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

TL;DR

This paper employs PDEs to decompose polynomial algebras over $E_7$, derives a combinatorial identity for module dimensions, and links irreducible modules to the fundamental invariant differential operator.

Contribution

It introduces a novel PDE-based method for decomposing $E_7$ polynomial algebras and establishes new combinatorial and invariant differential operator results.

Findings

Decomposition of polynomial algebra over $E_7$ into irreducible modules.

A combinatorial identity relating module dimensions and binomial coefficients.

Irreducible submodules as solutions to the fundamental invariant differential operator.

Abstract

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.