Blocks of Tame Representation Type and Related Algebras
This monograph studies algebras that are associated to blocks of tame representation type. Over the past few years, a range of new results have been obtained and a comprehensive account of these is provided here to- gether with some new proofs of known re
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Karin Erdmann
Blocks ofTame Representation Type and Related Algebras
Springer-Verlag BerlinHeidelberg NewYork London ParisTokyo Hong Kong
Author
Karin Erdmann Mathematical Institute, Oxford University, 24-29 St. Giles, Oxford OX 1 3LB, England
Mathematics Subject Classification (1980): 20C20, 20C05, 20C15, 16A46, 16A48, 16A64 ISBN 3-540-52709-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52709-5 Springer-Verlag New York Berlin Heidelberg
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Introduction In these notes we shall study algebras which are associated to blocks of tame representation type. These are the 2-blocks whose defect groups are dihedral or semidihedral or (generalized) quaternion. Over the last few years, a range of new results on a class of algebras including such blocks have been obtained. The algebras are essentially defined in terms of their stable Auslander-Reiten quivers (which we shall describe later), and it has been proved that any such algebra is Morita equivalent to one of the algebras in a small list which is explicitly given by generators and relations. In particular, this describes tame blocks; and allows one to extend classical results on the arithmetic properties of these blocks. The aim here is to provide a comprehensive account of these developments. We include new results, also some new proofs of known results; and the original work has been revised. We also present some general theory on algebras, including study of particular classes of algebras, which we think is important to understand the subject.
Suppose G is a finite group and K is a field of characteristic p; we assume that K is algebraically closed. The group algebra KG algebras,
is a direct sum of indecomposable
KG
B1 $ ... $ Bn, and the Bi are the blocks of KG. Equivalently, the identity of KG is a sum of orthogonal centrally primitive idempotents e i, and Bi
eiKG. One main topic in modular representation theory is the study of such blocks, as algebras, and their module categories. It is algebra and is
known that a block is
in particular self- injective.
a symmetric
When the prime p divides the group
order, then the blocks of KG are usually not semisimple. An analogue of the role played for KG by a Sylow p-subgroup of G is grOflp
of
a
block.
the
defect
In our context, we define a defect group of a block B
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