Generalised Hermite Constants, Voronoi Theory and Heights on Flag Varieties

TL;DR

This paper investigates generalized Hermite constants through heights on flag varieties, introducing notions of extremality, and establishing bounds and relations with reduction theories to advance understanding in lattice geometry.

Contribution

It develops a height function on flag varieties and introduces concepts of perfection and eutaxy for extremality, extending classical Hermite constant theory.

Findings

Defined a natural height on flag varieties.

Introduced notions of perfection and eutaxy for extremality.

Established bounds and relations with reduction theories.

Abstract

This paper explores the study of the general Hermite constant associated to the general linear group and its irreducible representations, as defined by T. Watanabe. To that end, a height, which naturally applies to flag varieties, is built and notions of perfection and eutaxy characterising extremality are introduced. Finally we acquaint some relations (e.g. with Korkine--Zolotareff reduction), upper bounds and computation relative to these constants.