Verma modules and are regular, then there exists a unique dominant weight and unique elements w, w′ of the Weyl group W such that and where is the affine action of the Weyl group. If the weights are further integral, then there exists a nonzero homomorphism if and only if in the Bruhat ordering of the Weyl group. Jordan–Hölder series Let be a sequence of -modules so that the quotient B/A is irreducible with highest weight μ. Then there exists a nonzero homomorphism . An easy consequence of this is, that for any highest weight modules such that there exists a nonzero homomorphism . Bernstein–Gelfand–Gelfand resolution Let be a finite-dimensional irreducible representation of the Lie algebra with highest weight λ. We know from the section about homomorphisms of Verma modules that there exists a homomorphism if and only if in the Bruhat ordering of the Weyl group. The following theorem describes a resolution of in terms of Verma modules (it was proved by Bernstein–Gelfand–Gelfand in 1975) : There exists an exact sequence of -homomorphisms where n is the length of the largest element of the Weyl group. A similar resolution exists for generalized Verma modules as well. It is denoted shortly as the BGG resolution. See also Classifying finite-dimensional representations of Lie algebras Theorem of the highest weight Generalized Verma module Weyl module Notes References . . . . . Representation theory of Lie algebras