For p > 2, odd Jordan block sizes of the images of regular unipotent elements from subsystem subgroups of type A2 in irreducible p-restricted representations for groups of type Ar over the field of characteristic p, the weights of which are locally small with respect to p, are found. The weight is called locally small if the double sum of its two neighboring coefficients is less than p. This result is part of a more general program investigating the behavior of unipotent elements in representations of the classical algebraic groups. It can be used to solve recognition problems for representations or linear groups by the presence of certain elements.
Similar content being viewed by others
References
N. Bourbaki, Groupes et Algèbres de Lie, Chaps. VII–VIII, Hermann (1975).
B. Braden, “Restricted representations of classical Lie algebras of type A 2 and B 2 ,” Bull. Amer. Math. Soc., 73, 482–486 (1967).
J. Brundan, A. Kleshchev, and I. Suprunenko, “Semisimple restrictions from GL(n) to GL(n − 1),” J. reine angew. Math., 500, 83–112 (1998).
J. C. Jantzen, Representations of Algebraic Groups, 2-nd ed., Providence (2003).
A. A. Osinovskaya, “Restrictions of irreducible representations of the Lie algebra \( \mathfrak{s}\mathfrak{l} \) 3 to \( \mathfrak{s}\mathfrak{l} \) 2-subalgebras and the Jordan block structure of nilpotent elements,” Vestsi NAN Belarusi, Ser. Fiz.-Mat. Navuk, No. 2, 52–55 (2000).
A. A. Osinovskaya, “On the 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 and I. D. Suprunenko, “On the Jordan block structure of images of some unipotent elements in modular irreducible representations of the classical algebraic groups,” J. Algebra, 273, No. 2, 586–600 (2004).
A. A. Osinovskaya and I. D. Suprunenko, “Unipotent elements from subsystem subgroups of type A 3 in representations of the special linear group,” Doklady NAN Belarusi, 56, No. 4, 11–15 (2012).
G. M. Seitz, “Unipotent elements, tilting modules, and saturation,” Inv. Math., 141, No. 3, 467–503 (2000).
S. Smith, “Irreducible modules and parabolic subgroups,” J. Algebra, 75, 286–289 (1982).
R. Steinberg, Lectures on Chevalley Groups, Yale Univ. Press (1968).
I. D. Suprunenko, “The invariance of the set of weights of irreducible representations of algebraic groups and Lie algebras of type A l with restricted highest weights under reduction modulo p,” Vestsi AN BSSR, Ser. Fiz.-Mat. Navuk, No. 2, 18–22 (1983).
I. D. Suprunenko, “The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic,” Memoirs of AMS, 200, No. 939(2009).
M. V. Velichko, “On the behavior of root elements in irreducible representations of simple algebraic groups,” Trudy Instituta Matematiki, 13, No. 2, 116–121 (2005).
M. V. Velichko, “On the behavior of root elements in modular representations of symplectic groups,” Trudy Instituta Matematiki, 14, No. 2, 28–34 (2006).
M. V. Velichko, “Properties of small unipotent elements in modular representations of classical algebraic groups,” PhD Thesis, Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 430, 2014, pp. 202–218.
Rights and permissions
About this article
Cite this article
Osinovskaya, A.A. Regular Unipotent Elements from Subsystem Subgroups of Type A 2 in Representations of the Special Linear Groups. J Math Sci 219, 473–483 (2016). https://doi.org/10.1007/s10958-016-3120-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-016-3120-7