  We study the spectrum of the long-range supersymmetric su$(m)$ $t$-$J$ model
of Kuramoto and Yokoyama in the presence of an external magnetic field and a
charge chemical potential. To this end, we first establish the precise
equivalence of a large class of models of this type to a family of su$(1|m)$
spin chains with long-range exchange interactions and a suitable chemical
potential term. We exploit this equivalence to compute in closed form the
partition function of the long-range $t$-$J$ model, which we then relate to
that of an inhomogeneous vertex model with simple interactions. From the
structure of this partition function we are able to deduce an exact formula for
the restricted partition function of the long-range $t$-$J$ model in subspaces
with well-defined magnon content in terms of its analogue for the equivalent
vertex model. This yields a complete analytical description of the spectrum in
the latter subspaces, including the precise degeneracy of each level, by means
of the supersymmetric version of Haldane's motifs and their related skew Young
tableaux. As an application, we determine the structure of the motifs
associated with the ground state of the spin $1/2$ model in the thermodynamic
limit in terms of the magnetic field strength and the charge chemical
potential. This leads to a complete characterization of the distinct ground
state phases, determined by their spin content, in terms of the magnetic field
strength and the charge chemical potential.
