  We introduce an algebraic method to study local stability in the Newtonian
$n$-body problem when certain symmetries are present. We use representation
theory of groups to simplify the calculations of certain eigenvalue problems.
The method should be applicable in many cases, we give two main examples here:
the square central configurations with four equal masses, and the equilateral
triangular configurations with three equal masses plus an additional mass of
arbitrary size at the center. We explicitly found the eigenvalues of certain
8x8 Hessians in these examples, with only some simple calculations of traces.
We also studied the local stability properties of corresponding relative
equilibria in the four-body problems.
