Matrix form of Lie algebra highest weight representation

[vega12]

I am currently working with Lie algebras and my research requires me to have matrix representations for any given Lie algebra and highest weight. I solved this problem with a program for cases where all weights in a representation have multiplicity 1 by finding how $E_\alpha$ acts on each node of this tree. However, when a node has multiplicity greater than one, I run into problems. For instance, the $\omega_2$ highest weight representation of $D_4$ is dimension 28, but I only have 25 unique nodes, and so my corresponding matrices are only 25 dimensional. They properly satisfy the appropriate commutation relations, but are not the right dimension. The same thing happens with, for example, $C_4$ and highest weight $\omega_2$.

How can one handle multiplicities properly to get the matrix representation with the correct dimension? Any references would be very much appreciated. Thank you!

 

[fresh_42]

Here is an example. And you do not mean Lie algebras, you meant semisimple or simple Lie algebras:
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-basics/