  A novel combined unitary and symmetric group approach is used to study the
spin-$\frac{1}{2}$ Heisenberg model and related Fermionic systems in a
spin-adapted representation, using a linearly-parameterised Ansatz for the
many-body wave function. We show that a more compact ground state wave function
representation is obtained when combining the symmetric group, $\mathcal{S}_n$,
in the form of permutations of the underlying lattice site ordering, with the
cumulative spin-coupling based on the unitary group, $\mathrm{U}(n)$. In
one-dimensional systems the observed compression of the wave function is
reminiscent of block-spin renormalization group approaches, and allows us to
study larger lattices (here taken up to 80 sites) with the spin-adapted full
configuration interaction quantum Monte Carlo method, which benefits from the
sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates
that emerge from the reordering. We find that in an optimal lattice ordering
the configuration state function with highest weight already captures with high
accuracy the spin-spin correlation function of the exact ground state wave
function. This feature is found for more general lattice models, such as the
Hubbard model, and ab initio quantum chemical models, in this work exemplified
by a one-dimensional hydrogen chain. We also provide numerical evidence that
the optimal lattice ordering for the unitary group approach is not generally
equivalent to the optimal ordering obtained for methods based on matrix-product
states, such as the density-matrix renormalization group approach.
