THE JORDAN BLOCK STRUCTURE OF THE IMAGES OF UNIPOTENT ELEMENTS IN IRREDUCIBLE MODULAR REPRESENTATIONS OF CLASSICAL ALGEBRAICAL GROUPS OF SMALL DIMENSIONS
Abstract
For unipotent elements of prime order, the Jordan block structure of their images in the infinitesimally irreducible representations of the classical algebraic groups in odd characteristic, whose dimensions are at most 100, is determined. The approach proposed can be applied for solving a similar problem for representations of larger dimensions. Detailed information on small cases is important for stating reasonable conjectures on the behavior of unipotent elements in irreducible representations of the classical algebraic groups.
About the Authors
T. S. BUSEL
Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk
Belarus
Belarus
I. D. SUPRUNENKO
Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk
Belarus
Belarus
References
1. Seitz, G. M. Unipotent elements, tilting modules, and saturation / G. M. Seitz // Invent. Math. – 2000. – Vol. 141, N 3. – P. 467–502.
2. Velichko, M. V. On the behaviour of the root elements in irreducible representations of simple algebraic groups / M. V. Velichko // Тр. Ин-та математики. – 2005. – T. 13, № 2. – C. 116–121.
3. Jantzen, J. C. Representations of algebraic groups (2 ed.) / J. C. Jantzen // Mathematical Surveys and Monographs. – 2003. – Vol. 107. – 576 p.
4. Залесский, А. Е. Срезанные симметрические степени естественных реализаций групп SLm(P) и Spm(P) и их ограничения на подгруппы / А. Е. Залесский, И. Д. Супруненко // Сиб. математ. журн. – 1990. – T. 31, № 4. – C. 33–46.
5. Seitz, G. M. The maximal subgroups of classical algebraic groups / G. M. Seitz // Memoirs Amer. Math. Soc. – 1987. – Vol. 67, N 365. – P. iv–286.
6. Супруненко, И. Д. О блочной структуре регулярных унипотентных элементов из подсистемных подгрупп типа A1 × A2 в представлениях специальной линейной группы / И. Д. Супруненко // Зап. научн. семин. ПОМИ. – 2011. – Т. 388. – С. 247–269.
7. Donkin, S. On tilting modules for algebraic groups / S. Donkin // Math. Zeits. – 1993. – Vol. 212. – P. 39–60.
8. Залесский, А. Е. Представления размерности (pn + 1) / 2 симплектической группы степени 2n над полем характеристики р / А. Е. Залесский, И. Д. Супруненко // Вести АН БССР. Сер. физ.- .- мат. наук. – 1987. – № 6. – Р. 9–15.
9. Lubeck, F. Small degree representations of finite Chevalley groups in defining characteristic / F. Lubeck // LMS // J. Comput. Math. – 2001. – Vol. 4. – P. 135–169.
10. Suprunenko, I. D. The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic / I. D. Suprunenko // Memoirs Amer. Math. Soc. – 2009. – Vol. 200, N 939. – 154 p.
11. Seitz, G. M. Unipotent and nilpotent classes in simple algebraic groups and Lie algebras / G. M. Seitz, M. W. Liebeck // Mathematical Surveys and Monographs. – 2012. – Vol. 180. – 380 p.
12. Хамфриc, Дж. Введение в теорию алгебр Ли и их представлений / Дж. Хамфрис. – М.: МЦНМО, 2003. – 216 с.
13. Humphreys, J. Modular representations of finite groups of Lie types / J. Humphreys. – UK: LMS Lecture Note Series 326, 2006. – 233 p.
Review
Views: 868
Email this article 