Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications

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.