There are no extremal eutactic stars other than root systems

TL;DR

This paper proves that the only extremal eutactic stars on integral lattices are root systems, confirming a conjecture and clarifying the structure of such configurations.

Contribution

It establishes that all extremal eutactic stars are derived from root systems, answering a question posed by Skoruppa.

Findings

Root systems are the only extremal eutactic stars.

The result confirms the conjecture linking extremal eutactic stars to root systems.

The paper provides a complete classification of extremal eutactic stars.

Abstract

A eutactic star on an integral lattice is called extremal if it induces a holomorphic Jacobi form of lattice index and singular weight via the theta block. The famous Macdonald identities imply that root systems are extremal as eutactic stars. In this paper we prove that every extremal eutactic star arises as a root system. This answers a question posed by Skoruppa.