Source for highest weight vectors for $\text{SL}_n(\mathbb{C})$ representations

Question

I have a list of a lot of irreducible $\text{SL}_n(\mathbb{C})$ representations and need to know what all of their highest weight vectors are. For example, using the notation from Fulton and Harris, I know $e_1 \wedge e_2$ is a highest weight vector for $\Gamma_{0,1,0,...,0}$ and $(e_1 \wedge e_2) \otimes e_1 \otimes e_n^*$ is a highest weight vector for $\Gamma_{1,1,0,...,0,1}$.

Are there any sources out there that document specifically what the highest weight vectors are for specific irreducible representations? The absolute best case scenario would be a table with this information, but any papers that work with explicit highest weight vectors would be helpful.

If not, I'll need to determine the highest weight vectors myself (tips for doing this would be appreciated), but it would be nice to have a source to check.

Answer 1

If you are trying to recognize the irreducible representations in some larger ambient representation, then the form of the highest weight vector(s) will depend heavily on that scenario. So your question could have many nonstandard answers depending on how you are realizing your representations.

However, a nice standard way to construct highest weight vectors for irreducible representations is via tensor products, which is how you hinted at getting the highest weight vector for $\Gamma_{1,1,0,\dots,0,1}$. Namely (using fundamental weight notation instead of Fulton--Harris's $\Gamma$ notation), if you are interested in the irreducible representation with highest weight $\Lambda=a_1 \varpi_1+a_2\varpi_2+\cdots+a_{n-1}\varpi_{n-1}$, and you have a fixed/preferred construction for each of the representations of fundamental highest weight $\varpi_i$ with highest weight vector $v_i$, then you can realize a highest weight vector of weight $\Lambda$ as $$ v_1 \otimes v_1 \otimes \cdots \otimes v_1 \otimes v_2 \otimes \cdots \otimes v_{n-1} \otimes \cdots \otimes v_{n-1} $$ where each $v_i$ appears $a_i$ times. This will in fact be (up to scalar) the unique highest weight vector of weight $\Lambda$ in the big tensor product of all of the fundamental weight representations appearing the appropriate amount of times.