Canonical bases of invariant polynomials for the irreducible reflection groups of types $E_6$, $E_7$, and $E_8$

Abstract: Given a rank $n$ irreducible finite reflection group $W$, the $W$-invariant polynomial functions defined in ${\mathbb R}^n$ can be written as polynomials of $n$ algebraically independent homogeneous polynomial functions, $p_1(x),\ldots,p_n(x)$, called basic invariant polynomials. Their degrees are well known and typical of the given group $W$. The polynomial $p_1(x)$ has the lowest degree, equal to 2. It has been proved that it is possible to choose all the other $n-1$ basic invariant polynomials in such a way that they satisfy a certain system of differential equations, including the Laplace equations $\triangle p_a(x)=0,\ a=2,\ldots,n$, and so are harmonic functions. Bases of this kind are called canonical. Explicit formulas for canonical bases of invariant polynomials have been found for all irreducible finite reflection groups, except for those of types $E_6$, $E_7$ and $E_8$. Those for the groups of types $E_6$, $E_7$ and $E_8$ are determined in this article.