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Transactions of the American Mathematical Society

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Perfect isometries and Murnaghan-Nakayama rules


Authors: Olivier Brunat and Jean-Baptiste Gramain
Journal: Trans. Amer. Math. Soc. 369 (2017), 7657-7718
MSC (2010): Primary 20C30, 20C15; Secondary 20C20
DOI: https://doi.org/10.1090/tran/6860
Published electronically: May 11, 2017
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Abstract: This article is concerned with perfect isometries between blocks of finite groups. Generalizing a method of Enguehard to show that any two $ p$-blocks of (possibly different) symmetric groups with the same weight are perfectly isometric, we prove analogues of this result for $ p$-blocks of alternating groups (where the blocks must also have the same sign when $ p$ is odd), of double covers of alternating and symmetric groups (for $ p$ odd, and where we obtain crossover isometries when the blocks have opposite signs), of complex reflection groups $ G(d,1,n)$ (for $ d$ prime to $ p$), of Weyl groups of type $ B$ and $ D$ (for $ p$ odd), and of certain wreath products. In order to do this, we need to generalize the theory of blocks in a way which should be of independent interest.


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Additional Information

Olivier Brunat
Affiliation: Université Paris-Diderot Paris 7, Institut de mathématiques de Jussieu – Paris Rive Gauche, UFR de mathématiques, Case 7012, 75205 Paris Cedex 13, France
Email: olivier.brunat@imj-prg.fr

Jean-Baptiste Gramain
Affiliation: Institute of Mathematics, University of Aberdeen, King’s College, Fraser Noble Building, Aberdeen AB24 3UE, United Kingdom
Email: jbgramain@abdn.ac.uk

DOI: https://doi.org/10.1090/tran/6860
Received by editor(s): April 2, 2014
Received by editor(s) in revised form: November 10, 2014, March 10, 2015, June 12, 2015, July 6, 2015, August 28, 2015, September 23, 2015, October 8, 2015, and October 27, 2015
Published electronically: May 11, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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