The theorem describes the representations of semisimple Lie groups; that is, it gives an explicit construction of the representations given by the theorem of highest weight .
If the constraint of to is a dominant integral element , then that representation of is whose greatest weight is the constraint of to .
Otherwise is .
Example: Representations of the SL (2, C)
For the Borel group we can choose the subgroup of the upper triangular matrices. Every 1-dimensional representation is of the form
for an integer .
The Fahnenvarietät is with homogeneous coordinates and the sections of the line bundle for displaying correspond to the homogeneous polynomials of degree in the coordinates . These form an (n + 1) -dimensional vector space with a basis . So you get a representation of the dimension and rediscover the well-known proposition that there is an unambiguous, irreducible representation of for every dimension .