I.D. Suprunenko, T.S. Busel, A.A. Osinovskaya. Special factors in the restrictions of irreducible modules of classical groups to subsystem subgroups with two simple components ... P. 259-273

For restrictions of $p$-restricted irreducible modules of classical algebraic groups in odd characteristic $p$ with highest weights that are relatively large with respect to $p$ to a subsystem subgroup $H$ of maximal rank with two main components $H_1$ and $H_2$ under slight constraints restrictions on the ranks of the subgroups $H_1$ and $H_2$, a lower bound is found for the number of composition factors that are $p$-large for the subgroup $H_1$ and not too small for $H_2$; the bound grows as the highest weight increases. On this basis, lower bounds are obtained for the number of Jordan blocks of maximal size for the images of certain unipotent elements in the corresponding representations of the groups.

Keywords: classical algebraic groups, modular representations, restrictions, composition factors, unipotent elements, Jordan blocks

Received June 30, 2023

Revised October 10, 2023

Accepted October 16, 2023

Funding Agency: This work was supported by the Belarusian Republican Foundation for Fundamental Research (project no. F21-054).

Irina Dmitrievna Suprunenko, Dr. Phys.-Math. Sci., Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 220072 Belarus, e-mail: anna@im.bas-net.by

Tatiana Sergeevna Busel, Cand. Sci. (Phys.-Math.), Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 220072 Belarus, e-mail: tbusel@gmail.com

Anna Aleksandrovna Osinovskaya, Cand. Sci. (Phys.-Math.), Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 220072 Belarus, e-mail: anna@im.bas-net.by

REFERENCES

1.    Bourbaki N. Groupes et algèbres de Lie, Chapitres 4,5 et 6. Berlin: Springer, 2007, 282 p. ISBN: 978-3-540-34490-2 . Translated to Russian under the title “Gruppy i algebry Li”: gl. IV-VI. Moscow: Mir Publ., 1972, 334 p.

2.   Suprunenko I.D. On the behaviour of unipotent elements in modular representations of classical groups with large highest weights. Dokl. Natsional’noi Akademii Nauk Belarusi, 2009, vol. 53, no. 1, pp. 27–32 (in Russian).

3.    Steinberg R. Lectures on Chevalley groups. Ser. University Lecture Series. 2016. 160 p. ISBN: 978-1-4704-3105-1 .

4.    Burness T.C., Ghandour S., Marion C., Testerman D.M. Irreducible almost simple subgroups of classical algebraic groups. Memoirs of the AMS, 2015, vol. 236, 110 p. ISBN: 978-1-4704-1046-9

5.   Burness T.C., Ghandour S., Testerman D.M. Irreducible geometric subgroups of classical algebraic groups. Memoirs of the AMS, 2015, vol. 239, 100 p. ISBN: 978-1-4704-1494-8

6.   Cavallin M., Testerman D.M. A new family of irreducible subgroups of the orthogonal algebraic groups. Trans. Amer. Math. Soc. Ser. B., 2019, vol. 6, no. 2, pp. 45–79. doi: 10.1090/btran/28

7.    Ghandour S. Irreducible disconnected subgroups of exceptional algebraic groups. J. Algebra, 2010, vol. 323, pp. 2671–2709. doi: 10.1016/j.jalgebra.2010.02.018

8.    Korhonen M. Reductive overgroups of distinguished unipotent elements in simple algebraic groups. Ph.D. Thesis, Lausanne: EPFL, 2017, 241 p. doi: 10.5075/epfl-thesis-8362

9.    Liebeck M.W., Seitz G.M., Testerman D.M. Distinguished unipotent elements and multiplicity-free subgroups of simple algebraic groups. Pacific J. Math., 2015, vol. 279, no. 1–2, pp. 357–382. doi: 10.2140/pjm.2015.279.357

10.    Lubeck F. Small degree representations of finite Chevalley groups in defining characteristic. LMS J. Comput. Math., 2001, vol. 4, pp. 135–169. doi: 10.1112/S1461157000000838

11.    Seitz G.M. The maximal subgroups of classical algebraic groups. Memoirs of the AMS., 1987, vol. 365, 286 p. ISBN: 978-1-4704-0781-0 .

12.    Smith S. Irreducible modules and parabolic subgroups. J. Algebra, 1982, vol. 75, pp. 286–289. doi: 10.1016/0021-8693(82)90076-X

13.    Suprunenko I.D. On Jordan blocks of elements of order p in irreducible representations of classical groups with p-large highest weights. J. Algebra, 1997, vol. 191, pp. 589–627. doi: 10.1006/jabr.1996.6916

14.    Suprunenko I.D. The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic. Memoirs of the Amer. Math. Soc., 2009, vol. 200, no. 939, 154 p. ISBN: 978-1-4704-0553-3 .

15.    Suprunenko I.D. Special composition factors in restrictions of representations of special linear and symplectic groups to subsystem subgroups with two simple components. Tr. Instituta Matematiki, 2018, vol. 26, no. 1, pp. 113–133.

16.    Testerman D.M. Irreducible subgroups of exceptional algebraic groups. Memoirs of the AMS, 1988, vol. 390, 190 p.

Cite this article as: I.D. Suprunenko, T.S. Busel, A.A. Osinovskaya. Special factors in the restrictions of irreducible modules of classical groups to subsystem subgroups with two simple components. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 259–273.