The construction of trigonometric invariants for Weyl groups and the derivation of corresponding exactly solvable Sutherland models

TL;DR

This paper develops a method to construct trigonometric invariants for Weyl groups, enabling the formulation of exactly solvable Sutherland models, with a specific application to the F4 case.

Contribution

It introduces a new algorithmic approach to generate polynomial bases of trigonometric invariants for Weyl groups, facilitating the creation of solvable models.

Findings

Constructed trigonometric invariants for Weyl group orbits.

Derived exactly solvable Sutherland models using these invariants.

Applied the method specifically to the F4 Weyl group case.

Abstract

Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the coroot lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The invariants of the basis can be used as coordinates in any cell of the coroot space and lead to an exactly solvable model of Sutherland type. We apply this construction to the $F_4$ case