Extreme lattices and vexillar designs

2008/12/14 by Meyer, Bertrand
#05B30 #14M15 #20C15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0812.2659

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 (E₈, \La₂₄, Barnes-Wall lattices, Thompson-Smith lattice for instance) that were out of reach until now.

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