A moduletheoretic approach to Clifford theory for blocks
Author:
S. J. Witherspoon
Journal:
Proc. Amer. Math. Soc. 128 (2000), 661670
MSC (1991):
Primary 20C20, 20C25
Published electronically:
July 8, 1999
MathSciNet review:
1646212
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This work concerns a generalization of Clifford theory to blocks of groupgraded algebras. A moduletheoretic approach is taken to prove a onetoone correspondence between the blocks of a fully groupgraded algebra covering a given block of its identity component, and conjugacy classes of blocks of a twisted group algebra. In particular, this applies to blocks of a finite group covering blocks of a normal subgroup.
 1.
J.
L. Alperin and David
W. Burry, Block theory with modules, J. Algebra
65 (1980), no. 1, 225–233. MR 578804
(81k:20018), http://dx.doi.org/10.1016/00218693(80)902471
 2.
D.
J. Benson, Representations and cohomology. I, Cambridge
Studies in Advanced Mathematics, vol. 30, Cambridge University Press,
Cambridge, 1991. Basic representation theory of finite groups and
associative algebras. MR 1110581
(92m:20005)
 3.
Paul
R. Boisen, The representation theory of fully groupgraded
algebras, J. Algebra 151 (1992), no. 1,
160–179. MR 1182020
(93i:20005), http://dx.doi.org/10.1016/00218693(92)90137B
 4.
A. H. CLIFFORD, Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), pp. 661670.
 5.
M.
Cohen and S.
Montgomery, Groupgraded rings, smash products,
and group actions, Trans. Amer. Math. Soc.
282 (1984), no. 1,
237–258. MR
728711 (85i:16002), http://dx.doi.org/10.1090/S00029947198407287114
 6.
Charles
W. Curtis and Irving
Reiner, Methods of representation theory. Vol. I, John Wiley
& Sons Inc., New York, 1981. With applications to finite groups and
orders; Pure and Applied Mathematics; A WileyInterscience Publication. MR 632548
(82i:20001)
 7.
Charles
W. Curtis and Irving
Reiner, Methods of representation theory. Vol. II, Pure and
Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1987.
With applications to finite groups and orders; A WileyInterscience
Publication. MR
892316 (88f:20002)
 8.
E.
C. Dade, Compounding Clifford’s theory, Ann. of Math.
(2) 91 (1970), 236–290. MR 0262384
(41 #6992)
 9.
E.
C. Dade, Block extensions, Illinois J. Math.
17 (1973), 198–272. MR 0327884
(48 #6226)
 10.
Everett
C. Dade, Groupgraded rings and modules, Math. Z.
174 (1980), no. 3, 241–262. MR 593823
(82c:16028), http://dx.doi.org/10.1007/BF01161413
 11.
Everett
C. Dade, The equivalence of various generalizations of group rings
and modules, Math. Z. 181 (1982), no. 3,
335–344. MR
678889 (84a:16018), http://dx.doi.org/10.1007/BF01161981
 12.
Everett
C. Dade, Counting characters in blocks. I, Invent. Math.
109 (1992), no. 1, 187–210. MR 1168370
(93g:20021), http://dx.doi.org/10.1007/BF01232023
 13.
Everett
C. Dade, Counting characters in blocks. II, J. Reine Angew.
Math. 448 (1994), 97–190. MR 1266748
(95a:20007), http://dx.doi.org/10.1515/crll.1994.448.97
 14.
, Clifford theory for blocks. Preprint, 1995.
 15.
E.
C. Dade, Counting characters in blocks. II.9, Representation
theory of finite groups (Columbus, OH, 1995) Ohio State Univ. Math. Res.
Inst. Publ., vol. 6, de Gruyter, Berlin, 1997, pp. 45–59.
MR
1611009 (99b:20016)
 16.
Harald
Ellers, Cliques of irreducible representations, quotient groups,
and Brauer’s theorems on blocks, Canad. J. Math.
47 (1995), no. 5, 929–945. MR 1350642
(96g:20013), http://dx.doi.org/10.4153/CJM1995048x
 17.
