  We consider homogeneous multiaffine polynomials whose coefficients are the
Pl\"ucker coordinates of a point $V$ of the Grassmannian. We show that such a
polynomial is stable (with respect to the upper half plane) if and only if $V$
is in the totally nonnegative part of the Grassmannian. To prove this, we
consider an action of matrices on multiaffine polynomials. We show that a
matrix $A$ preserves stability of polynomials if and only if $A$ is totally
nonnegative. The proofs are applications of classical theory of totally
nonnegative matrices, and the generalized P\'olya-Schur theory of Borcea and
Br\"and\'en.
