Certain imprimitive reflection groups and their generic versions
HTML articles powered by AMS MathViewer
- by Jian-yi Shi PDF
- Trans. Amer. Math. Soc. 354 (2002), 2115-2129 Request permission
Abstract:
The present paper is concerned with the connection between the imprimitive reflection groups $G(m,m,n)$, $m\in \mathbb {N}$, and the affine Weyl group $\widetilde {A}_{n-1}$. We show that $\widetilde {A}_{n-1}$ is a generic version of the groups $G(m,m,n)$, $m\in \mathbb {N}$. We introduce some new presentations of these groups which are shown to have some group-theoretic advantages. Then we define the Hecke algebras of these groups and of their braid versions, each in two ways according to two presentations. Finally we give a new description for the affine root system $\overline {\Phi }$ of $\widetilde {A}_{n-1}$ such that the action of $\widetilde {A}_{n-1}$ on $\overline {\Phi }$ is compatible with that of $G(m,m,n)$ on its root system in some sense.References
- Kirsten Bremke and Gunter Malle, Root systems and length functions, Geom. Dedicata 72 (1998), no. 1, 83–97. MR 1644163, DOI 10.1023/A:1005082508279
- Michel Broué, Gunter Malle, and Raphaël Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127–190. MR 1637497
- Arjeh M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379–436. MR 422448, DOI 10.24033/asens.1313
- M. C. Hughes, Complex reflection groups, Comm. Algebra 18 (1990), no. 12, 3999–4029. MR 1084439, DOI 10.1080/00927879008824120
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR 573434
- George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
- Jian Yi Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, Lecture Notes in Mathematics, vol. 1179, Springer-Verlag, Berlin, 1986. MR 835214, DOI 10.1007/BFb0074968
- J. Y. Shi, Conjugacy relation for the Coxeter elements in certain Coxeter groups, Adv. in Math. 161 (2001), 1-19.
Additional Information
- Jian-yi Shi
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556; Department of Mathematics, East China Normal University, Shanghai, 200062, P.R.C.
- MR Author ID: 231063
- Email: jyshi@math.ecnu.edu.cn
- Received by editor(s): November 9, 1999
- Received by editor(s) in revised form: May 24, 2001
- Published electronically: January 7, 2002
- Additional Notes: Supported partly by University of Notre Dame, and partly by the National Science Foundation of China, the Science Foundation of the University Doctorial Program of CNEC and the City Foundation for the Selected Academic Research of Shanghai.
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 2115-2129
- MSC (2000): Primary 20F55
- DOI: https://doi.org/10.1090/S0002-9947-02-02941-0
- MathSciNet review: 1881032
Dedicated: Dedicated to Professor Cao Xi-hua on his 80th birthday