Partial differential equation approach to E 7

By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E₇ into a sum of irreducible submodules. This essentially gives a partial differential equation proof of a combinatorial identity on the dimensions of certain irreducible modules of E₇. We also determine two three-parameter families of irreducible submodules in the solution space of Cartan's well-known fourth-order E₇-invariant partial differential equation.