Polynomials Associated with Equilibrium Positions in Calogero-Moser Systems

TL;DR

This paper introduces new polynomials linked to equilibrium positions in Calogero-Moser systems for exceptional and non-crystallographic root systems, extending known classical polynomial relationships.

Contribution

It defines and derives new polynomials for exceptional and non-crystallographic root systems, expanding the mathematical framework of Calogero-Moser systems.

Findings

Polynomials for exceptional root systems are characterized.

These polynomials share properties with classical orthogonal polynomials.

They do not possess orthogonality but have similar structural features.

Abstract

In a previous paper (Corrigan-Sasaki), many remarkable properties of classical Calogero and Sutherland systems at equilibrium are reported. For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". The equilibrium positions of Calogero and Sutherland systems for the classical root systems (A_r, B_r, C_r and D_r) correspond to the zeros of Hermite, Laguerre, Jacobi and Chebyshev polynomials. Here we define and derive the corresponding polynomials for the exceptional (E_6, E_7, E_8, F_4 and G_2) and non-crystallographic (I_2(m), H_3 and H_4) root systems. They do not have orthogonality but share many other properties with the above mentioned classical polynomials.