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Splendid Morita equivalences for principal 2-blocks with dihedral defect groups

  • Shigeo Koshitani
  • Caroline LassueurEmail author
Article
  • 12 Downloads

Abstract

Given a dihedral 2-group P of order at least 8, we classify the splendid Morita equivalence classes of principal 2-blocks with defect groups isomorphic to P. To this end we construct explicit stable equivalences of Morita type induced by specific Scott modules using Brauer indecomposability and gluing methods; we then determine when these stable equivalences are actually Morita equivalences, and hence automatically splendid Morita equivalences. Finally, we compute the generalised decomposition numbers in each case.

Keywords

Puig’s finiteness conjecture Morita equivalence Splendid Morita equivalence Stable equivalence of Morita type Scott module Brauer indecomposability Generalised decomposition numbers Dihedral 2-group 

Mathematics Subject Classification

16D90 20C20 20C15 20C33 

Notes

Acknowledgements

The authors thank Richard Lyons and Ronald Solomon for the proof of Lemma 4.2. The authors are also grateful to Naoko Kunugi and Tetsuro Okuyama for their useful pieces of advice, and to Gunter Malle for his careful reading of a preliminary version of this work. Part of this work was carried out while the first author was visiting the TU Kaiserslautern in May and August 2017, who thanks the Department of Mathematics of the TU Kaiserslautern for the hospitality. The second author gratefully acknowledges financial support by the funding body TU Nachwuchsring of the TU Kaiserslautern for the year 2016 when this work started. Part of this work was carried out during the workshop ”New Perspective in Representation Theory of Finite Groups” in October 2017 at the Banff International Research Station (BIRS). The authors thank the organisers of the workshop. Finally, the authors would like to thank the referees for their careful reading and useful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Frontier ScienceChiba UniversityChibaJapan
  2. 2.FB MathematikTU KaiserslauternKaiserslauternGermany

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