subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out unphysical components, leaving only a single spin. Construction of pure spin representations for any (under ) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints. Dual representations The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation: The set of weights of the dual representation of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation. Two irreducible representations are isomorphic if and only if they have the same highest weight. For each semisimple Lie algebra there exists a unique element of the Weyl group such that if is a dominant integral weight, then is again a dominant integral weight. If is an irreducible representation with highest weight , then has highest weight . Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If is an element of the Weyl group of a semisimple Lie algebra, then . In the case of the Weyl group is . It follows that each is isomorphic to its dual The root system of is shown in the figure to the right. The Weyl group is generated by where is reflection in the plane orthogonal to as ranges over all roots. Inspection shows that so . Using the fact that if are Lie algebra representations and , then , the conclusion for is Complex conjugate representations If is a representation of a Lie algebra, then is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication. In general, every irreducible representation of can be written uniquely as , where with holomorphic (complex linear) and anti-holomorphic (conjugate linear). For since is holomorphic, is anti-holomorphic. Direct examination of the explicit expressions for and in equation below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression also allows for identification of and for as Using the above identities (interpreted as pointwise addition of functions), for yields where the statement for the