an example, let $\mathfrak{g}=B\_2= \mathfrak{L}\_{\mathbb{C}}(SO(5))$. How do I work out the highest weight for the fundamental and adjoint representation?


Thanks in advance!



Highest weight of a representation of a Lie Algebra
To obtain the highest weight of a semisimple Lie algebra $\mathfrak{g}$ you first have to choose Cartan subalgebra $\mathfrak{h} \leq \mathfrak{g}$ and then set of positive roots (or alternatively choose a Borel subalgebra). Then the highest weight vectors of your representation $V$ are given by linear system 
$$
\rho(X) v = 0,\quad \forall X \in \bigoplus\_{\alpha > 0} \mathfrak{g}\_\alpha
$$
The set of such vectors $v \in V$ then gives you the number of irreducible constituents of your representation. The highest weight of each such subrepresentation is given by eigenvalues of $H\_\alpha \in \mathfrak{h}$ for $\alpha$ simple. 


