Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines. (Growth series and Ehrhart polynomials associated to root lattices)
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.
scientific article; zbMATH DE number 1314942
scientific article