Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariable Polynomials. II. $E_7$ case

TL;DR

This paper demonstrates that the $E_7$ trigonometric Olshanetsky-Perelomov Hamiltonian, expressed via Fundamental Trigonometric Invariants, is algebraic and preserves a specific polynomial space structure, advancing understanding of multivariable polynomial models.

Contribution

It shows the algebraic form of the $E_7$ Hamiltonian in terms of FTI and identifies its polynomial space preservation, linking trigonometric and rational models.

Findings

Hamiltonian has polynomial coefficients in FTI variables

Preserves a specific polynomial flag with characteristic vector

Aligns with the minimal characteristic vector for the rational model

Abstract

It is shown that the $E_7$ 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 the infinite flag of polynomial spaces with the characteristic vector $\vec \alpha = (1,2,2,2,3,3,4)$. Its flag coincides with one of the minimal characteristic vector for the $E_7$ rational model.