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Rationality and reciprocity for the greedy normal form of a Coxeter group

Author: Richard Scott
Journal: Trans. Amer. Math. Soc. 363 (2011), 385-415
MSC (2010): Primary 20F55, 20F10, 05A15
Published electronically: August 30, 2010
MathSciNet review: 2719687
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Abstract: We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series of Coxeter groups as well as some new ones.

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  • 1. Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266 (2006d:05001)
  • 2. N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968. MR 0240238 (39:1590)
  • 3. Brigitte Brink and Robert B. Howlett, A finiteness property and an automatic structure for Coxeter groups, Math. Ann. 296 (1993), no. 1, 179-190. MR 1213378 (94d:20045)
  • 4. W. A. Casselman, Automata to perform basic calculations in Coxeter groups, Representations of groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 35-58. MR 1357194 (96i:20050)
  • 5. N. Changey, Rationalité des séries de croissance complète des groupes de Coxeter, Mémoire DEA, 1997.
  • 6. Ruth Charney and Michael Davis, Reciprocity of growth functions of Coxeter groups, Geom. Dedicata 39 (1991), no. 3, 373-378. MR 1123152 (92h:20067)
  • 7. Michael W. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. (2) 117 (1983), no. 2, 293-324. MR 690848 (86d:57025)
  • 8. -, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008. MR 2360474
  • 9. Michael W. Davis, Jan Dymara, Tadeusz Januszkiewicz, Boris Okun, Weighted $ L\sp 2$-cohomology of Coxeter groups, Geom. Topol. 11 (2007), 47-138. MR 2287919 (2008g:20084)
  • 10. Jan Dymara, Thin buildings, Geom. Topol. 10 (2006), 667-694 (electronic). MR 2240901 (2007h:20027)
  • 11. William J. Floyd and Steven P. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88 (1987), no. 1, 1-29. MR 877003 (88m:22023)
  • 12. R. Glover and R. Scott, Automatic growth series for right-angled Coxeter groups, Involve 2 (2009), no. 4, 370-384. MR 2579557
  • 13. R. Grigorchuk and P. de la Harpe, On problems related to growth, entropy, and spectrum in group theory, J. Dynam. Control Systems 3 (1997), no. 1, 51-89. MR 1436550 (98d:20039)
  • 14. Rostislav Grigorchuk and Tatiana Nagnibeda, Complete growth functions of hyperbolic groups, Invent. Math. 130 (1997), no. 1, 159-188. MR 1471889 (98i:20038)
  • 15. James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • 16. F. Liardet, Croissance dans les groupes virtuellement abéliens, Thèse, University of Genève, 1996.
  • 17. M. J. Mamaghani, Complete growth series of Coxeter groups with more than three generators, Bull. Iranian Math. Soc. 29 (2003), no. 1, 65-76, 88. MR 2046307
  • 18. Graham Niblo and Lawrence Reeves, Groups acting on $ {\rm CAT}(0)$ cube complexes, Geom. Topol. 1 (1997), approx. 7 pp. (electronic). MR 1432323 (98d:57005)
  • 19. Christophe Reutenauer, A survey of noncommutative rational series, Formal power series and algebraic combinatorics (New Brunswick, NJ, 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 159-169. MR 1363511 (96k:05012)
  • 20. Arto Salomaa and Matti Soittola, Automata-theoretic aspects of formal power series, Springer-Verlag, New York, 1978, Texts and Monographs in Computer Science. MR 0483721 (58:3698)
  • 21. M. P. Schützenberger, On the definition of a family of automata, Information and Control 4 (1961), 245-270. MR 0135680 (24:B1725)
  • 22. Jean-Pierre Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 77-169. MR 0385006 (52:5876)
  • 23. Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997, With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original. MR 1442260 (98a:05001)
  • 24. Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728 (37:6288)

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Additional Information

Richard Scott
Affiliation: Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053

Received by editor(s): October 22, 2008
Received by editor(s) in revised form: April 21, 2009
Published electronically: August 30, 2010
Additional Notes: The author thanks MSRI for its support and hospitality during the writing of this paper.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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