Hi I am trying to find weights for representation of the lie algebra $sl(n,c)$ .
I understand how to find weights for representation of $sl(3,c)$ on $C^3$. i.e the fundamental representations. I can do this by first finding roots via adjoint representation. and then by hand find weights.
however i am totally stuck on finding weights of representation in different vector spaces.
for example how do i get weights of $sl(3,c)$ on vector space $C^5$ or $C^4$. really stuck on even associating a 5X5 matrix to the basis of cartan subalgebra of $sl(3,c)$.
I am self studying this topic. this question is just my idea cant find any examples or anything related to this in the textbooks or internet.
Note: I know how to find roots, fundamental weights however i am stuck on how to find weights of a particular representations that is the reason for asking this question.
Thanks for any help.