426 research outputs found
Universal deformation rings and generalized quaternion defect groups
We determine the universal deformation ring 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.Comment: 20 pages, 6 figures. The paper has been updated as follows: The
results remain true for more general 2-modular blocks with generalized
quaternion defect groups (see the introduction and Hypothesis 3.1). Sections
4 and 5 have been swapped
Large universal deformation rings
We provide a series of examples of finite groups G and mod p representations
V of G whose stable endomorphisms are all given by scalars such that V has a
universal deformation ring R(G,V) which is large in the sense that
R(G,V)/pR(G,V) is isomorphic to a power series algebra in one variable.Comment: 9 pages, 5 figures; for Prop. 2, the description of the groups and
the proof have slightly change
Universal deformation rings of modules over Frobenius algebras
Let be a field, and let be a finite dimensional -algebra. We
prove that if is a self-injective algebra, then every finitely
generated -module whose stable endomorphism ring is isomorphic to
has a universal deformation ring which is a complete local
commutative Noetherian -algebra with residue field . If is also
a Frobenius algebra, we show that is stable under taking
syzygies. We investigate a particular Frobenius algebra of dihedral
type, as introduced by Erdmann, and we determine for every
finitely generated -module whose stable endomorphism ring is
isomorphic to .Comment: 25 pages, 2 figures. Some typos have been fixed, the outline of the
paper has been changed to improve readabilit
Dihedral blocks with two simple modules
Let be an algebraically closed field of characteristic 2, and let be
a finite group. Suppose is a block of with dihedral defect groups such
that there are precisely two isomorphism classes of simple -modules. The
description by Erdmann of the quiver and relations of the basic algebra of
is usually only given up to a certain parameter which is either 0 or 1. In
this article, we show that if there exists a central extension
of by a group of order 2 together with a block of with
generalized quaternion defect groups such that is contained in the image of
under the natural surjection from onto . As a special
case, we obtain that if for some odd
prime power and is the principal block of .Comment: 11 pages, 5 figures. The arguments work also for non-principal
blocks. The paper has been changed accordingly; in particular, the word
"principal" was removed from the titl
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.Comment: 37 pages, 13 figures. Changed introduction, updated reference
Universal deformation rings of modules for algebras of dihedral type of polynomial growth
Let k be an algebraically closed field, and let \Lambda\ be an algebra of
dihedral type of polynomial growth as classified by Erdmann and Skowro\'{n}ski.
We describe all finitely generated \Lambda-modules V whose stable endomorphism
rings are isomorphic to k and determine their universal deformation rings
R(\Lambda,V). We prove that only three isomorphism types occur for
R(\Lambda,V): k, k[[t]]/(t^2) and k[[t]].Comment: 11 pages, 2 figure
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 S_4 denote the symmetric group on 4
letters. We determine the universal deformation ring R(S_4,V) for every
kS_4-module V which has stable endomorphism ring k and show that R(S_4,V) is
isomorphic to either k, or W[t]/(t^2,2t), 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.Comment: 12 pages, 2 figure
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