Polynomial Representation of E6 and Its Combinatorial and PDE Implications

Details Polynomial Representation of E6 and Its Combinatorial and PDE Implications Polynomial Representation of E6 and Its Combinatorial and PDE Implications

2008-11-10

arxiv.org/abs/0811.1399v1

Mathematics

Representation Theory

24pages

Scientific paper

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.

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