Advertisement

Algebras and Representation Theory

, Volume 22, Issue 6, pp 1387–1397 | Cite as

The Brauer Indecomposability of Scott Modules for the Quadratic Group Qd(p)

  • Shigeo Koshitani
  • İpek TuvayEmail author
Article
  • 33 Downloads

Abstract

Let k be an algebraically closed field of prime characteristic p and P a finite p-group. We compute the Scott kG-module with vertex P when \(\mathcal {F}\) is a constrained fusion system on P and G is Park’s group for \(\mathcal {F}\). In the case that \(\mathcal {F}\) is a fusion system of the quadratic group \(\operatorname {Qd}(p)=(\mathbb {Z}/p\times \mathbb {Z}/p)\rtimes {\text {SL}}(2,p)\) on a Sylow p-subgroup P of Qd(p) and G is Park’s group for \(\mathcal {F}\), we prove that the Scott kG-module with vertex P is Brauer indecomposable.

Keywords

Scott module Constrained fusion system The quadratic group 

Mathematics Subject Classification (2010)

20C05 20C20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors would like to thank the referees for their careful reading of the first manuscript and for valuable comments. A part of this work was done while the second author was visiting Chiba University in July 2017. She would like to thank the Center for Frontier Science, Chiba University for their hospitality. She would like to thank also Naoko Kunugi for her hospitality.

Funding Information

The first author was supported in part by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)15K04776, 2015–2018. The second author was partially supported by the Center for Frontier Science, Chiba University and Mimar Sinan Fine Arts University Scientific Research Project Unit with project number 2017/22.

References

  1. 1.
    Aschbacher, M., Kessar, R., Oliver, B.: Fusion Systems in Algebra and Topology. London Math. Soc Lecture Notes vol. 391. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  2. 2.
    Broto, C., Levi, R., Oliver, B.: The homotopy theory of fusion systems. J. Amer. Math. Soc. 16(4), 779–856 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Broto, C., Castellana, N., Grodal, J., Levi, R., Oliver, B.: Subgroup families controlling p-local finite groups. Proc. Lond. Math. Soc. (3) 91(2), 325–354 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Broué, M.: On Scott modules and p-permutation modules: an approach through the Brauer morphism. Proc. Amer. Math. Soc. 93, 401–408 (1985)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Broué, M., Puig, L.: Characters and local structure in G-algebras. J. Algebra 63, 306–317 (1980)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Glauberman, G.: A characteristic subgroup of a p-stable group. Canad. J. Math. 20, 1101–1135 (1968)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gorenstein, D.: Finite Groups. Harper and Row, New York (1968)zbMATHGoogle Scholar
  8. 8.
    Ishioka, H., Kunugi, N.: Brauer indecomposability of Scott modules. J. Algebra 470, 441–449 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    James, G., Kerber, A.: The Representation theory of the symmetric group. Addison-Wesley, Massachusetts (1981)zbMATHGoogle Scholar
  10. 10.
    Kessar, R., Koshitani, S., Linckelmann, M.: On the Brauer indecomposability of Scott modules. Quarterly J. Math. 66, 895–903 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kessar, R., Kunugi, N., Mitsuhashi, N.: On saturated fusion systems and Brauer indecomposability of Scott modules. J. Algebra 340, 90–103 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Koshitani, S., Linckelmann, M.: The indecomposability of a certain bimodule given by the Brauer construction. J. Algebra 285, 726–729 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Linckelmann, M. In: Geck, M., Testerman, D., Thévenaz, J. (eds.) : Introduction to fusion systems. Group representation theory, pp 79–113. EPFL Press, Lausanne (2007)Google Scholar
  14. 14.
    Nagao, H., Tsushima, Y.: Representations of finite groups. Academic Press, New York (1988)zbMATHGoogle Scholar
  15. 15.
    Park, S.: Realizing a fusion system by a single finite group. Arch. Math. 94, 405–410 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Thévenaz, J.: G-algebras and modular representation theory. Oxford Science Publications, Oxford University Press, New York (1995)zbMATHGoogle Scholar
  17. 17.
    Tuvay, İ.: On Brauer indecomposability of Scott modules of Park-type groups. J. Group Theory 17, 1071–1079 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Center of Frontier ScienceChiba UniversityChibaJapan
  2. 2.Department of MathematicsMimar Sinan Fine Arts UniversityIstanbulTurkey

Personalised recommendations