On blocks with cyclic defect group and their head orders
Gabriele Nebe
它表明，第8节Plesken的1983年讲义描述块循环亏损群森田等价。特别地，这样的块由其平面嵌入的Brauer树确定。应用根理想化过程，明确计算这样的块的头顺序。
It is shown that Section 8 of Plesken's 1983 lecture notes describes blocks of cyclic defect group up to Morita equivalence. In particular such a block is determined by its planar embedded Brauer tree. Applying the radical idealizer process, the head order of such blocks is calculated explicitly.
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On character degrees of cyclic and Klein four defect blocks
本文证明 Giannelli 关于高度零特征标次数的猜想适用于缺陷群为循环群或 Klein 四群的块。
The purpose of this note is to provide a reference for the fact that the strong Frobenius number, in the sense of Eaton and Livesey, of a block of a finite group with a cyclic defect group is equal to one. This answers a question of Farrell and Kessar.
In this short note we study the center ZB of a block B of a finite group over an algebraically closed field of prime characteristic through its Loewy length llZB. A result of Okuyama in 1981 gave an upper bound for llZB in terms of defect group of B. The purpose of this note is to improve this bound for non-cyclic defect groups.
We describe the ordinary characters of trivial source modules lying in blocks with cyclic defect groups relying on their recent classification in terms of paths on the Brauer tree by G.~Hiss and the second author. In particular, we show how to recover the exceptional constituents of such characters using the source algebra of the block.
Broué's Abelian Defect Conjecture predicts interesting derived equivalences between derived categories of modular representations of finite groups. We investigate a generalization of Broué's Conjecture to ring spectrum coefficients and prove this generalization in the cyclic defect case, following an argument of Rouquier.
We demonstrate that the blocks of a profinite group whose defect groups are cyclic have a Brauer tree algebra structure analogous to the case of finite groups. We show further that the Brauer tree of a block with defect group $\mathbb{Z}_{p}$ is of star type.
We study blocks with an abelian defect group and a cyclic inertial quotient acting freely but not transitively. We prove that when p=2, such blocks are inertial, i.e. basic Morita equivalent to their Brauer correspondent. Together with a result of the second author on Singer cycle actions on homocyclic defect groups, this completes the classification of 2-blocks with a cyclic inertial quotient acting freely on an abelian defect group.
The inductive Alperin-McKay and blockwise Alperin weight conditions for blocks with cyclic defect groups
我们证明了第二作者对有限拟单群的块具有循环亏群所引入的归纳分块Alperin重量（BAW）和归纳Alperin-McKay（AM）条件。此外，我们建立了一个标准，描述的条件下，与阿贝尔缺陷群块的诱导AM条件意味着这些块的诱导BAW条件。
The self-dual indecomposable modules in blocks with cyclic defect groups
本文研究具有循环亏群的有限群 $p$-块中的自对偶不可分解模，并确定每个模是辛型还是正交型。
Relying on the classification of the indecomposable liftable modules in arbitrary blocks with non-trivial cyclic defect groups we give a complete classification of the trivial source modules lying in such blocks, describing in particular their associated path on the Brauer tree of the block in the sense of Janusz (1969). The appendix contains a description of the minimal distance from an arbitrary non-projective indecomposable liftable module to the boundary of the stable Auslander-Reiten quiver of the block.
Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed, using the canonical character of $B$. We also investigate the Frobenius-Schur indicators of the non-exceptional characters in $B$. For a finite group which has cyclic Sylow $p$-subgroups, we show that the number of irreducible characters with Frobenius-Schur indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements i…
The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups
本文研究具有循环亏群的块的完美p-置换双模表示环的环论结构，证明嘉当矩阵含初等因子1时[B]非本原，并给出本原分解及标量扩张后的直积分解。
Cartan matrices and Brauer's k(B)-conjecture
我们证明了Brauer的k（B）-猜想对亚循环2-群通过循环群的中心扩张的亏群成立。这同样适用于包含指数至多为9的中心循环子群的亏群。特别地，k（B）-猜想对于2-块缺陷至多为4和3-块缺陷至多为3成立。作为一个副产品，我们得到了具有最小非亚循环缺陷群的2-块的块不变量。
On the source algebra equivalence class of blocks with cyclic defect groups, III
本文研究了具有循环缺陷群的p-块的源代数等价类，通过分类不变量W(B)来确定其等价类。
On the source algebra equivalence class of blocks with cyclic defect groups, II
本文研究了循环p-群的端排列模块在拟单群p-块下的分类，通过将问题转化为一般线性群和单位ary群的分类来完成。
We consider $2$-blocks of finite groups with defect group $D=Q \times R$ and inertial quotient $\mathbb{E}$ where $Q \cong (C_{2^m})^n$, $R \cong C_{2^r}$, and $\mathbb{E}$ contains a Singer cycle of $\operatorname{Aut}(Q)$ (an element of order $2^n-1$). We classify such blocks up to Morita equivalence when either $\mathbb{E}$ is cyclic or $r=1$. We achieve a partial classification when $r>1$ and $E$ is non-cyclic.
