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ROOT POLYTOPES AND GROWTH SERIES OF ROOT LATTICES FEDERICO ARDILA, MATTHIAS BECK, SERKAN HOSTEN, JULIAN PFEIFLE,
 

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 f-and h-vectors. 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)

  

Source: Ardila, Federico - Department of Mathematics, San Francisco State University

 

Collections: Mathematics