Images of regular unipotent elements from subsystem subgroups of type A2 and B2 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. Bibliography: 17 titles.
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References
N. Bourbaki, Groupes et Algèbres de Lie, Chaps. VII–VIII, Paris, Hermann (1975).
J. C. Jantzen, Representations of Algebraic Groups, 2nd edition, AMS, Providence (2003).
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).
A. A. Osinovskaya, “Restrictions of irreducible representations of classical algebraic groups to root A 1-subgroups,” Commun. Algebra, 31, 2357–2379 (2003).
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).
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).
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).
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).
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).
S. Smith, “Irreducible modules and parabolic subgroups,” J. Algebra, 75, 286–289 (1982).
R. Steinberg, Lectures on Chevalley groups, Yale Univ. Math. Dept., New Haven (1968).
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).
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).
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).
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).
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).
D. P. Želobenko, “Classical groups. Spectral analysis of finite-dimensional representations,” Usp. Mat. Nauk, 17, No. 1, 27–120 (1962).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 159–178.
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Osinovskaya, A.A. Regular unipotent elements from subsystem subgroups of rank 2 in modular representations of classical groups. J Math Sci 156, 943–953 (2009). https://doi.org/10.1007/s10958-009-9300-y
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DOI: https://doi.org/10.1007/s10958-009-9300-y