Osinovskaya, A. A. Regular unipotent elements from subsystem subgroups of rank 2 in modular representations of classical groups. (English. Russian original) Zbl 1183.20048 J. Math. Sci., New York 156, No. 6, 943-953 (2009); translation from Zap. Nauchn. Semin. POMI 356, 159-178 (2008). Summary: Images of regular unipotent elements from subsystem subgroups of type \(A_2\) and \(B_2\) in irreducible modular representations of classical groups are studied. For images of such elements and representations with locally small highest weights, all sizes of Jordan blocks of one and the same parity are found. Cited in 1 Document MSC: 20G05 Representation theory for linear algebraic groups Keywords:subsystem subgroups; irreducible modular representations; classical groups; highest weights; Jordan blocks × Cite Format Result Cite Review PDF Full Text: DOI References: [1] N. Bourbaki, Groupes et Algèbres de Lie, Chaps. VII–VIII, Paris, Hermann (1975). · Zbl 0329.17002 [2] J. C. Jantzen, Representations of Algebraic Groups, 2nd edition, AMS, Providence (2003). · Zbl 1034.20041 [3] A. A. Osinovskaya, ”Nilpotent elements in irreducible representations of simple Lie algebras of small rank,” Preprint/ Inst. Math. Nat. Acad. Sci. Belarus, No 5(554), Minsk (1999). [4] A. A. Osinovskaya, ”Restrictions of irreducible representations of classical algebraic groups to root A 1-subgroups,” Commun. Algebra, 31, 2357–2379 (2003). · Zbl 1065.20058 · doi:10.1081/AGB-120019001 [5] A. A. Osinovskaya and I. D. Suprunenko, ”On the Jordan block structure of images of some unipotent elements in modular irreducible representations of classical algebraic groups,” J. Algebra, 273, 586–600 (2004). · Zbl 1068.20045 · doi:10.1016/j.jalgebra.2003.06.001 [6] A. A. Osinovskaya, ”On restrictions of modular irreducible representations of algebraic groups of type A n to naturally embedded subgroups of type A 2,” J. Group Theory, 8, 43–92 (2005). · Zbl 1103.20042 · doi:10.1515/jgth.2005.8.1.43 [7] A. A. Osinovskaya, ”Restrictions of representations of algebraic groups of types B n and D n to subgroups of type A 2,” J. Algebra Appl, 4, 467–479 (2005). · Zbl 1104.20046 · doi:10.1142/S0219498805001344 [8] A. A. Osinovskaya and I. D. Suprunenko, ”The block structure of unipotent elements from subsystem sub-groups of type A 3 in special modular representations of groups of type A n ,” Dokl. NAN Belarusi, 51, No. 6, 25–29 (2007). · Zbl 1204.20054 [9] A. A. Premet, ”Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic,” Math USSR-Sb., 61, No. 1, 167–183 (1988). · Zbl 0669.20035 · doi:10.1070/SM1988v061n01ABEH003200 [10] S. Smith, ”Irreducible modules and parabolic subgroups,” J. Algebra, 75, 286–289 (1982). · Zbl 0496.20030 · doi:10.1016/0021-8693(82)90076-X [11] R. Steinberg, Lectures on Chevalley groups, Yale Univ. Math. Dept., New Haven (1968). · Zbl 1196.22001 [12] I. D. Suprunenko, ”Minimal polynomials of elements of order p in irreducible representations of Chevalley groups over fields of characteristic p,” Sib. Advances Math., 6, No. 4, 97–150 (1996). · Zbl 0941.20049 [13] I. D. Suprunenko, ”On Jordan blocks of elements of order p in irreducible representations of classical groups with p-large highest weights,” J. Algebra, 273, 589–627 (1997). · Zbl 0893.20032 · doi:10.1006/jabr.1996.6916 [14] P. H. Tiep and A. E. Zalesskii, ”Mod p reducibility of unramified representations of finite groups of Lie type,” Proc. London Math. Soc., 84, 439–472 (2002). · Zbl 1020.20008 · doi:10.1112/plms/84.2.439 [15] M. V. Velichko, ”On the behavior of root elements in irreducible representations of simple algebraic groups,” Tr. Inst. Mat. Belarusi, 13, No. 2, 116–121 (2005). · Zbl 1165.20315 [16] M. V. Velichko, ”On the behaviour of root elements in the modular representations of symplectic groups,” Tr. Inst. Mat. Belarusi, 14, No. 2, 28–34 (2006). [17] D. P. Želobenko, ”Classical groups. Spectral analysis of finite-dimensional representations,” Usp. Mat. Nauk, 17, No. 1, 27–120 (1962). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.