Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines

Roland Bacher; Pierre de La Harpe; Boris Venkov

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 3, page 727-762
  • ISSN: 0373-0956

Abstract

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Given a root system R in one of the families A, B, C, D, F, G and the free abelian group that it generates, we compute explicitly the growth series of this group with respect to R . The results can be interpreted in terms of the Ehrhart polynomial of the convex hull of R .

How to cite

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Bacher, Roland, La Harpe, Pierre de, and Venkov, Boris. "Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines." Annales de l'institut Fourier 49.3 (1999): 727-762. <http://eudml.org/doc/75360>.

@article{Bacher1999,
abstract = {Étant donnés un système de racines $R$ d’une des familles A, B, C, D, F, G et le groupe abélien libre qu’il engendre, on calcule explicitement la série de croissance de ce groupe relativement à $R.$ Les résultats s’interprètent en termes du polynôme d’Ehrhart de l’enveloppe convexe de $R$.},
author = {Bacher, Roland, La Harpe, Pierre de, Venkov, Boris},
journal = {Annales de l'institut Fourier},
keywords = {Coxeter groups; Ehrhart polynomials; root systems; words in groups},
language = {fre},
number = {3},
pages = {727-762},
publisher = {Association des Annales de l'Institut Fourier},
title = {Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines},
url = {http://eudml.org/doc/75360},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Bacher, Roland
AU - La Harpe, Pierre de
AU - Venkov, Boris
TI - Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 3
SP - 727
EP - 762
AB - Étant donnés un système de racines $R$ d’une des familles A, B, C, D, F, G et le groupe abélien libre qu’il engendre, on calcule explicitement la série de croissance de ce groupe relativement à $R.$ Les résultats s’interprètent en termes du polynôme d’Ehrhart de l’enveloppe convexe de $R$.
LA - fre
KW - Coxeter groups; Ehrhart polynomials; root systems; words in groups
UR - http://eudml.org/doc/75360
ER -

References

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  1. [BaG] M. BAAKE et U. GRIMM, Coordination sequences for root lattices and related graphs, Zeitschrift für Kristallographie, 212 (1997), 253-256. Zbl0921.05058MR2000h:05110
  2. [BHV] R. BACHER, P. de la HARPE et B. VENKOV, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 Sér. I (1997), 1137-1142. Zbl0917.52006MR98k:52029
  3. [Ben] M. BENSON, Growth series of finite extensions of ℤn are rational, Inventiones Math., 73 (1983), 251-269. Zbl0498.20022MR85e:20026
  4. [Bil] N. BILLINGTON, Growth of groups and graded algebras : erratum, Communications in Algebra, 13(3) (1985), 753-755. Zbl0558.20024MR86e:20039b
  5. [Bou] N. BOURBAKI, Groupes et algèbres de Lie (chapitres 4, 5 et 6), Hermann, 1968. 
  6. [Br1] M. BRION, Points entiers dans les polytopes convexes, Séminaire Bourbaki, mars 1994, Astérisque 227, Soc. Math. France (1995), 145-169. Zbl0847.52015MR96e:11123
  7. [Br2] M. BRION, Polytopes convexes entiers, Gazette des mathématiciens, 67 (janvier 1996), 21-42. Zbl0878.52005MR97d:52022
  8. [CoG] J.H. CONWAY et R.K. GUY, The book of numbers, Springer, 1996. Zbl0866.00001MR98g:00004
  9. [CS1] J.H. CONWAY et N.J.A. SLOANE, The cell structure of certain lattices, in “Miscellanea Mathematics”, édité par P. Hilton, F. Hirzebruch et R. Remmert, Springer (1991), 72-107. Zbl0738.52014
  10. [CS2] J.H. CONWAY et N.J.A. SLOANE, Sphere packings, lattices and groups (second edition), Springer, 1993. 
  11. [CS3] J.H. CONWAY et N.J.A. SLOANE, Low-dimensional lattices. VII Coordination sequences, Proc. R. Soc. Lond., Ser. A, 453 (1997), 2369-2389. Zbl1066.11505MR98j:11051
  12. [Cox] H.S.M. COXETER, The polytopes with regular-prismatic vertex figures, Phil. Trans. Royal Soc. of London, Ser. A, 229 (1930), 329-425. Zbl56.1119.03JFM56.1119.03
  13. [Ebe] W. EBELING, Lattices and codes (a course partially based on lectures by F. Hirzebruch), Vieweg, 1994. Zbl0805.11048
  14. [Ehr] EHRHART, Sur un problème de géométrie diophantienne linéaire, J. reine angew. Math., 226 (1967), 1-29. Zbl0155.37503MR35 #4184
  15. [GKP] R.L. GRAHAM, D.E. KNUTH et O. PATASHNIK, Concrete mathematics, Addison-Wesley, 1991. 
  16. [HiC] D. HILBERT and S. COHN-VOSSEN, Geometry and the imagination, Chelsea Publishing Company, 1952 and 1983. Zbl0047.38806
  17. [Kla] D.A. KLARNER, Mathematical crystal growth II, Discrete Applied Math., 3 (1981), 113-117. Zbl0466.05007MR83a:05018
  18. [KoM] A.I. KOSTRIKIN and Yu. I. MANIN, Linear algebra and geometry, Gordon and Breach, 1989. 
  19. [Pa1] D.I. PANYUSHEV, Multiplicities of weights of some representations and convex polytopes, Functional Analysis and its Applications, 28-4 (1994), 293-295. Zbl0866.22015MR95k:22017
  20. [Pa2] D.I. PANYUSHEV, Cones of highest weight vectors, weight polytopes, and Lusztig's q-analog, Transformation Groups, 2 (1997), 91-115. Zbl0891.22013MR98j:22017
  21. [PoS] G. PÓLYA and G. SZEGÖ, Problems and theorems in analysis, volume I and II, Springer, 1972 and 1976. 
  22. [Ser] J-P. SERRE, Cours d'arithmétique, P.U.F., 1970. Zbl0225.12002MR41 #138
  23. [Sta] R.P. STANLEY, Enumerative combinatorics, Volume I, Wadsworth and Brooks, 1986. Zbl0608.05001MR87j:05003

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