Geometry of the Jantzen region in Lusztig’s conjecture
HTML articles powered by AMS MathViewer
- by Brian D. Boe PDF
- Math. Comp. 70 (2001), 1265-1280 Request permission
Abstract:
The Lusztig Conjecture expresses the character of a finite-dimensional irreducible representation of a reductive algebraic group $G$ in prime characteristic as a linear combination of characters of Weyl modules for $G$. The representations described by the conjecture are in one-to-one correspondence with the (finitely many) alcoves in the intersection of the dominant cone and the so-called Jantzen region. Each alcove has a length, defined to be the number of alcove walls (hyperplanes) separating it from the fundamental alcove (the unique alcove in the dominant cone whose closure contains the origin). This article determines the maximum length of an alcove in the intersection of the dominant cone with the Jantzen region.References
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- 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 [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- 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
- Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
- George Lusztig, Hecke algebras and Jantzen’s generic decomposition patterns, Adv. in Math. 37 (1980), no. 2, 121–164. MR 591724, DOI 10.1016/0001-8708(80)90031-6
- George Lusztig, Some problems in the representation theory of finite Chevalley groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 313–317. MR 604598
- M. Schönert et. al., GAP — Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, fifth edition, 1995.
- Wolfgang Soergel, Conjectures de Lusztig, Astérisque 237 (1996), Exp. No. 793, 3, 75–85 (French, with French summary). Séminaire Bourbaki, Vol. 1994/95. MR 1423620
Additional Information
- Brian D. Boe
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602-7403
- Email: brian@math.uga.edu
- Received by editor(s): May 22, 1998
- Received by editor(s) in revised form: July 6, 1999
- Published electronically: March 14, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 70 (2001), 1265-1280
- MSC (2000): Primary 20G05; Secondary 20F55, 51F15
- DOI: https://doi.org/10.1090/S0025-5718-00-01220-5
- MathSciNet review: 1709146