Sutherland Models for Complex Reflection Groups

TL;DR

This paper extends integrable Sutherland models to complex reflection groups, introduces internal degrees of freedom for dynamical spin chains, and demonstrates their integrability using new Dunkl operators.

Contribution

It associates integrable Sutherland models to complex reflection groups and develops Dunkl operators for these systems, expanding the scope beyond real reflection groups.

Findings

Constructed integrable models for complex reflection groups.

Introduced dynamical spin chains with internal degrees of freedom.

Proved integrability using new Dunkl operators for wreath products.

Abstract

There are known to be integrable Sutherland models associated to every real root system -- or, which is almost equivalent, to every real reflection group. Real reflection groups are special cases of complex reflection groups. In this paper we associate certain integrable Sutherland models to the classical family of complex reflection groups. Internal degrees of freedom are introduced, defining dynamical spin chains, and the freezing limit taken to obtain static chains of Haldane-Shastry type. By considering the relation of these models to the usual BC_N case, we are led to systems with both real and complex reflection groups as symmetries. We demonstrate their integrability by means of new Dunkl operators, associated to wreath products of dihedral groups.