Root Polytopes and Growth Series of Root Lattices

Abstract

The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices An, Cn, and Dn, and we compute their f -and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway, Mallows, and Sloane and Baake and Grimm and were proved by Conway and Sloane and Bacher, de la Harpe, and Venkov. We also prove the formula for the growth series of the root lattice Bn, which requires a modification of our technique. Introduction.A lattice L is a discrete subgroup of R n for some n ∈ Z >0 . The rank of a lattice is the dimension of the subspace spanned by the lattice. We say that a lattice L is generated as a monoid by a finite collection of vectors M = {a 1 , . . . , a r } if each u ∈ L is a nonnegative integer combination of the vectors in M. For convenience, we often write the vectors from M as columns of a matrix M ∈ R n×r , and to make the connection between L and M more transparent, we refer to the lattice generated by M as L M . The word length of u with respect to M, denoted w(u), is min( c i ) taken over all expressions u = c i a i with c i ∈ Z ≥0 . The growth function S(k) counts the number of elements u ∈ L with word length w(u) = k with respect to M. We define the growth series to be the generating function G(x) := k≥0 S(k) x k . It is a

Root Polytopes and Borel Subalgebras

“…Finally, we are going to prove that we can find α ∈ Φ which is simultaneously not orthogonal to V 1 and V 2 : this will conclude the proof, since contradicts (1). By the previous step, and Proposition 2.3, we can find three subsets Π, Π 1 , Π 2 ⊆ Φ, which are respectively bases of Φ, Φ 1 and Φ 2 , and such that Π 1 , Π 2 ⊆ Π.…”

“…In this paper we only consider the complete root polytope; our results have direct applications to the study of the positive root polytopes of types A n and C n (see [4]). In [1], some properties of the complete root polytopes are provided for the classical types through a case by case analysis, using the usual coordinate descriptions of the root systems. In our paper, we give case free statements and proofs for all types.…”

Interlacing Ehrhart polynomials of reflexive polytopes

“…Indeed, other reflexive polytopes exhibit similar properties. In particular, among other families, we present proofs for classical root polytopes of type C [1] and the duals of Stasheff polytopes [18].…”

“…Furthermore, for c > −2 there is one zero of f + (c − 2)g with imaginary part larger than Im(β 2k−2 ) and with imaginary part smaller than Im(β 1 …”

Roots of Ehrhart polynomials arising from graphs

“…, ±σ(e n )}. Note that if G is the complete graph K d , the symmetric edge polytope P ± K d coincides with the root polytope of the lattice A d defined in [1].…”