Root polytopes and Borel subalgebras

TL;DR

This paper provides a uniform explicit description of root polytopes associated with finite crystallographic root systems, analyzes their face structures, and explores connections with Borel subalgebras in Lie algebras, along with enumerative results.

Contribution

It offers a new explicit description of root polytopes and their face structures, linking them to Borel subalgebras in a uniform way across all types.

Findings

Explicit description of root polytopes for all finite crystallographic root systems

Analysis of the face lattice and combinatorial structure of these polytopes

Enumerative results related to the faces and structure of the polytopes

Abstract

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.