Extreme lattices and vexillar designs

TL;DR

This paper introduces vexillar designs on flag varieties, linking group actions to lattice extremality, and demonstrates how these concepts identify new extreme lattices like E8 and Barnes-Wall.

Contribution

It defines vexillar designs on flag varieties and connects group orbit conditions to lattice extremality, enabling the identification of new extreme lattices.

Findings

Vexillar designs generalize spherical designs to flag varieties.

Conditions on group orbits imply lattice extremality.

New examples of extreme lattices are identified, including E8 and Barnes-Wall.

Abstract

We define a notion of vexillar design for the flag variety in the spirit of the spherical designs introduced by Delsarte, Goethals and Seidel. For a finite subgroup of the orthogonal group, we explain how conditions on the group have the orbits of any flag under the group action be a design and point out why the minima of a lattice in the sense of the general Hermite constant forming a 4-design implies being extreme. The reasoning proves useful to show the extremality of many new expected examples ($E_8$, $\La_{24}$, Barnes-Wall lattices, Thompson-Smith lattice for instance) that were out of reach until now.