Clifford theory of characters in induced blocks
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- by Shigeo Koshitani and Britta Späth PDF
- Proc. Amer. Math. Soc. 143 (2015), 3687-3702 Request permission
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.References
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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
- MR Author ID: 202274
- Email: koshitan@math.s.chiba-u.ac.jp
- Britta Späth
- Affiliation: Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
- Email: spaeth@mathematik.uni-kl.de
- 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.
- Communicated by: Pham Huu Tiep
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3687-3702
- MSC (2010): Primary 20C20; Secondary 20C15
- DOI: https://doi.org/10.1090/proc/12431
- MathSciNet review: 3359562
Dedicated: Dedicated to the memory of Masafumi Murai