Walter
Feit, The representation theory of finite groups,
NorthHolland Mathematical Library, vol. 25, NorthHolland Publishing
Co., Amsterdam, 1982. MR 661045
(83g:20001)
 18.
Jeremy
Haefner, Graded equivalence theory with applications, J.
Algebra 172 (1995), no. 2, 385–424. MR 1322411
(96f:16052), http://dx.doi.org/10.1016/S00218693(05)800092
 19.
Yôichi
Miyashita, On Galois extensions and crossed products, J. Fac.
Sci. Hokkaido Univ. Ser. I 21 (1970), 97–121. MR 0271163
(42 #6046)
 20.
C.
Năstăsescu and F.
van Oystaeyen, Graded ring theory, NorthHolland Mathematical
Library, vol. 28, NorthHolland Publishing Co., Amsterdam, 1982. MR 676974
(84i:16002)
 1.
 J. L. ALPERIN AND D. W. BURRY, Block theory with modules, J. Algebra 65 (1980), pp. 661670. MR 81k:20018
 2.
 D. J. BENSON, Representations and Cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge University Press, 1991. MR 92m:20005
 3.
 P. R. BOISEN, The representation theory of fully groupgraded algebras, J. Algebra 151 (1992), pp. 661670. MR 93i:20005
 4.
 A. H. CLIFFORD, Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), pp. 661670.
 5.
 M. COHEN AND S. MONTGOMERY, Groupgraded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), pp. 661670. MR 85i:16002
 6.
 C. W. CURTIS AND I. REINER, Methods of Representation Theory with Applications to Finite Groups and Orders, Volume I, Wiley, 1981. MR 82i:20001
 7.
 , Methods of Representation Theory with Applications to Finite Groups and Orders, Volume II, Wiley, 1987. MR 88f:20002
 8.
 E. C. DADE, Compounding Clifford's theory, Ann. of Math. 91 (1970), pp. 661670. MR 41:6992
 9.
 , Block extensions, Illinois J. Math. 17 (1973), pp. 661670. MR 48:6226
 10.
 , Groupgraded rings and modules, Math. Z. 174 (1980), pp. 661670. MR 82c:16028
 11.
 , The equivalence of various generalizations of group rings and modules, Math. Z. 181 (1982), pp. 661670. MR 84a:16018
 12.
 , Counting characters in blocks, I, Inv. Math. 109 (1992), pp. 661670. MR 93g:20021
 13.
 , Counting characters in blocks, II, J. reine angew. Math. 448 (1994), pp. 661670. MR 95a:20007
 14.
 , Clifford theory for blocks. Preprint, 1995.
 15.
 , Counting characters in blocks, 2.9, in Representation Theory of Finite Groups, Proceedings of a Special Research Quarter at the Ohio State University, de Gruyter, 1997.MR 99b:20016
 16.
 H. ELLERS, Cliques of irreducible representations, quotient groups, and Brauer's theorems on blocks, Can. J. Math. 47 (5) (1995), pp. 929945. MR 96g:20013
 17.
 W. FEIT, The Representation Theory of Finite Groups, NorthHolland, 1982. MR 83g:20001
 18.
 J. HAEFNER, Graded equivalence theory with applications, J. Algebra 172 (1995), pp. 661670. MR 96f:16052
 19.
 Y. MIYASHITA, On Galois extensions and crossed products, J. Fac. Sci. Hokkaido Univ. Ser. I 21 (1970), pp. 661670. MR 42:6046
 20.
 C. NSTSESCU AND F. VAN OYSTAEYEN, Graded Ring Theory, NorthHolland, 1982. MR 84i:16002
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
20C20,
20C25
Retrieve articles in all journals
with MSC (1991):
20C20,
20C25
Additional Information
S. J. Witherspoon
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Address at time of publication:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email:
sjw@math.toronto.edu, sjw@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S0002993999052247
PII:
S 00029939(99)052247
Received by editor(s):
April 20, 1998
Published electronically:
July 8, 1999
Additional Notes:
Research supported in part by NSERC grant # OGP0170281.
Communicated by:
Ronald M. Solomon
Article copyright:
© Copyright 1999 American Mathematical Society
