  One of the main problems in any quantum resource theory is the
characterization of the conversions between resources by means of the free
operations of the theory. In this work, we advance on this characterization
within the quantum coherence resource theory by introducing the generalized
coherence vector of an arbitrary quantum state. The generalized coherence
vector is a probability vector that can be interpreted as a concave roof
extension of the pure states coherence vector. We show that it completely
characterizes the notions of being incoherent, as well as being maximally
coherent. Moreover, using this notion and the majorization relation, we obtain
a necessary condition for the conversion of general quantum states by means of
incoherent operations. These results generalize the necessary conditions of
conversions for pure states given in the literature, and show that the tools of
the majorization lattice are useful also in the general case. Finally, we
introduce a family of coherence quantifiers by considering concave and
symmetric functions applied to the generalized coherence vector. We compare
this proposal with the convex roof measure of coherence and others quantifiers
given in the literature.
