A module-theoretic approach to Clifford theory for blocks

Witherspoon, S.

doi:10.1090/S0002-9939-99-05224-7

A module-theoretic approach to Clifford theory for blocks
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by S. J. Witherspoon PDF

Proc. Amer. Math. Soc. 128 (2000), 661-670 Request permission

Abstract:

This work concerns a generalization of Clifford theory to blocks of group-graded algebras. A module-theoretic approach is taken to prove a one-to-one correspondence between the blocks of a fully group-graded 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.

References

J. L. Alperin and David W. Burry, Block theory with modules, J. Algebra 65 (1980), no. 1, 225–233. MR 578804, DOI 10.1016/0021-8693(80)90247-1
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
Paul R. Boisen, The representation theory of fully group-graded algebras, J. Algebra 151 (1992), no. 1, 160–179. MR 1182020, DOI 10.1016/0021-8693(92)90137-B
A. H. Clifford, Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), pp. 533–550.
M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), no. 1, 237–258. MR 728711, DOI 10.1090/S0002-9947-1984-0728711-4
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
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 Wiley-Interscience Publication. MR 892316
E. C. Dade, Compounding Clifford’s theory, Ann. of Math. (2) 91 (1970), 236–290. MR 262384, DOI 10.2307/1970606
E. C. Dade, Block extensions, Illinois J. Math. 17 (1973), 198–272. MR 327884
Everett C. Dade, Group-graded rings and modules, Math. Z. 174 (1980), no. 3, 241–262. MR 593823, DOI 10.1007/BF01161413
Everett C. Dade, The equivalence of various generalizations of group rings and modules, Math. Z. 181 (1982), no. 3, 335–344. MR 678889, DOI 10.1007/BF01161981
Everett C. Dade, Counting characters in blocks. I, Invent. Math. 109 (1992), no. 1, 187–210. MR 1168370, DOI 10.1007/BF01232023
Everett C. Dade, Counting characters in blocks. II, J. Reine Angew. Math. 448 (1994), 97–190. MR 1266748, DOI 10.1515/crll.1994.448.97
E. C. Dade, Clifford theory for blocks. Preprint, 1995.
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
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, DOI 10.4153/CJM-1995-048-x
Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
Jeremy Haefner, Graded equivalence theory with applications, J. Algebra 172 (1995), no. 2, 385–424. MR 1322411, DOI 10.1016/S0021-8693(05)80009-2
Yôichi Miyashita, On Galois extensions and crossed products, J. Fac. Sci. Hokkaido Univ. Ser. I 21 (1970), 97–121. MR 0271163
C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 676974