  In this paper, we use partial differential equations to find the
decomposition of the polynomial algebra over the basic irreducible module of
$E_7$ into a sum of irreducible submodules. Moreover, we obtain a combinatorial
identity, saying that the dimensions of certain irreducible modules of $E_7$
are correlated by the binomial coefficients of fifty-five. Furthermore, we
prove that two families of irreducible submodules with three integral
parameters are solutions of the fundamental invariant differential operator
corresponding to Cartan's unique quartic $E_7$ invariant.
