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Séries de croissance et polynômes d’Ehrhart associés aux réseaux de racines. (Growth series and Ehrhart polynomials associated to root lattices). (French) Zbl 0920.05076

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.

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
20F65 Geometric group theory
11H06 Lattices and convex bodies (number-theoretic aspects)
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