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Algebras and semidihedral defect groups. I. (English) Zbl 0648.20007

Throughout G denotes a finite group and F an algebraically closed field of characteristic 2. A block ideal B of the group algebra FG is of tame representation type, if the defect groups \(\delta (B)=_ GD\) of B are dihedral, semidihedral or generalized quaternion groups of order \(| D| =2\) n. In a very impressive series of highly original papers the author has described the structure of the indecomposable projective B- modules P belonging to a tame 2-block ideal B of FG. In this part of the series D is assumed to be a semidihedral group.
The classification is done in two steps. Collecting known results form group representation theory and proving some new results on the stable Auslander-Reiten quiver of a 2-block B with a semidihedral defect group D it is shown that the block B is Morita equivalent to an F-algebra \(\Lambda\) of semidihedral type in the sense of the following definition.
The F-algebra \(\Lambda\) is of semidihedral type if it has the following properties: (1) \(\Lambda\) is symmetric, indecomposable, of finite or tame representation type, and has at most three non isomorphic simple modules. (2) The stable Auslander-Reiten quiver of \(\Lambda\) has only the following components: a) tubes of rank at most 3, the number of 3-tubes being at most one, b) \(ZD_{\infty}\) and \(ZA^{\infty}_{\infty}\), and both types occur. (3) The Cartan matrix of \(\Lambda\) is non-singular. The main results of the paper give a complete classification of the algebras \(\Lambda\) of semidihedral type with \(l(\Lambda)=1\) or \(l(\Lambda)=2\) many simple \(\Lambda\)-modules. In the second part of this paper the case \(l(\Lambda)=3\) is dealt with.
If \(l(\Lambda)=2\), then it is shown here that there are five different families of F-algebras \(\Lambda\) of semidihedral type, four of which are realized by blocks B with semidihedral defect groups. In the case \(l(\Lambda)=1\) it is shown that \(\Lambda\) is Morita equivalent to the algebra F[X,Y]/I, where the two-sided ideal I is generated by \(\{\) X 2- (YX) nY, Y 2, \((YX)^{n+1}-(XY)^{n+1}\}.\)
In each case the decomposition numbers and the Cartan invariants of the blocks B of G are determined. There is a very surprising correspondence between the results on the algebras and the results on the group representations of these blocks.
Reviewer: G.Michler

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C20 Modular representations and characters
16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras
16S34 Group rings
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