Polytopes of roots of type A_N

Abstract

Polytopes of roots of type A n -i are investigated, which we call P n . The polytopes, F+, of positive roots and the origin have been considered in relation to other branches of mathematics [4], We show that exactly n copies of P* forms a disjoint cover of P n . Moreover, those n copies of P+ can be obtained by letting the elements of a subgroup of the symmetric group S n generated by an n-cycle act on P+. We also characterise the faces of P n and some facets of P+, 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.

Simion’s Type B Associahedron is a Pulling Triangulation of the Legendre Polytope

“…, ±2e n+1 is an (n + 1)-dimensional cross-polytope. The intersection of this cross-polytope with the hyperplane x 1 + x 2 + · · · + x n+1 = 0 is an n-dimensional centrally symmetric polytope P n first studied by Cho [7]. It is called the Legendre polytope in the work of Hetyei [13], since the polynomial n j=0 f j−1 · ((x − 1)/2) j is the nth Legendre polynomial, where f i is the number of i-dimensional faces in any pulling triangulation of the boundary of P n .…”

“…The type A root polytope P + n is the convex hull of the origin and the set of points {e i − e j : 1 ≤ i < j ≤ n + 1}. Cho [7] gave a decomposition of the Legendre polytope P n into n + 1 copies of P + n as follows. The symmetric group S n+1 acts on the Euclidean space R n+1 by permuting the coordinates, that is, the permutation σ ∈ S n+1 sends the basis vector e i into e σ(i) .…”

“…Hence the permutation σ acts on the Legendre polytope P n by sending each e i − e j into e σ(i) − e σ(j) . Cho's main result [7,Theorem 16] is the following decomposition.…”

“…For the valid set of arrows A shown in Figure 2, in the first step we add the letters H 2 and H 7 to M (A). In the second step we add the letters D 3 , U 9 , D 4 , and U 7 to M (A), and we remove the forward arrows (3,9) and (4,7). We also replace (3,8) with (9,8) and (4, 5) with (7,5).…”

GIT stability of linear maps on projective space with marked points

“…A point p is GIT stable if and only if its weight polytope contains the origin. We find that the weight polytopes arising from the conjugation action of SL N +1 on P(Mat N +1 ) are root polytopes of A type [5]. The vertices of these polytopes are root vectors of the A N lattice.…”