Lattice Polytopes and Root Systems

Details Lattice Polytopes and Root Systems Lattice Polytopes and Root Systems

2006-09-28

arxiv.org/abs/math/0609809v1

Mathematics

Combinatorics

14 pages, 1 Figure, 1 Table

Scientific paper

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.

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