A Correspondence between Generic Alcoved Polytopes and Subdivisions of the Root Polytope of Lie Type A

Alcoved polytopes of Lie type A are polytopes whose facets are orthognal to the roots of root system An−1. An alcoved polytope of type An−1 is generic if it has two facets orthogonal to each root in An−1. In this paper, we prove that there is one-to-one correspondence between equivalence classes of generic alcoved polytopes of type An−1 and regular central subdivisions of the root polytope of An−1. We apply this result to give an explicit classification for small n.