Unipotent elements of nonprime order in representations of the classical algebraic groups: two big Jordan blocks

I. D. Suprunenko

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 193-241

Résumé

For irreducible rational representations of the classical algebraic groups in characteristic $p>2$ that are not equivalent to a composition of a group morphism and the standard representation, it is proved that usually the image of a unipotent element of order $p^{s+1}>p$ has at least two Jordan blocks of size $>p^s$; all exceptions are indicated explicitly. As a corollary, irreducible rational representations of these groups whose images contain unipotent elements with just one Jordan block of size $>1$ are classified.

Bibliographie

[1] A. Borel, “Properties and linear representations of Chevalley groups”, Seminar on Algebraic Groups and Related Finite Groups, Lect. Notes Math., 131, eds. A. Borel et al., Springer, 1970, 1–55 | DOI | MR

[2] N. Bourbaki, Groupes et algèbres de Lie, Chaps. IV–VI, Hermann, Paris, 1968 | MR

[3] N. Bourbaki, Groupes et algèbres de Lie, Chaps. VII–VIII, Hermann, Paris, 1975 | MR | Zbl

[4] R. W. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley, Chichester, 1985 | MR | Zbl

[5] W. Feit, The representation theory of finite groups, North-Holland, Amsterdam, 1982 | MR | Zbl

[6] P. M. Gudivok, V. P. Rudko, Tensor products of representations of finite groups, Uzhgorod Univ., Uzhgorod, 1985 (in Russian) | Zbl

[7] J. C. Jantzen, Darstellungen halbeinfacher algebraicher Gruppen und zugeordnetekontravariante Formen, Bonner Math. Schr., 67, 1973 | MR | Zbl

[8] J. C. Jantzen, Representations of Algebraic Groups, Second Edition, AMS, Providence, 2003 | MR | Zbl

[9] M. W. Liebeck, G. M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Math. Surv. Monogr., 180, AMS, Providence, 2012 | MR | Zbl

[10] F. Lubeck, “Small degree representations of finite Chevalley groups in defining characteristic”, London Math. Soc. J. Comput. Math., 4 (2001), 135–169 | MR | Zbl

[11] A. A. Premet, I. D. Suprunenko, “Quadratic modules for Chevalley groups over fields of odd characteristic”, Math. Nachrichten, 110 (1983), 65–96 | DOI | MR | Zbl

[12] J. Saxl, G. M. Seitz, “Subgroups of algebraic groups containing regular unipotent elements”, J. London Math. Soc. (2), 55 (1997), 370–386 | DOI | MR | Zbl

[13] G. M. Seitz, The maximal subgroups of classical algebraic groups, Memoirs of the AMS, 67, no. 365, 1987 | DOI | MR

[14] S. Smith, “Irreducible modules and parabolic subgroups”, J. Algebra, 75 (1982), 286–289 | DOI | MR | Zbl

[15] T. A. Springer, R. Steinberg, “Conjugacy classes”, Seminar on algebraic groups and related finite groups, Lect. Notes Math., 131, eds. A. Borel et al., Springer, 1970, 167–266 | DOI | MR

[16] R. Steinberg, “Representations of algebraic groups”, Nagoya Math. J., 22 (1963), 33–56 | MR | Zbl

[17] R. Steinberg, Lectures on Chevalley groups, Mimeographed Lect. Notes, Yale Univ. Math. Dept., New Haven, Conn., 1968 | MR | Zbl

[18] I. D. Suprunenko, “Minimal polynomials of elements of order $p$ in irreducible representations of Chevalley groups over fields of characteristic $p$”, Siberian Adv. Math., 6 (1996), 97–150 | MR | Zbl

[19] I. D. Suprunenko, “On Jordan blocks of elements of order $p$ in irreducible representations of classical groups with $p$-large highest weights”, J. Algebra, 191 (1997), 589–627 | DOI | MR | Zbl

[20] I. D. Suprunenko, “The second Jordan block and recognition of representations”, Dokl. NAN Belarusi, 47:1 (2003), 48–52 (in Russian) | MR | Zbl

[21] I. D. Suprunenko, “The minimal polynomials of unipotent elements in irreducible representations of the special linear group over a field of characteristic two”, Dokl. NAN Belarusi, 47:1 (2003), 9–13 (in Russian) | MR | Zbl

[22] I. D. Suprunenko, “Unipotent elements of prime order in representations of exceptional algebraic groups: the second Jordan block”, Dokl. NAN Belarusi, 49:4 (2005), 5–9 (in Russian) | MR | Zbl

[23] I. D. Suprunenko, “On the behaviour of unipotent elements in representations of the classical groups with large highest weights”, Dokl. NAN Belarusi, 49:5 (2005), 11–15 (in Russian) | MR | Zbl

[24] I. D. Suprunenko, The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic, Memoirs of the AMS, 200, no. 939, 2009 | DOI | MR

[25] I. D. Suprunenko, “On the block structure of regular unipotent elements from subsystem subgroups of type $A_1\times A_2$ in representations of the special linear group”, Zap. Nauchn. Semin. POMI, 388, 2011, 247–269 (in Russian) | MR

[26] I. D. Suprunenko, “Unipotent elements of nonprime order in representations of the classical groups: two big Jordan blocks”, Dokl. NAN Belarusi, 55:4 (2011), 21–26 (in Russian) | MR | Zbl

[27] I. D. Suprunenko, A. E. Zalesskii, “Truncated symmetric powers of the natural realizations of the groups $SL_m(P)$ and $Sp_m(P)$ and their restrictions to subgroups”, Siber. Math. J., 31 (1990), 555–566 | MR | Zbl

[28] I. D. Suprunenko, A. E. Zalesskii, “On restricting representations of simple algebraic groups to semisimple subgroups with two simple components”, Trudy Inst. Matem., 13:2 (2005), 109–115

[29] D. Testerman, A. E. Zalesski, “Irreducibility in algebraic groups and regular unipotent elements”, Proceedings of the Amer. Math. Soc. (to appear)

[30] A. Wagner, “Groups generated by elations”, Abh. Math. Sem. Univ. Hamburg, 41 (1974), 190–205 | DOI | MR | Zbl

[31] A. E. Zalesskii, V. N. Serezhkin, “Linear groups generated by transvections”, Math. USSR-Izv., 10:1 (1976), 25–46 | DOI | MR | Zbl

[32] A. E. Zalesskii, I. D. Suprunenko, “Representations of dimensions $(p^n\pm1)/2$ of the symplectic group of degree $2n$ over a field of characteristic $p$”, Vesti AN BSSR. Ser. Fiz.-Mat. nauk, 1987, no. 6, 9–15 (in Russian) | MR