Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariate Polynomials. III. $E_8$ case

TL;DR

This paper demonstrates that the $E_8$ trigonometric Olshanetsky-Perelomov Hamiltonian can be expressed in algebraic form using Fundamental Trigonometric Invariants, preserving polynomial spaces and enabling explicit eigenfunction calculation.

Contribution

It provides the explicit algebraic form of the $E_8$ Hamiltonian in new variables and shows its preservation of polynomial flags, advancing understanding of $E_8$ trigonometric models.

Findings

Hamiltonian expressed with polynomial coefficients in new variables

Preserves two infinite polynomial flags marked by Weyl vectors and highest root

Explicit examples of eigenfunctions are provided

Abstract

It is shown that the $E_8$ trigonometric Olshanetsky-Perelomov Hamiltonian, when written in terms of the Fundamental Trigonometric Invariants (FTI), is in algebraic form, i.e., has polynomial coefficients, and preserves two infinite flags of polynomial spaces marked by the Weyl (co)-vector and $E_8$ highest root (both in the basis of simple roots) as characteristic vectors. The explicit form of the Hamiltonian in new variables has been obtained both by direct calculation and by means of the orbit function technique. It is shown a triangularity of the Hamiltonian in the bases of orbit functions and of algebraic monomials ordered through Weyl heights. Examples of first eigenfunctions are presented.