How Can I Understand the Representation Theory of Lie Algebras?

[Jim Kata]

I'm trying to derive the Gell Mann matrices for fun, but I really don't know representation theory. Can somebody help me? I have Hall's and Humphrey's books, but only online versions and staring at them makes me go crosseyed, and I'm to broke to buy a real book on representation theory. Let me tell you what I kind of know, and maybe you can help me from there.

So, I know the root system of su(3) is $$A_2$$, ok? Now the Weyl group for $$A_2$$ is given by dihedral three, the symmetries of an equilateral triangle, fine. I don't really understand how you go from looking at $$A_2$$ to determining that its Weyl group is dihedral three. I understand it has rotational symmetry like cyclic six, but whatever. From the root diagrams, I should be able to determine the weights of the representation and hence the matrices that make up the Cartan subalgebra of su(3), I guess the vertices of the triangle being the weights I think. The elements of the Weyl group act like ladder operators on the Cartan subalgebra?
Can someone simply spell out the algorithm from looking at root diagrams to finding the Weyl group, Cartan subalgebras and determining the representations of the lie algebras. I understand su(2) pretty well. I also heard (this weeks finds John Baez) that the root systems of $$A_n$$ were related to the alternating group, but I don't see it. Dihedral three is not alternating two.

 

[Jim Kata]

I may have quoted Baez wrong, I think he was talking about the root systems of $$A_n$$ being the same as the symmetries of an n simplex