contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight If the highest weight is dominant and integral, one then proves that this irreducible quotient is actually finite dimensional. As an example, consider the case discussed above. If the highest weight is "dominant integral"—meaning simply that it is a non-negative integer—then and the span of the elements is invariant. The quotient representation is then irreducible with dimension . The quotient representation is spanned by linearly independent vectors . The action of is the same as in the Verma module, except that in the quotient, as compared to in the Verma module. The Verma module itself is irreducible if and only if is antidominant. Consequently, when is integral, is irreducible if and only if none of the coordinates of in the basis of fundamental weights is from the set , while in general, this condition is necessary but insufficient for to be irreducible. Other properties The Verma module is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight . In other word, there exist an element w of the Weyl group W such that where is the affine action of the Weyl group. The Verma module is called singular, if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight so that is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights). Homomorphisms of Verma modules For any two weights a non-trivial homomorphism may exist only if and are linked with an affine action of the Weyl group of the Lie algebra . This follows easily from the Harish-Chandra theorem on infinitesimal central characters. Each homomorphism of Verma modules is injective and the dimension for any . So, there exists a nonzero if and only if is isomorphic to a (unique) submodule of . The full classification of Verma module homomorphisms was done by Bernstein–Gelfand–Gelfand and Verma and can be summed up in the following statement: There exists a nonzero homomorphism if and only if there exists a sequence of weights such that for some positive roots (and is the corresponding root reflection and is the sum of all fundamental weights) and for each is a natural number ( is the coroot associated to the root ). If the