We compute the dimensions of $\operatorname{Ext}_G^n(V, W)$ for all irreducible $V$, $W$ lying in $r$-blocks of cyclic defect in the simple groups $\operatorname{Sz}(q)$, $\operatorname{PSU}_3(q)$ and $\operatorname{{}^2G}_2(q)$ in cross characteristic, obtaining in particular the dimensions of all cohomology groups for such modules. Along the way, we also obtain an analogous result for any $r$-block of cyclic defect whose Brauer tree is either a star or line (open polygon).
Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander-Reiten quiver of the principal block of a finite group with non-cyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$, we prove that simple modules in the principal block all lie at the end of their components
In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the `groups' $I_2(n,q)$, $H_3(q)$ and $H_4(q)$. The degrees of the unipotent characters of these objects were given by Lusztig, and using the general theory of perverse equivalences we can reconstruct the Brauer trees that would be consistent with Deligne--Lusztig theory and the geometric version of Broué's conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Broué's conjecture are consistent with both the mathematics and philo…
Brauer's Height Zero Conjecture for metacyclic defect groups
本文证明了Brauer的零高度猜想对具有亚循环亏群的有限群的p-块成立。若亏群是非交换群且含有循环极大子群，则得到了p-共轭和p-有理不可约特征标的分布。假设p=3，则Alperin-McKay猜想成立。一路上，我们验证了一些其他的假设。最后，我们更仔细地考虑了奇素数的p^3阶指数p^2的超特殊亏群。对于惯性指数为2的块体，本文通过计算k_0（B）证明了Galois-Alperin-McKay猜想。那么对于p \le 11，Alperin的权猜想也成立。这改进了[Gao，2012]，[Holloway-Koshitani-Kunugi，2010]和[Henri，2005]的一些结果。
Blocks with defect group D_{2^n} x C_{2^m}
本文确定了亏群为D_{2^n} \times C_{2^m}的块的数值不变量，其中D_{2^n}表示2^n阶二面体群，C_{2^m}表示2^m阶循环群.这推广了Brauer在m=0时的结果。因此，我们证明了Brauer的k（B）-猜想，Olsson的猜想（和更一般的Eaton的猜想），Brauer的零高度猜想，Alperin-McKay猜想，Alperin的重量猜想和Robinson的普通重量猜想。此外，我们表明，胶合问题有一个独特的解决方案，在这种情况下。
The objective of this research paper is to study the relationship between a block of a finite group and a defect group of such block. We define a new notion which is called a strongly $k(D)$-block and give a necessary and sufficient condition of a block with a cyclic defect group to be a $k(D)$-block in term of its inertial index.
We calculate the Picard groups for principal blocks $B$ with TI defect groups and cyclic inertial quotient. The methods used generalize results on self stable equivalences and take advantage of the existence of equivalences given by Green correspondence in this setting. In particular, we show that $\text{Pic}(B)=\mathcal{E}(B)$, giving more evidence for a conjecture on basic auto-Morita equivalences.
We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.
Perverse Equivalences and Broué's Conjecture II: The Cyclic Case
利用最近的反常等价理论和Deligne-Lusztig簇研究了Lie型群的Broué交换亏群猜想.我们的方法是直接分析某些Deligne-Lusztig簇（Broué猜想的几何形式）引起的反常等价;这使用了这些簇的上同调，以及来自分圆Hecke代数的信息。我们从一个猜想上同调的这些Deligne-Lusztig品种，证明各种理想的性质，然后用它来证明存在的反常等价预测的几何形式的Broué猜想每当缺陷组是循环的（除了可能为两个块的布劳尔树是未知的）。这是一个必要的第一步，以证明布鲁埃猜想一般，作为反常等价建立归纳从各种列维小组。这篇文章是最新的一系列拉斐尔Rouquier和作者的最终目的是证明Broué猜想的幂幺块群的李型。
We prove that the number of irreducible real characters in a nilpotent block of a finite group is locally determined. We further conjecture that the Frobenius-Schur indicators of those characters can be computed for p=2 in terms of the extended defect group. We derive this from a more general conjecture on the Frobenius-Schur indicator of projective indecomposable characters of 2-blocks with one simple module. This extends results of Murray on 2-blocks with cyclic and dihedral defect groups.
