Blocks of central group extensions
Authors:
Shigeo Koshitani and Naoko Kunugi
Journal:
Proc. Amer. Math. Soc. 133 (2005), 2126
MSC (2000):
Primary 20C20, 20C05, 20C11
Published electronically:
July 26, 2004
MathSciNet review:
2085148
Fulltext PDF Free Access
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Abstract: Let and be finite groups that have a common central subgroup for a prime number , and let and respectively be blocks of and induced by blocks and respectively of and , both of which have the same defect group. We prove that if and are Morita equivalent via a certain special bimodule, then such a Morita equivalence lifts to a Morita equivalence between and .
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J. Thévenaz, Algebras and modular representation theory, Clarendon Press, Oxford, 1995. MR 96j:20017
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Y. Usami, M. Nakabayashi, Morita equivalent principal blocks of the Chevalley group , Proc. London Math. Soc.(3) 86 (2003), 397434. MR 2004c:20025
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W. Willems, On the projectives of a group algebra, Math. Zeit. 171 (1980), 163174. MR 81g:20007
 1.
 J. L. Alperin, Local representation theory, Cambridge Univ. Press, Cambridge, 1986. MR 87i:20002
 2.
 B. Külshammer, T. Okuyama, A. Watanabe, A lifting theorem with applications to blocks and source algebras, J. Algebra 232 (2000), 299309. MR 2001g:20013
 3.
 H. Nagao, Y. Tsushima, Representations of finite groups, Academic Press, New York, 1988. MR 90h:20008
 4.
 L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77116. MR 89e:20023
 5.
 L. Puig, On the local structure of Morita and Rickard equivalences between Brauer blocks, Birkhäuser, Basel, 1999. MR 2001d:20006
 6.
 L. Puig, Source algebras of central group extensions, J. Algebra 235 (2001), 359398. MR 2001k:20006
 7.
 G. R. Robinson, On projective summands of induced modules, J. Algebra 122 (1989), 106111. MR 90c:20011
 8.
 R. Rouquier, The derived category of blocks with cyclic defect groups, in Derived equivalences for group rings, S. König and A. Zimmermann, Lecture Notes in Mathematics, Vol. 1685, Springer, Berlin, 1998, pp. 199220. MR 2000g:16018
 9.
 R. Rouquier, Block theory via stable and Rickard equivalences, in Modular representation theory of finite groups, M. J. Collins, B. J. Parshall, L. L. Scott (Eds.), de Gruyter, Berlin, 2001, pp. 101146.MR 2003g:20018
 10.
 J. Thévenaz, Algebras and modular representation theory, Clarendon Press, Oxford, 1995. MR 96j:20017
 11.
 Y. Usami, M. Nakabayashi, Morita equivalent principal blocks of the Chevalley group , Proc. London Math. Soc.(3) 86 (2003), 397434. MR 2004c:20025
 12.
 W. Willems, On the projectives of a group algebra, Math. Zeit. 171 (1980), 163174. MR 81g:20007
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Additional Information
Shigeo Koshitani
Affiliation:
Department of Mathematics, Faculty of Science, Chiba University, Yayoicho, Inageku, Chiba, 2638522, Japan
Email:
koshitan@math.s.chibau.ac.jp
Naoko Kunugi
Affiliation:
Department of Mathematics, Aichi University of Education, Hirosawa, Igayacho, Kariya, 4488542, Japan
Email:
nkunugi@auecc.aichiedu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993904075094
PII:
S 00029939(04)075094
Keywords:
$p$block,
Morita equivalence,
central extension
Received by editor(s):
April 25, 2003
Received by editor(s) in revised form:
September 8, 2003
Published electronically:
July 26, 2004
Communicated by:
Jonathan I. Hall
Article copyright:
© Copyright 2004 American Mathematical Society
