Frauke M. Bleher

Universal deformation rings and dihedral defect groups

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group and B is a block of kG with dihedral defect group D, which is Morita equivalent to the principal 2-modular block of a finite simple group. We determine the universal deformation ring R(G, V) for every kG-module V which belongs to B and has stable endomorphism ring k. It follows that R(G, V) is always isomorphic to a subquotient ring of WD. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections.

Universal deformation rings and Klein four defect groups

In this paper, the universal deformation rings of certain modular representations of a finite group are determined. The representations under consideration are those which are associated to blocks with Klein four defect groups and whose stable endomorphisms are given by scalars. It turns out that these universal deformation rings are always subquotient rings of the group ring of a Klein four group over the ring of Witt vectors.

Universal deformation rings and generalized quaternion defect groups

Abstract We determine the universal deformation rings R ( G , V ) of certain mod 2 representations V of a finite group G which belong to a 2-modular block of G whose defect groups are isomorphic to a generalized quaternion group D. We show that for these V, a question raised by the author and Chinburg concerning the relation of R ( G , V ) to D has an affirmative answer. We also show that R ( G , V ) is a complete intersection even though R ( G / N , V ) need not be for certain normal subgroups N of G which act trivially on V.

Autoequivalences of blocks and a conjecture of Zassenhaus

Abstract In this paper, we show that for every finite group with cyclic Sylow p-subgroups the principal p-block B is rigid with respect to the trivial simple module. This means that each autoequivalence which fixes the trivial simple module fixes the isomorphism class of each finitely generated B-module. As a consequence each augmentation preserving automorphism of the integral group ring of PSL(2, p), p a rational prime, is given by a group automorphism followed by a conjugation in QPSL(2, p). In particular this proves a conjecture of Zassenhaus for these groups. Finally we show the same statement for a couple of other simple groups by different methods.

Universal deformation rings of modules over Frobenius algebras

Abstract Let k be a field, and let Λ be a finite dimensional k-algebra. We prove that if Λ is a self-injective algebra, then every finitely generated Λ-module V whose stable endomorphism ring is isomorphic to k has a universal deformation ring R ( Λ , V ) which is a complete local commutative Noetherian k-algebra with residue field k. If Λ is also a Frobenius algebra, we show that R ( Λ , V ) is stable under taking syzygies. We investigate a particular Frobenius algebra Λ 0 of dihedral type, as introduced by Erdmann, and we determine R ( Λ 0 , V ) for every finitely generated Λ 0 -module V whose stable endomorphism ring is isomorphic to k.

Universal deformation rings and dihedral 2-groups

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose that D is a dihedral 2-group. We prove that the universal deformation ring R(D, V) of an endo-trivial kD-module V is always isomorphic to W [/2x/2]. As a consequence, we obtain a similar result for modules V with stable endomorphism ring k belonging to an arbitrary nilpotent block with defect group D. This confirms, for such V, conjectures on the ring structure of the universal deformation ring of V that had previously been shown for V belonging to cyclic blocks or to blocks with Klein four defect groups.

Universal Deformation Rings and Dihedral Blocks with Two Simple Modules

Abstract Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group and B is a block of kG with a dihedral defect group D such that there are precisely two isomorphism classes of simple B-modules. We determine the universal deformation ring R ( G , V ) for every finitely generated kG-module V which belongs to B and whose stable endomorphism ring is isomorphic to k. The description by Erdmann of the quiver and relations of the basic algebra of B is usually only determined up to a certain parameter c which is either 0 or 1. We show that R ( G , V ) is isomorphic to a subquotient ring of WD for all V as above if and only if c = 0 , giving an answer to a question raised by the first author and Chinburg in this case. Moreover, we prove that c = 0 if and only if B is Morita equivalent to a principal block.

Tensor products and a conjecture of Zassenhaus

These conjectures should be seen as a stronger version of the integral isomorphism problem, i.e. the question whether 7ZG ~ 2gH implies G ~ H. Obviously (ZC) implies (ZCAut). If the isomorphism problem has a positive answer for G, then also the converse is true. Note that the isomorphism problem has a positive answer for all finite simple groups [12]. Thus, if G is finite simple, only the validity of (ZCAut) has to be shown in order to prove (ZC) for G. Using the ordinary character table there is an equivalent statement to (ZCAut): (ZCAut) holds for G if, and only if, for every a ~ Aut,(7lG) there exists z ~ Aut (G) which acts in the same way on the ordinary character table as ~ does. These two conjectures of Zassenhaus led to new considerations in recent years. One of the first results in this context is due to Peterson who proved that the symmetric groups S, are elementary represented which means that for S. (ZCAut) is valid [15]. It was probably this theorem that led to the formulation of the two conjectures. In general, the conjecture (ZC) is not valid, as shown by Roggenkamp and Scott who constructed a metabelian counterexample [18, IX w i]. Nevertheless, they were able to prove that (ZC) holds for nilpotent groups and for groups whose generalized Fitting group is a p-group [16, 17]. However, there is very limited knowledge with respect to simple non-abelian groups. The main result of this paper is the proof of the following two theorems.

Universal Deformation Rings of Modules for Algebras of Dihedral Type of Polynomial Growth

Let k be an algebraically closed field, and let Λ be an algebra of dihedral type of polynomial growth as classified by Erdmann and Skowroński. We describe all finitely generated Λ-modules V whose stable endomorphism rings are isomorphic to k and determine their universal deformation rings R(Λ, V). We prove that only three isomorphism types occur for R(Λ, V): k, k[[t]]/(t2) and k[[t]].

Universal Deformation Rings for the Symmetric Group S (4)

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S4,V) for every kS4-module V which has stable endomorphism ring k and show that R(S4,V) is isomorphic to either k, or W[t]/(t2,2t), or the group ring W[ℤ/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.