  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.
