Regular Lattice Polytopes and Root Systems

Nicolas Ressayre (1) , Pierre-Louis Montagard (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Consider a lattice in a real finite dimensional vector space. Here, we are interested in the lattice polytopes, that is the convex hulls of finite subsets of the lattice. Consider the group $G$ of the affine real transformations which map the lattice onto itself. Replacing the group of euclidean motions by the group $G$ one can define the notion of regular lattice polytopes. More precisely, a lattice polytope is said to be regular if the subgroup of $G$ which preserves the polytope acts transitively on the set of its complete flags. Recently, Karpenkov obtained a classification of the regular lattice polytopes. Here we obtain this classification by a more conceptual method. Another difference is that Karpenkov uses in an essential way the classification of the euclidean regular polytopes, but we don't.

Domaines: Combinatoire [math.CO]; Géométrie algébrique [math.AG]

Nicolas Ressayre, Pierre-Louis Montagard. Regular Lattice Polytopes and Root Systems. Bulletin of the London Mathematical Society, 2009, 41, pp.227-241. ⟨10.1112/blms/bdn120⟩. ⟨hal-00102048⟩