danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2019-63-5-519-525dan-647Research ArticleМАТЕМАТИКАMATHEMATICSМинимальные полиномы унипотентных элементов непростого порядка в неприводимых представлениях исключительных алгебраических групп в некоторых хороших характеристикахMinimal polynomials of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic groups in some good characteristicsБуселТ. С.BuselTatsiana S.

Бусел Татьяна Сергеевна - кандидат физико-математических наук, научный сотрудник.

Ул. Сурганова, 11, 220072, Минск

Busel Tatsiana Sergeevna - Ph. D. (Physics and Mathematics), Researcher

11, Surganov Str., 220072, Minsk

tbusel@im.bas-net.byСупруненкоИ. Д.SuprunenkoIrina D.

Супруненко Ирина Дмитриевна - доктор физико-математических наук, главный научный сотрудник.

Ул. Сурганова, 11, 220072, Минск

Suprunenko Irina Dmitrievna - D. Sc. (Physics and Mathematics), Chief researcher.

suprunenko@im.bas-net.byТестерманД.TestermanDonna

Тестерман Донна - профессор.

MA B3 434 (Station 8), CH-1015 Лозанна

Testerman Donna - Professor.

MA B3 434 (Station 8), CH-1015 Lausanne

donna.testerman@epfl.chИнститут математики, Национальная академия наук БеларусиБеларусьInstitute of Mathematics, National Academy of Sciences of BelarusBelarusИнститут математики, Федеральная политехническая школа ЛозанныШвейцарияInstitute of Mathematics, Ecole Polytechnique Federale de LausanneSwitzerland201906112019635519525Copyright © Бусел Т.С., Супруненко И.Д., Тестерман Д., 20192019Бусел Т.С., Супруненко И.Д., Тестерман Д.Busel T.S., Suprunenko I.D., Testerman D.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/647

Представлено академиком В. И. Янчевским

В ряде случаев найдены минимальные многочлены образов унипотентных элементов непростого порядка в неприводимых представлениях исключительных алгебраических групп в хороших характеристиках. Установлено, что если p > 5 для группы типа Е8 и p > 3 для других исключительных алгебраических групп, то для неприводимых представлений этих групп в характеристике p с большими относительно характеристики старшими весами степень минимального многочлена образа унипотентного элемента равна порядку этого элемента.

Communicated by Academician Vyacheslav I. Yanchevskii

In a number of cases the minimal polynomials of the images of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic groups in good characteristics are found. It is proved that if p > 5 for a group of type E8 and p > 3 for other exceptional algebraic groups, then for irreducible representations of these groups in characteristic p with large highest weights with respect to p, the degree of the minimal polynomial of the image of a unipotent element is equal to the order of this element.

исключительные алгебраические группыунипотентные элементынеприводимые представленияexceptional algebraic groupsunipotent elementsirreducible representationsРабота первого и второго авторов поддержана БРФФИ (проект № Ф17МС-021). Значительная часть этих исследований была проведена, когда второй автор посещала Федеральную политехническую школу Лозанны (EPFL) в 2017 и 2018 гг. Она благодарна EPFL за гостеприимствоThe first and the second authors have been supported by the Belarusian Republican Foundation for Fundamental Research (Project no. Ф17МС-021). A substantial part of this research was done when the second author visited the Ecole Politechnique Federale de Lausanne in 2017 and 2018. She thanks the EPFL for the hospitalityReferencesSuprunenko I. D. The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic. Memoirs of the American Mathematical Society, 2009, vol. 200, no. 939. https://doi.org/10.1090/memo/0939Suprunenko I. D. The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic. Memoirs of the American Mathematical Society, 2009, vol. 200, no. 939. https://doi.org/10.1090/memo/0939Suprunenko I. D. Minimal polynomials of elements of orderp in irreducible representations of Chevalley groups over fields of characteristic p. Siberian Advances in Mathematics, 1996, vol. 6, pp. 97-150.Suprunenko I. D. Minimal polynomials of elements of orderp in irreducible representations of Chevalley groups over fields of characteristic p. Siberian Advances in Mathematics, 1996, vol. 6, pp. 97-150.Steinberg R. Lectures on Chevalley groups. University Lecture Series, 2016, vol. 66. https://doi.org/10.1090/ulect/066Steinberg R. Lectures on Chevalley groups. University Lecture Series, 2016, vol. 66. https://doi.org/10.1090/ulect/066Lawther R. Jordan block sizes of unipotent elements in exceptional algebraic groups. Communications in Algebra, 1995, vol. 23, no. 11, pp. 4125-4156. https://doi.org/10.1080/00927879508825454Lawther R. Jordan block sizes of unipotent elements in exceptional algebraic groups. Communications in Algebra, 1995, vol. 23, no. 11, pp. 4125-4156. https://doi.org/10.1080/00927879508825454Steinberg R. Representations of algebraic groups. Nagoya Mathematical Journal, 1963, vol. 22, pp. 33-56. https://doi.org/10.1017/s0027763000011016Steinberg R. Representations of algebraic groups. Nagoya Mathematical Journal, 1963, vol. 22, pp. 33-56. https://doi.org/10.1017/s0027763000011016James G. D. The representation theory of the symmetric groups. Lecture Notes in Mathematics. Vol. 682. Berlin, 1978. https://doi.org/10.1007/BFb0067708James G. D. The representation theory of the symmetric groups. Lecture Notes in Mathematics. Vol. 682. Berlin, 1978. https://doi.org/10.1007/BFb0067708Feit W. The representation theory offinite groups. North-Holland, Amsterdam, 1982. https://doi.org/10.1016/s0924-6509(08)x7025-4Feit W. The representation theory offinite groups. North-Holland, Amsterdam, 1982. https://doi.org/10.1016/s0924-6509(08)x7025-4Gudivok P M., Rudko V. P Tensor products of representations offinite groups. Uzhgorod, 1985 (in Russian).Gudivok P M., Rudko V. P Tensor products of representations offinite groups. Uzhgorod, 1985 (in Russian).Suprunenko I. D. Unipotent elements of nonprime order in representations of the classical algebraic groups: two big Jordan blocks. Journal of Mathematical Sciences, 2014, vol. 199, no. 3, pp. 350-374. https://doi.org/10.1007/s10958-014-1863-6Suprunenko I. D. Unipotent elements of nonprime order in representations of the classical algebraic groups: two big Jordan blocks. Journal of Mathematical Sciences, 2014, vol. 199, no. 3, pp. 350-374. https://doi.org/10.1007/s10958-014-1863-6Spaltenstein N. Classes unipotentes et sous-groupes de Borel. Lecture Notes in Mathematics. Vol. 946. Berlin, Heidelberg, 1982. https://doi.org/10.1007/bfb0096302Spaltenstein N. Classes unipotentes et sous-groupes de Borel. Lecture Notes in Mathematics. Vol. 946. Berlin, Heidelberg, 1982. https://doi.org/10.1007/bfb0096302Seitz G. M. The maximal subgroups of classical algebraic groups. Memoirs of the American Mathematical Society, 1987, vol. 67, no. 365. https://doi.org/10.1090/memo/0365Seitz G. M. The maximal subgroups of classical algebraic groups. Memoirs of the American Mathematical Society, 1987, vol. 67, no. 365. https://doi.org/10.1090/memo/0365

The authors declare that there are no conflicts of interest present.