Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Trivial source bimodule rings for blocks and -permutation equivalences

Author(s): Markus Linckelmann
Journal: Trans. Amer. Math. Soc. 361 (2009), 1279-1316.
MSC (2000): Primary 20C20
Posted: October 10, 2008
MathSciNet review: 2457399
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We associate with any -block of a finite group a Grothendieck ring of certain -permutation bimodules. We extend the notion of -permutation equivalences introduced by Boltje and Xu (2006) to source algebras of -blocks of finite groups. We show that a -permutation equivalence between two source algebras , of blocks with a common defect group and same local structure induces an isotypy.

References:

1.: J. L. Alperin, M. Broué, Local methods in block theory, Ann. of Math. 110 (1979), 143-157. MR 541333 (80f:20010)
2.: R. Boltje, Chain complexes for Alperin's weight conjecture and Dade's ordinary conjecture in the abelian defect group case, J. Group Theory, to appear.
3.: R. Boltje, B. Külshammer, A generalised Brauer construction and linear source modules, Trans. Amer. Math. Soc. 363 (2000), 3411-3428. MR 1694281 (2000j:20017)
4.: R. Boltje, B. Xu, On -permutation equivalences: between Rickard equivalences and isotypies, preprint (2006).
5.: S. Bouc, Résolutions de foncteurs de Mackey, in: ``Groups, representations: cohomology, group actions and topology'' (eds.: A. Adem, J. Carlson, S. Priddy, P. Webb), Proc. Symp. Pure Math. 63 (1998), 31-83. MR 1603131 (99b:20015)
6.: C. Broto, R. Levi, R. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779-856. MR 1992826 (2004k:55016)
7.: M. Broué, Brauer coefficients of -subgroups associated with a -block of a finite group, J. Algebra 56 (1979), 365-383. MR 528581 (80d:20013)
8.: M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181-182 (1990), 61-92. MR 1051243 (91i:20006)
9.: M. Broué, L. Puig, Characters and local structure in -algebras, J. Algebra 63 (1980), 306-317. MR 570714 (81j:20021)
10.: M. Broué, L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117-128. MR 558864 (81d:20011)
11.: W. Feit, The representation theory of finite groups, North-Holland, Amsterdam, 1982. MR 661045 (83g:20001)
12.: R. Kessar, M. Linckelmann, On blocks of strongly -solvable groups, Archiv Math., to appear. MR 2283678 (2007k:20023)
13.: M. Linckelmann, The source algebras of blocks with a Klein four defect group, J. Algebra 167 (1994), 821-854. MR 1287072 (95h:20014)
14.: M. Linckelmann, On stable equivalences of Morita type, in: S. König, A. Zimmermann: ``Derived Equivalences for Group Rings'', Springer Lecture Notes in Mathematics 1685 (1998), 221-232. MR 1649847
15.: M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, Turkish J. Math. 22 (1998), 93-107. MR 1631770 (99k:20021)
16.: M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebras and Repr. Theory 2 (1999), 107-135. MR 1702272 (2000h:20024)
17.: M. Linckelmann, On splendid derived and stable equivalences of blocks of finite groups, J. Algebra 242 (2001), 819-843. MR 1848975 (2002i:20013)
18.: M. Linckelmann, Simple fusion systems and the Solomon -local group, J. Algebra 296 (2006), 385-401. MR 2201048 (2006i:20024)
19.: M. Linckelmann, Introduction to fusion systems, in: Group Representation Theory (editors: M. Geck, D. Testerman, J. Thévenaz), EPFL Press, Lausanne (2007), 79-113. MR 2336638
20.: A. Marcuş, On equivalences between blocks of group algebras: reduction to simple components, J. Algebra 184 (1996), 372-396. MR 1409219 (97m:20012)
21.: L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265-292. MR 607966 (82d:20015)
22.: L. Puig, Local fusion in block source algebras, J. Algebra 104 (1986), 358-365. MR 866781 (88c:20020)
23.: L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77-116. MR 943924 (89e:20023)
24.: L. Puig, On the local structure of Morita and Rickard equivalences between Brauer blocks, Progress in Mathematics 178 (Boston, Mass.), Basel: Birkhäuser, 1999. MR 1707300 (2001d:20006)
25.: J. Rickard, Splendid equivalence: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996), 331-358. MR 1367082 (97b:20011)
26.: R. Rouquier, The derived category of blocks with cyclic defect groups, in: S. König, A. Zimmermann: ``Derived Equivalences for Group Rings'', Springer Lecture Notes in Mathematics 1685 (1998), 199-220. MR 1649846
27.: R. Rouquier, Block theory via stable and Rickard equivalences; in: Modular Representation Theory of Finite Groups, deGruyter, Berlin, 2001, pp. 101-146. MR 1889341 (2003g:20018)
28.: J. Thévenaz, -algebras and Modular Representation Theory, Oxford Science Publications, Clarendon, Oxford, 1995. MR 1365077 (96j:20017)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|