of this basis are called descendant states, and their linear combinations are also called descendant states. For generic values of the charges, the Verma module is the only highest weight representation. For special values of the charges that depend on the algebra's central charge, there exist other highest weight representations, called degenerate representations. Degenerate representations exist if the Verma module is reducible, and they are quotients of the Verma module by its nontrivial submodules. Degenerate representations If a Verma module is reducible, any indecomposible submodule is itself a highest weight representation, and is generated by a state that is both descendant and primary, called a null state or null vector. A degenerate representation is obtained by setting one or more null vectors to zero. Setting all the null vectors to zero leads to an irreducible representation. The structures and characters of irreducible representations can be deduced by Drinfeld-Sokolov reduction from representations of affine Lie algebras. The existence of null vectors is possible only under -dependent constraints on the charge . A Verma module can have only finitely many null vectors that are not descendants of other null vectors. If we start from a Verma module that has a maximal number of null vectors, and set all these null vectors to zero, we obtain an irreducible representation called a fully degenerate representation. For example, in the case of the algebra W(3), the Verma module with vanishing charges has the three null vectors at levels 1, 1 and 2. Setting these null vectors to zero yields a fully degenerate representation called the vacuum module. The simplest nontrivial fully degenerate representation of W(3) has vanishing null vectors at levels 1, 2 and 3, whose expressions are explicitly known. An alternative characterization of a fully degenerate representation is that its fusion product with any Verma module is a sum of finitely many indecomposable representations. Case of W(N) It is convenient to parametrize highest-weight representations not by the set of charges , but by an element of the weight space of , called the momentum. Let be the simple roots of , with a scalar product given by the Cartan matrix of , whose nonzero elements are . The positive simple roots are sums of any number of consecutive simple roots, and the Weyl vector is their half-sum , which obeys . The fundamental weights are defined by . Then the momentum is a vector