  We initiate a study of determinantal representations with symmetry. We show
that Grenet's determinantal representation for the permanent is optimal among
determinantal representations respecting left multiplication by permutation and
diagonal matrices (roughly half the symmetry group of the permanent). In
particular, if any optimal determinantal representation of the permanent must
be polynomially related to one with such symmetry, then Valiant's conjecture on
permanent v. determinant is true.
