  It is demonstrated that decompositions of integrable highest weight modules
of a simple Lie algebra with respect to its reductive subalgebra obey the set
of algebraic relations leading to the recursive properties for the
corresponding branching coefficients. These properties are encoded in the
special element (of the formal algebra E(g)) Gamma_(a ->g) that describes the
injection and is called the fan. In the simplest case, when "a" is a Cartan
subalgebra of g, the recursion procedure generates the weight diagram of the
module L(g). When applied to a reduction of highest weight modules the
recursion described by the fan provides a highly effective tool to obtain the
explicit values of branching coefficients.
