  I define a set of wavefunctions for SU(N) lattice antiferromagnets, analogous
to the valence bond solid states of Affleck, Kennedy, Lieb, and Tasaki (AKLT),
in which the singlets are extended over N-site simplices. As with the valence
bond solids, the new simplex solid (SS) states are extinguished by certain
local projection operators, allowing us to construct Hamiltonians with local
interactions which render the SS states exact ground states. Using a coherent
state representation, we show that the quantum correlations in each SS state
are calculable as the finite temperature correlations of an associated
classical model, with N-spin interactions, on the same lattice. In three and
higher dimensions, the SS states can spontaneously break SU(N) and exhibit
N-sublattice long-ranged order, as a function of a discrete parameter which
fixes the local representation of SU(N). I analyze this transition using a
classical mean field approach. For N>2 the ordered state is selected via an
"order by disorder" mechanism. As in the AKLT case, the bulk representations
fractionalize at an edge, and the ground state entropy is proportional to the
volume of the boundary.
