Polynomial Representation of E6 and Its Combinatorial and PDE Implications

TL;DR

This paper explores the polynomial algebra over E6, decomposing it into irreducible modules using PDEs, identifying fundamental invariants, and establishing combinatorial identities related to module dimensions.

Contribution

It introduces a novel PDE-based method for decomposing polynomial algebras over E6 and identifies the unique fundamental invariant related to Dickson's form.

Findings

Decomposition of polynomial algebra over E6 into irreducible modules.

Identification of the unique fundamental invariant polynomial.

Derivation of a combinatorial identity involving module dimensions.

Abstract

In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of E6 into a sum of irreducible submodules. It turns out that the cubic polynomial invariant corresponding to the Dicksons' invariant trilinear form is the unique fundamental invariant. Moreover, we obtain a combinatorial identity saying that the dimensions of certain irreducible modules of E6 are correlated by the binomial coefficients of twenty-six. Furthermore, we find all the polynomial solutions for the invariant differential operator corresponding to the Dickson trilinear form in terms of the irreducible submodules.