- #1
Jim Kata
- 197
- 10
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.
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.
