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

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