The indecomposability of a certain bimodule given by the Brauer construction

Markus Linckelmann, S. Koshitani

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11 Citations (Scopus)

Abstract

Broue's abelian defect conjecture [Asterisque 181/182 (1990) 61-92. 6.2] predicts for a p-block of a finite group G with an abelian defect group P a derived equivalence between the block algebra and its Brauer correspondent. By a result of Rickard [J. London Math. Soc. 43 (1991) 37-48], such a derived equivalence would in particular imply a stable equivalence induced by tensoring with a suitable bimodule-and it appears that these stable equivalences in turn tend to be obtained by "gluing" together Morita equivalences at the local levels of the considered blocks see. e.g., [M. Broue, Equivalences of blocks of group algebras, in: V Dlab, L.L. Scott (Eds.). Finite Dimensional Algebras and Related Topics, Kluwer Acad. Publ., 1994, pp. 1-26, 6.3], [M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, J. Algebra 242 (2001) 819-843, 3.1] [J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996) 331-358, 4.1], and [R. Rouquier, Block theory via stable and Rickard equivalences, in: M.J. Collins, B.J. Parshall, L.L. Scott (Eds.), Modular Representation Theory of Finite Groups., de Gruyter, Berlin, 2001, pp. 101-146, 5.6, A.4.1]. This note provides a technical indecomposability result which is intended to verify in suitable circumstances the hypotheses that are necessary to apply gluing results as mentioned above. This is used in [S. Koshitani, N. Kunugi, K. Waki, Broue's abelian defect group conjecture for the Held group and the sporadic Suzuki group, J. Algebra 279 (2004)638-666] to show that Broue's abelian defect group conjecture holds for nonprincipal blocks of the simple Held group and the sporadic Suzuki group. (c) 2004 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)726-729
Number of pages3
JournalJournal of Algebra
Volume285
Issue number2
DOIs
Publication statusPublished - 2005

Keywords

  • Broue's conjecture
  • Brauer construction
  • block
  • Brauer pair

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