the representation theory of semisimple Lie algebras (or the closely related representation theory of compact Lie groups), the weights of the dual representation are the negatives of the weights of the original representation. (See the figure.) Now, for a given Lie algebra, if it should happen that operator is an element of the Weyl group, then the weights of every representation are automatically invariant under the map . For such Lie algebras, every irreducible representation will be isomorphic to its dual. (This is the situation for SU(2), where the Weyl group is .) Lie algebras with this property include the odd orthogonal Lie algebras (type ) and the symplectic Lie algebras (type ). If, for a given Lie algebra, is not in the Weyl group, then the dual of an irreducible representation will generically not be isomorphic to the original representation. To understand how this works, we note that there is always a unique Weyl group element mapping the negative of the fundamental Weyl chamber to the fundamental Weyl chamber. Then if we have an irreducible representation with highest weight , the lowest weight of the dual representation will be . It then follows that the highest weight of the dual representation will be . Since we are assuming is not in the Weyl group, cannot be , which means that the map is not the identity. Of course, it may still happen that for certain special choices of , we might have . The adjoint representation, for example, is always isomorphic to its dual. In the case of SU(3) (or its complexified Lie algebra, ), we may choose a base consisting of two roots at an angle of 120 degrees, so that the third positive root is . In this case, the element is the reflection about the line perpendicular to . Then the map is the reflection about the line through . The self-dual representations are then the ones that lie along the line through . These are the representations with labels of the form , which are the representations whose weight diagrams are regular hexagons. Motivation In representation theory, both vectors in and linear functionals in are considered as column vectors so that the representation can act (by matrix multiplication) from the left. Given a basis for and the dual basis for , the action of a linear functional on , can be expressed by matrix multiplication,