  Based on the UrQMD model, we have investigated the influence of the symmetry
potential on the negatively and positively charged $\pi$ and $\Sigma$ hyperon
production ratios in heavy ion collisions at the SIS energies. We find that, in
addition to $\pi^-/\pi^+$ ratio, the $\Sigma^-/\Sigma^+$ ratio can be taken as
a sensitive probe for investigating the density dependence of the symmetry
potential of nuclear matter at high densities (1-4 times of normal baryon
density). This sensitivity of the symmetry potential to both the $\pi^-/\pi^+$
and $\Sigma^-/\Sigma^+$ ratios is found to depend strongly on the incident beam
energy. Furthermore, the $\Sigma^-/\Sigma^+$ ratio is shown to carry the
information about the isospin-dependent part of the $\Sigma$ hyperon
single-particle potential.