Let $p$ be a prime number, $k$ an algebraically closed field of characteristic $p$, $\tilde{G}$ a finite group, and $G$ a normal subgroup of $\tilde{G}$ having a $p$-power index in $\tilde{G}$. Moreover let $B$ be a block of $kG$ with a cyclic defect group and $\tilde{B}$ be the unique block of $k\tilde{G}$ covering $B$. We study tilting complexes over the block $\tilde{B}$ and show that the block $\tilde{B}$ is a tilting-discrete algebra. Moreover we show that the set of all tilting complexes over $\tilde{B}$ is isomorphic to that over $B$ as partially ordered sets.
Defect Invariant Nakayama Algebras
本文证明了存在唯一的循环 Nakayama 代数使其合流过滤代数与给定代数缺陷不变，并据此分类了极小 Auslander-Gorenstein 等代数，获得了 cluster-tilting 对象。
Blocks with quaternion defect group over a 2-adic ring: the case \tilde{A}_4
除了具有循环亏群或Klein四亏群的块之外，一般不知道素特征域上的块代数的Morita等价类是否决定了p-adic环上相应块代数的Morita等价类.我们证明了这是当缺陷群是8阶四元数和块代数特征为2的代数闭域k是森田等价于$k\tilde A_4$的情况下。主要成分是Erdmann的驯服块的分类和Cabanes和Picaronny的驯服块之间的完美等距的工作。
Real characters in blocks
我们考虑Brauer的k（B）猜想，Olsson的猜想和Eaton的猜想的实版本.本文证明了Eaton猜想的实版本，即对于具有循环亏群的群的2-块和具有平凡实核的群的主2-块。我们还刻画了块B的亏群的G-类、实G-类和有理G-类。
Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a "large" family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even the isomorphism type of D is determined in this situation. Moreover, it can read off from the character table whether |D/D'|=4 where D' denotes the commutator subgroup of D. We also propose a new cha…
Hyperfocal subalgebras of hyperfocal abelian Frobenius blocks
本文引入了超焦点阿贝尔弗罗贝尼乌斯块，研究其超焦点亚代数与弗罗贝尼乌斯群群代数的莫rita型稳定等价，探讨了超焦点子群为克莱因四元组和循环子群的块结构。
Virtual Morita equivalences and Brauer character bijections
本文将 Kessar 和 Linckelmann 关于 Morita 等价与 Brauer 特征双射的定理推广至虚拟 Morita 等价，并由此证明 Navarro 对 Alperin 权猜想的精化在多种特定缺陷群及线性群块中成立。
We show that the subgroup of the Picard group of a $p$-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over the a complete discrete valuation ring ${\mathcal O}$ of characteristic zero with an algebraic closure $k$ of ${\mathbb F}_p$ as residue field is a colimit of finite Picard groups of blocks over $p$-adic subrings of ${\mathcal O}$. We apply the results to blocks with an abelian defect group and Frobenius …
In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author. As a consequence, the Brauer trees of principal $\ell$-blocks of finite groups are known for $\ell>71$.
Universal deformation rings and dihedral 2-groups
设$k$是特征为2的代数闭域，$W$是$k$上的无穷Witt向量环.设$D$是二面体2-群.证明了内平凡$kD$ -模$V$的泛变形环$R(D,V)$总是同构于$W[\mathbb{Z}/2\times\mathbb{Z}/2]$ .作为结果，我们得到了一个类似的结果，模$V$与稳定的自同态环$k$属于一个任意的幂零块与亏损群$D$。这证实了这种$V$的环结构上的通用变形环的$V$，这已被证明为$V$属于循环块或块克莱因四个缺陷群。
Brauer's problem 21 for principal blocks
本文证明主块情形下 Brauer 第 21 号问题成立，并将含 3 个不可约特征标的块的亏群问题在多数情形归约为单群问题。
A local conjecture on Brauer character degrees of finite groups
最近，第二作者提出了关于有限群的不可约Brauer特征标的次数的一个新猜想。本文对有限群的块B提出了这个猜想的一个“局部”版本，给出了属于块B的不可约Brauer特征标的最大度的一个下界，它是关于块B的维数和不可约Brauer特征标的不变量，如亏损和不可约Brauer特征标的个数.我们还提出了一个较弱的版本，这个猜想轻微的重新制定导致有趣的公开问题的痕迹块的嘉当矩阵。然后，我们证明了强猜想是真实的块与一个简单的模块，块的p-可解群和块与循环亏损群。它也适用于零星群、对称群或李型群的块的许多进一步的例子。我们还表明，弱猜想是真实的驯服表示类型的块。
We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal p-block of a finite group G the normalized unit group of the integral group ring of G contains an element of order pq if and only if so does G, for q a prime different from p. Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.
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