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arXiv · 1207.3429

Root polytopes and abelian ideals

Paola Cellini·Mario Marietti
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Abstract

We study the root polytope $\mathcal P_\Phi$ of a finite irreducible crystallographic root system $\Phi$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\Phi$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of $\mathcal P_\Phi$ and analyze its relation with the facets of $\mathcal P_\Phi$. For $\Phi$ of type $A_n$ or $C_n$, we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of $\mathcal P_\Phi$. We show that this triangulation restricts to a triangulation of the positive root polytope $\mathcal P_\Phi^+$.

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Paola Cellini, Mario Marietti. 2012-07-14. Root polytopes and abelian ideals. https://doi.org/10.1007/s10801-013-0458-5

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