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

TL;DR

This paper determines explicit formulas for canonical bases of invariant polynomials for the exceptional irreducible finite reflection groups of types E6, E7, and E8, completing the classification for all such groups.

Contribution

It provides the explicit formulas for canonical bases of invariant polynomials for the groups of types E6, E7, and E8, which were previously unknown.

Findings

Explicit formulas for E6, E7, and E8 canonical bases are derived.

Completes the classification of canonical bases for all irreducible finite reflection groups.

Enhances understanding of invariant polynomial structures in exceptional groups.

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.