Weyl Group in SU(3)

The representations of sl(3;C) are invariant under the adjoint action of SU(3). Let π be a finite-dimensional representation of sl(3;C) acting on a vector space V and let Π be the associated representation of SU(3) acting on the same space. For any A ∈ SU(3), we can define a new representation π_A of sl(3;C), acting on the same vector space V, by setting π_A(X) = π(AXA^-1). Since the adjoint action of A on sl(3;C) is a Lie algebra automorphism, this is a representation of sl(3;C). Π(A) is an intertwining map between (π, V) and (π_A, V). We say that adjoint action of SU(3) is a symmetry of the set of equivalence classes of representations of sl(3;C). 

The two-dimensional subspace h of sl(3;C) spanned by H₁ and H₂ is called a Cartan subalgebra. In general, the adjoint action of A  ∈ SU(3) will not preserve the space h and so the equivalence of π and π_A does not tell us anything about the weights of π. However, there are elements A in SU(3) for which Ad_A does preserve h. These elements make up the Weyl group for SU(3) and give rise to a symmetry of the set of weights of any representation π. 

Let N be the subgroup of SU(3) consisting of those A ∈ SU(3) such that Ad_A(H) is an element of h for all H in h. And let Z be the subgroup of SU(3) consisting of those  A  ∈ SU(3) such that Ad_A(H) = H for all H ∈ h. The Weyl group of SU(3), denoted by W, is the quotient group N/Z. 

The group Z consists precisely of the diagonal matrices inside SU(3), namely the matrices of the form A = (e^iθ,0,0 : 0,e^iΦ,0 : 0,0,e^-i(θ+Φ)) for θ and Φ in R. The group N consists of precisely those matrices A ∈ SU(3) such that for each k = 1,2,3, there exist l ∈ {1,2,3} and θ ∈ R such that Ae_k = e^iθe_l. Here e₁, e₂,e₃ is the standard basis for C³. The Weyl group W = N/Z is isomorphic to the permutation group on three elements. 

In order to show that Weyl group is a symmetry of the weights of any finite-dimensional representation of sl(3;C), we need to adopt a less basis-dependent view of weights. A vector v is an eigenvector of π(H₁) and π(H₂) then it is also an eigenvector for π(H) for any element H of the space h spanned by H₁ and H₂. Furthermore, the eigenvalues must depend linearly on H. For π(J) = λ₂v, then π(aH + bJ)v = (aπ(H) + bπ(J))v = (aλ₁ + bλ₂)v. We have the following basis-independent notion of weight :  a linear functional μ ∈ h^* is called a weight for π if there exists a nonzero vector v in V such that π(H)v = μ(H)v for all H in h. Such a vector v is called a weight vector with weight μ. So a weight is just a collection of simultaneously eigenvalues of all the elements H of h, which depends linearly on H and, therefore, define a linear functional on h. The reason for adopting this basis-independent approach is that the action of the Weyl group does not preserve the basis {H₁, H₂} for h.

In another words, the Weyl group is a group of linear transformation of h. This means that W acts on h, and we denote this action as w⋅H. We can define an associated action on the dual space h^*. Thus, for μ ∈ h^* and w ∈ W, we define w⋅μ to be the element of  h^* given by (w⋅μ)(H) = μ(w^-1⋅H).