Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines. (Growth series and Ehrhart polynomials associated to root lattices)
type: Item
instance of: scholarly article
title: Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines. (Growth series and Ehrhart polynomials associated to root lattices)
zbMATH Open document ID: 0920.05076
author: Roland Bacher
author: Pierre de la Harpe
author: B. Venkov
publication date: 1999-07-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75360
review text: For a group \(\Gamma\) of finite type and a finite system of generators \(S\) for \(\Gamma\) one can define the length \(l(\gamma)\) of \(\gamma \in \Gamma\) as the smallest length of a word in the elements of \(S \cup S^{-1}\) that represents \(\gamma\). Then one can define the growth series as the formal power series \[ \Sigma(\Gamma,S,z)=\sum_{\gamma \in \Gamma} z^{l(\gamma)}=\sum_{k=0}^{\infty} \sigma(k) z^k, \] where \(\sigma(k)=|\{ \gamma \in \Gamma\mid l(\gamma)=k\}|\). The purpose of this paper is to calculate explicitly \(\Sigma(\Gamma,S,z)\) for the case that \(\Gamma\) is the abelian group generated by a root system of the classical families \(A_n\), \(B_n\), \(C_n\), and \(D_n\). The authors exhibit a beautiful connection of the above with lattice point counting inside dilations of convex polytopes.
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 52B20
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 11H06
zbMATH DE Number: 1314942
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Ehrhart polynomials
zbMATH Keywords: root systems
zbMATH Keywords: words in groups
MaRDI profile type: Publication
OpenAlex ID: W2070338493
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DOI: 10.5802/AIF.1689
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1314942