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

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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.

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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

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.

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Authors and Affiliations

  1. Center of Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

    Shigeo Koshitani

  2. Department of Mathematics, Mimar Sinan Fine Arts University, 34380 Bomonti, Şişli, Istanbul, Turkey

    İpek Tuvay

Authors
  1. Shigeo Koshitani
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  2. İpek Tuvay
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Correspondence to İpek Tuvay.

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Presented by: Radha Kessar

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Koshitani, S., Tuvay, İ. The Brauer Indecomposability of Scott Modules for the Quadratic Group Qd(p). Algebr Represent Theor 22, 1387–1397 (2019). https://doi.org/10.1007/s10468-018-9825-1

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  • DOI: https://doi.org/10.1007/s10468-018-9825-1

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Mathematics Subject Classification (2010)