Geodesic
The Jordan block structure of the images of unipotent elements in irreducible modular representations of classical algebraic groups of small dimensions
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 306-454.

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For unipotent elements of prime order, the Jordan block structure of their images in 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 bigger dimensions. A 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.
Keywords: Jordan block sizes, representations of small dimensions.
Mots-clés : unipotent elements 
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T. S. Busel; I. D. Suprunenko. The Jordan block structure of the images of unipotent elements in irreducible modular representations of classical algebraic groups of small dimensions. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 306-454. http://geodesic.mathdoc.fr/item/SEMR_2023_20_1_a10/

[1] M.J.J. Barry, “Decomposing tensor products and exterior and symmetric squares”, J. Group Theory, 14:1 (2011), 59–82 | DOI | MR | Zbl

[2] A. Borel, “Properties and linear representations of Chevalley groups”, Semin. Algebr. Groups related Finite Groups Princeton 1968/69, Lect. Notes Math., 131, eds. A. Borel et al., Springer, Berlin etc, 1970, A1-A55 | MR | Zbl

[3] N. Bourbaki, Groupes et algèbres de Lie, Chaps. IV-VI, Hermann, Paris, 1968 | MR | Zbl

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

[5] B. Braden, “Restricted representations of classical Lie algebras of type $A_2$ and $B_2$”, Bull. Am. Math. Soc., 73 (1967), 482–486 | DOI | MR | Zbl

[6] J. Brundan, A. Kleshchev, I.D. Suprunenko, “Semisimple restrictions from $GL_{n}$ to $GL_{n-1}$”, J. Reine Angew. Math., 500 (1998), 83–112 | MR | Zbl

[7] R.W. Carter, E. Cline, “The submodule structure of Weyl modules for groups of type $A_1$”, Proc. Conf. finite Groups (Park City 1975), Academic Press, New York-London, 1976, 303–311 | DOI | MR | Zbl

[8] R.W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Wiley, Chichester-New York etc, 1985 | MR | Zbl

[9] S. Donkin, “On tilting modules for algebraic groups”, Math. Z., 212:1 (1993), 39–60 | DOI | MR | Zbl

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

[11] J.A. Green, Polynomial representations of $GL_n$, Lecture Notes in Mathematics, 830, Springer, Berlin, 2007 | MR | Zbl

[12] J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York etc, 1980 | MR | Zbl

[13] J.C. Jantzen, Representations of algebraic groups, Mathematical surveys and monographs, 107, American Mathematical Society, Providence, 2003 | MR | Zbl

[14] P. Kleidman, M. Liebeck, The subgroup structure of the finite classical groups, LMS Lecture Note Series, 129, Cambridge University Press, Cambridge etc., 1990 | MR | Zbl

[15] M. Korhonen, Reductive overgroups of distinguished unipotent elements in simple algebraic groups, Ph. D. thesis, École Polytechnique Fédérale de Lausanne, 2017

[16] M. Korhonen, “Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic”, Proc. Am. Math. Soc., 147:10 (2019), 4205–4219 | DOI | MR | Zbl

[17] R. Lawther, “Jordan block sizes of unipotent elements in exceptional algebraic groups”, Commun. Algebra, 25:11 (1995), 4125–4156 | DOI | MR | Zbl

[18] F. Lübeck, “Small degree representations of finite Chevalley groups in defining characteristic”, LMS J. Comput. Math., 4 (2001), 135–169 | DOI | MR | Zbl

[19] A.A. Osinovskaya, 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, 273:2 (2004), 586–600 | DOI | MR | Zbl

[20] A.A. Premet, “Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic”, Math. USSR, Sb., 61:1 (1988), 167–183 | DOI | MR | Zbl

[21] G.M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Am. Math. Soc., 365, American Mathematical Society, Providence, 1987 | MR | Zbl

[22] G.M. Seitz, “Unipotent elements, tilting modules, and saturation”, Invent. Math., 141:3 (2000), 467–502 | DOI | MR | Zbl

[23] M.W. Liebeck, G.M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Mathematical Surveys and Monographs, 180, American Mathematical Society, Providence, 2012 | DOI | MR | Zbl

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

[25] R. Steinberg, Lectures on Chevalley groups, Yale Univ. Math. Dept., New Haven, 1968 | MR | Zbl

[26] I.D. Suprunenko, “Preservation of weight systems of irreducible representations of an algebraic group and a Lie algebra of type $A_l$ with bounded highest weights for reduction modulo $p$”, Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, 1983:2 (1983), 18–22 | MR | Zbl

[27] I.D. Suprunenko, “Minimal polynomials of elements of order $p$ in irreducible representations of Chevalley groups over fields of characteristic $p$”, Sib. Adv. Math., 6:4 (1996), 97–150 | MR | Zbl

[28] I.D. Suprunenko, “The second Jordan block and recognition of presentations”, Dokl. Nats. Akad. Nauk Belarusi, 47:1 (2003), 48–52 | MR | Zbl

[29] I.D. Suprunenko, The minimal polynomials of unipotent elements in irreducible representations of the classical groups over fields of odd characteristic, Mem. Amer. Math. Soc., 939, American Mathematical Society, Providence, 2009 | MR | Zbl

[30] P.H. Tiep, A.E. Zalesskii, “Mod $p$ reducibility of unramified representations of finite groups of Lie type”, Proc. Lond. Math. Soc., III. Ser., 84:2 (2002), 439–472 | DOI | MR | Zbl

[31] M.V. Velichko, “On the behaviour of root elements in modular representations of symplectic groups”, Tr. Inst. Mat., Minsk, 14:2 (2006), 28–34 | MR | Zbl

[32] A.E. Zalesskii, I.D. Suprunenko, “Representations of dimension $(p^n \mp 1)/2$ of symplectic groups of degree $2n$ over a field of characteristic $p$”, Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, 1987:6 (1987), 9–15 | MR | Zbl

[33] A.E. Zalesskii, I.D. Suprunenko, “Reduced symmetric powers of natural realizations of the groups $SL_m (P)$ and $Sp_m (P)$ and their restrictions to subgroups”, Sib. Math. J., 31:4 (1990), 555–566 | DOI | MR | Zbl