Trigonometric $\vee$-systems and solutions of WDVV equations

TL;DR

This paper explores a class of trigonometric solutions to WDVV equations linked to vector collections with multiplicities, introducing new solutions via restrictions and analyzing their properties, including connections to root systems and Coxeter numbers.

Contribution

It introduces a new framework for trigonometric $ vee$-systems, demonstrates how to generate solutions through restrictions, and extends the understanding of Coxeter numbers in this context.

Findings

New solutions from restrictions of root systems

Subsystems of trigonometric $ vee$-systems are also $ vee$-systems

Determined a generalized Coxeter number for Weyl group representations

Abstract

We consider a class of trigonometric solutions of WDVV equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. Further, we consider a closely related notion of a trigonometric $\vee$-system and we show that their subsystems are also trigonometric $\vee$-systems. Finally, while reviewing the root system case we determine a version of (generalised) Coxeter number for the exterior square of the reflection representation of a Weyl group.