We study the trigonometric quantum spin-Calogero-Sutherland model, and the
Haldane-Shastry spin chain as a special case, using a Bethe-ansatz analysis. We
harness the model's Yangian symmetry to import the standard tools of
integrability for Heisenberg spin chains into the world of integrable
long-range models with spins. From the transfer matrix with a diagonal twist we
construct Heisenberg-style symmetries (Bethe algebra) that refine the usual
hierarchy of commuting Hamiltonians (quantum determinant) of the
spin-Calogero-Sutherland model. We compute the first few of these new conserved
charges explicitly, and diagonalise them by Bethe ansatz inside each
irreducible Yangian representation. This yields a new eigenbasis for the
spin-Calogero-Sutherland model that generalises the Yangian Gelfand-Tsetlin
basis of Takemura and Uglov. The Bethe-ansatz analysis involves non-generic
values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX
chain, with special attention to how the Bethe ansatz works in the presence of
fusion, may be of independent interest.Comment: 42 pages, 3 figure