Home Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block
Article
Licensed
Unlicensed Requires Authentication

Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block

  • Donna M. Testerman EMAIL logo and Alexandre E. Zalesski
Published/Copyright: August 17, 2017
Journal of Group Theory
From the journal Journal of Group Theory Volume 21 Issue 1

Abstract

Let G be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field F of characteristic p≥0, and let u∈G be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation of G. Then the Jordan normal form of ϕ⁢(u) contains at most one non-trivial block if and only if G is of type G2, u is a regular unipotent element and dim⁡ϕ≤7. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].

Communicated by Christopher W. Parker

Funding source: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Award Identifier / Grant number: 200021_146223

Funding statement: The first author was supported in part by Swiss National Science Foundation (grant no. ). A part of this work was carried out with the generous support of the Bernoulli Center, at the Swiss Federal Institute of Technology Lausanne, when the second author participated in the program “Local representation theory and simple groups” (2016).

Acknowledgements

We would like to thank Gunter Malle for reading an earlier version of the paper and also Irina Suprunenko for providing us with precise details of her own work.

References

[1] J. L. Alperin, Local Representation Theory, Cambridge Stud. Adv. Math. 11, Cambridge University Press, Cambridge, 1986. 10.1017/CBO9780511623592Search in Google Scholar

[2] H. H. Andersen, J. Jørgensen and P. Landrock, The projective indecomposable modules of SL⁢(2,pn), Proc. Lond. Math. Soc. (3) 46 (1983), no. 1, 38–52. 10.1112/plms/s3-46.1.38Search in Google Scholar

[3] N. Bourbaki, Éléments de Mathématique. XXVI. Groupes et Algèbres de Lie. Chapitre 1: Algèbres de Lie, Actualités Sci. Ind. 1285, Hermann, Paris, 1960. Search in Google Scholar

[4] D. Craven, Groups with a p-element acting with a single non-trivial Jordan block, preprint. Search in Google Scholar

[5] C. W. Curtis, W. M. Kantor and G. M. Seitz, The 2-transitive permutation representations of the finite Chevalley groups, Trans. Amer. Math. Soc. 218 (1976), 1–59. 10.1090/S0002-9947-1976-0422440-8Search in Google Scholar

[6] W. Feit, The Representation Theory of Finite Groups, North-Holland Math. Libr. 25, North-Holland Publishing, Amsterdam, 1982. Search in Google Scholar

[7] R. M. Guralnick and J. Saxl, Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), no. 2, 519–571. 10.1016/S0021-8693(03)00182-0Search in Google Scholar

[8] J. E. Humphreys, Modular Representations of Finite Groups of Lie Type, London Math. Soc. Lecture Note Ser. 326, Cambridge University Press, Cambridge, 2006. 10.1017/CBO9780511525940Search in Google Scholar

[9] R. Lawther, Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23 (1995), no. 11, 4125–4156. 10.1080/00927879508825454Search in Google Scholar

[10] F. Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169. 10.1112/S1461157000000838Search in Google Scholar

[11] F. Lübeck, Tables of weight multiplicities, on-line data, available at http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/WMSmall/index.html. Search in Google Scholar

[12] G. J. McNinch, Dimensional criteria for semisimplicity of representations, Proc. Lond. Math. Soc. (3) 76 (1998), no. 1, 95–149. 10.1112/S0024611598000045Search in Google Scholar

[13] H. Pollatsek, Irreducible groups generated by transvections over finite fields of characteristic two, J. Algebra 39 (1976), no. 1, 328–333. 10.1016/0021-8693(76)90080-6Search in Google Scholar

[14] R. Proud, J. Saxl and D. Testerman, Subgroups of type A1 containing a fixed unipotent element in an algebraic group, J. Algebra 231 (2000), no. 1, 53–66. 10.1006/jabr.2000.8353Search in Google Scholar

[15] G. M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365. 10.1090/memo/0365Search in Google Scholar

[16] S. D. Smith, Irreducible modules and parabolic subgroups, J. Algebra 75 (1982), no. 1, 286–289. 10.1016/0021-8693(82)90076-XSearch in Google Scholar

[17] R. Steinberg, Lectures on Chevalley Groups, Yale University, New Haven, 1968. Search in Google Scholar

[18] I. D. Suprunenko, Irreducible representations of simple algebraic groups containing matrices with big Jordan blocks, Proc. Lond. Math. Soc. (3) 71 (1995), no. 2, 281–332. 10.1112/plms/s3-71.2.281Search in Google Scholar

[19] I. D. Suprunenko, Unipotent elements of prime order in representations of exceptional algebraic groups: The second Jordan block, Dokl. Nats. Akad. Nauk Belarusi 49 (2005), no. 4, 5–9, 124. Search in Google Scholar

[20] I. D. Suprunenko, Unipotent elements of nonprime order in representations of classical groups: Two large Jordan blocks, Dokl. Nats. Akad. Nauk Belarusi 55 (2011), no. 4, 21–26, 125. Search in Google Scholar

[21] I. D. Suprunenko, Unipotent elements of nonprime order in representations of the classical algebraic groups: Two big Jordan blocks, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 414 (2013), 193–241. 10.1007/s10958-014-1863-6Search in Google Scholar

[22] D. M. Testerman, Irreducible subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 75 (1988), no. 390. 10.1090/memo/0390Search in Google Scholar

[23] D. M. Testerman, A1-type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups, J. Algebra 177 (1995), no. 1, 34–76. 10.1006/jabr.1995.1285Search in Google Scholar

[24] D. M. Testerman and A. E. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 (2013), no. 1, 13–28. 10.1090/S0002-9939-2012-11898-2Search in Google Scholar

[25] D. M. Testerman and A. E. Zalesski, Subgroups of simple algebraic groups containing regular tori, and irreducible representations with multiplicity 1 non-zero weights, Transform. Groups 20 (2015), no. 3, 831–861. 10.1007/s00031-015-9321-1Search in Google Scholar

Received: 2017-4-19
Revised: 2017-5-17
Published Online: 2017-8-17
Published in Print: 2018-1-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block
  3. Sandwich classification for GLn(R), O2n(R) and U2n(R,Λ) revisited
  4. Critical maximal subgroups and conjugacy of supplements in finite soluble groups
  5. Finite $\smash{p}$\hbox{-}groups of conjugate type {1,p3}
  6. Normal subgroups in limit groups of prime index
  7. Uncountable locally free groups and their group rings
  8. Full box spaces of free groups
  9. Almost congruence extension property for subgroups of free groups
  10. Solving equations of length seven over torsion-free groups
  11. On the structure of virtually nilpotent compact p-adic analytic groups
https://doi.org/10.1515/jgth-2017-0019

Articles in the same Issue

  1. Frontmatter
  2. Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block
  3. Sandwich classification for GLn(R), O2n(R) and U2n(R,Λ) revisited
  4. Critical maximal subgroups and conjugacy of supplements in finite soluble groups
  5. Finite $\smash{p}$\hbox{-}groups of conjugate type {1,p3}
  6. Normal subgroups in limit groups of prime index
  7. Uncountable locally free groups and their group rings
  8. Full box spaces of free groups
  9. Almost congruence extension property for subgroups of free groups
  10. Solving equations of length seven over torsion-free groups
  11. On the structure of virtually nilpotent compact p-adic analytic groups
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jgth-2017-0019/html
Scroll to top button