Polytopes of roots of type AN

Abstract

Polytopes of roots of type An−1 are investigated, which we call Pn. The polytopes, , of positive roots and the origin have been considered in relation to other branches of mathematics [4]. We show that exactly n copies of forms a disjoint cover of Pn. Moreover, those n copies of can be obtained by letting the elements of a subgroup of the symmetric group Sn generated by an n-cycle act on . We also characterise the faces of Pn and some facets of , which we believe to be useful in some optimisation problems. As by-products, we obtain an interesting identity on the number of lattice paths and a triangulation of the product of two simplices.