 
Summary: ROOT POLYTOPES AND GROWTH SERIES OF ROOT LATTICES
FEDERICO ARDILA, MATTHIAS BECK, SERKAN HOS¸TEN, JULIAN PFEIFLE,
AND KIM SEASHORE
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 compute their fand hvectors. This
leads us to recover formulae for the growth series of these root lattices, which were first
conjectured by ConwayMallowsSloane and BaakeGrimm and proved by ConwaySloane
and Bacherde la HarpeVenkov.
1. Introduction
A lattice L is a discrete subgroup of Rn 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 = {a1, . . . , ar} 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 Rn×r, and to make the connection between L and M
more transparent, we refer to the lattice generated by M as LM . The word length of u
with respect to M, denoted w(u), is min( ci) taken over all expressions u = ciai with
ci Z0. 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) := k0 S(k) xk. It is a rational function G(x) = h(x)
