  We study the action of the group of automorphisms of the projective plane on
the Maruyama scheme of sheaves $\mathcal{M}_{P^{2}}(r,c_{1,}c_{2})$ of rank $r$
and Chern classes $c_{1}=0$ and $c_{2}=n$ and obtain sufficient conditions for
unstability in the sense of Mumford's geometric invariant theory. The
conditions are in terms of the splitting behaviour of sheaves when restricted
to lines in the projective plane. A very strong parallel is observed with
Mumford's theory for the action of the automorphism group of the projective
plane on the spaces of curves of a fixed degree.
