  We study a natural conjecture regarding ferromagnetic ordering of energy
levels in the Heisenberg model which complements the Lieb-Mattis Theorem of
1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest
energies in each subspace of fixed total spin are strictly ordered according to
the total spin, with the lowest, i.e., the ground state, belonging to the
maximal total spin subspace. Our main result is a proof of this conjecture for
the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof
has two main ingredients. The first is an extension of a result of Koma and
Nachtergaele which shows that monotonicity as a function of the total spin
follows from the monotonicity of the ground state energy in each total spin
subspace as a function of the length of the chain. For the second part of the
proof we use the Temperley-Lieb algebra to calculate, in a suitable basis, the
matrix elements of the Hamiltonian restricted to each subspace of the highest
weight vectors with a given total spin. We then show that the positivity
properties of these matrix elements imply the necessary monotonicity in the
volume. Our method also shows that the first excited state of the XXX
ferromagnet on any finite tree has one less than maximal total spin.
