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Clifford theory of characters in induced blocks

Authors: Shigeo Koshitani and Britta Späth
Journal: Proc. Amer. Math. Soc. 143 (2015), 3687-3702
MSC (2010): Primary 20C20; Secondary 20C15
Published electronically: May 20, 2015
MathSciNet review: 3359562
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Abstract: We present a new criterion to predict if a character of a finite group extends. Let $ G$ be a finite group and $ p$ a prime. For $ N\vartriangleleft G$, we consider $ p$-blocks $ b$ and $ b'$ of $ N$ and $ \operatorname {N}_N(D)$, respectively, with $ (b')^N=b$, where $ D$ is a defect group of $ b'$. Under the assumption that $ G$ coincides with a normal subgroup $ G[b]$ of $ G$, which was introduced by Dade early in the 1970's, we give a character correspondence between the sets of all irreducible constituents of $ \phi ^G$ and those of $ (\phi ')^{\operatorname {N}_G(D)}$, where $ \phi $ and $ \phi '$ are irreducible Brauer characters in $ b$ and $ b'$, respectively. This implies a sort of generalization of the theorem of Harris-Knörr. An important tool is the existence of certain extensions that also help in checking the inductive Alperin-McKay and inductive Blockwise-Alperin-Weight conditions, due to the second author.

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Additional Information

Shigeo Koshitani
Affiliation: Department of Mathematics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

Britta Späth
Affiliation: Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany

Keywords: Clifford theory, induced block, Dade's group $G[b]$, Harris-Kn{\"o}rr
Received by editor(s): April 22, 2013
Received by editor(s) in revised form: October 8, 2013, and December 10, 2013
Published electronically: May 20, 2015
Additional Notes: The first author was supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)23540007, 2011–2014. The second author has been supported by the Deutsche Forschungsgemeinschaft, SPP 1388 and by the ERC Advanced Grant 291512.
Dedicated: Dedicated to the memory of Masafumi Murai
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society

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