On certain quotients of the Green rings of dihedral 2-groups

In this paper we study the properties of Green rings of dihedral 2-groups, and in particular certain quotients of these Green rings introduced by Benson and Carlson. It is shown that these quotients can be realised as group rings over . The properties of the corresponding groups are investigated: they are shown to be abelian, torsion-free and infinitely generated. We also show how taking products of elements of these groups is related to the structure of the Auslander–Reiten quivers for dihedral 2-groups.     

Universal deformation rings and cyclic blocks

In this paper we determine the universal deformation rings of certain modular representations of finite groups which belong to cyclic blocks. The representations we consider are those for which every endomorphism is stably equivalent to multiplication by a scalar. We then apply our results to study the counterparts for universal deformation rings of conjectures about embedding problems in Galois theory. Received July 19, 1999 / Revised May 13, 2000 / Published online October 30, 2000     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.     

Universal deformation rings and generalized quaternion defect groups

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.     

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 need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups. The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. To cite this article: F.M. Bleher, T. Chinburg, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Describing units of integral group rings of some 2-groups

Let R be a ring and M a right R-module. The module M is a CS-module or satifies (C₁) if every submodule is essential in a direct summand of M. In this note we investigate two generalizations of CS-modules.     

Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.     

Augmentation quotients for burnside rings of generalized dihedral groups

Let H be a finite abelian group of odd order, D be its generalized dihedral group, i.e., the semidirect product of C₂ acting on H by inverting elements, where C₂ is the cyclic group of order two. Let Ω (D) be the Burnside ring of D, Δ(D) be the augmentation ideal of Ω (D). Denote by Δⁿ(D) and Q_n(D) the nth power of Δ(D) and the nth consecutive quotient group Δⁿ(D)/Δⁿ⁺¹(D), respectively. This paper provides an explicit Z-basis for Δⁿ(D) and determines the isomorphism class of Q_n(D) for each positive integer n.     

Augmentation quotients for complex representation rings of dihedral groups

Denote by D _m the dihedral group of order 2m. Let ℛ(D _m ) be its complex representation ring, and let Δ(D _m ) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δ ⁿ (D _m )/Δ ⁿ⁺¹(D _m ) for each positive integer n.     

On isomorphisms between deformation rings and Hecke rings

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