Title: Theory of ground states for classical Heisenberg spin systems I,
Abstract: We formulate part I of a rigorous theory of ground states for classical,
finite, Heisenberg spin systems. The main result is that all ground states can
be constructed from the eigenvectors of a real, symmetric matrix with entries
comprising the coupling constants of the spin system as well as certain
Lagrange parameters. The eigenvectors correspond to the unique maximum of the
minimal eigenvalue considered as a function of the Lagrange parameters.
However, there are rare cases where all ground states obtained in this way have
unphysical dimensions $M>3$ and the theory would have to be extended. Further
results concern the degree of additional degeneracy, additional to the trivial
degeneracy of ground states due to rotations or reflections. The theory is
illustrated by a couple of elementary examples.