Root polytopes and Borel subalgebras. (English) Zbl 1337.17011

Cellini, Paola; Marietti, Mario

Int. Math. Res. Not. 2015, No. 12, 4392-4420 (2015).

Summary: Let \(\Phi\) be a finite crystallographic irreducible root system and \({\mathcal P}_{\Phi}\) be the convex hull of the roots in \(\Phi\). We give a uniform explicit description of the polytope \({\mathcal P}_{\Phi}\), analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results.

MSC:

17B22	Root systems
17B20	Simple, semisimple, reductive (super)algebras
52B05	Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)