Almost gauge-invariant states and the ground state of Yang-Mills theory

We consider the problem of the explicit description of the gauge-invariant subspace of pure lattice gauge theories in the Hamiltonian formulation, where the gauge group is either a compact Lie group or a finite group. The latter case is particularly interesting for quantum simulation. A basis of states where configurations are grouped according to their holonomies is shown to have several advantages over other descriptions. Using this basis, we compute some properties of interest for some non-Abelian finite groups on small lattices, and in particular we examine the question of whether a certain ansatz introduced long ago is a good approximation for the ground state.