On defect groups for generalized blocks of the symmetric group

Conjectures of Alperin and Broue for 2-blocks with elementary abelian defect groups of order 8

Using the classification of finite simple groups we prove Alperin's weight conjecture and the character theoretic version of Broue's abelian defect group conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

Blocks with abelian defect groups of rank 2 and one simple module

In this paper, we investigate the block that has an abelian defect group of rank $2$ and its Brauer correspondent has only one simple module. We will get an isotypy between the block and its Brauer correspondent. It will generalize the result of Kessar and Linckelmann (\cite{KL}).

Blocks with a generalized quaternion defect group and three simple modules over a 2-adic ring

We show that two blocks of generalized quaternion defect with three simple modules over a sufficiently large $2$-adic ring $\mathcal O$ are Morita-equivalent if and only if the corresponding blocks over the residue field of $\mathcal O$ are Morita-equivalent. As a corollary we show that any two blocks defined over $\mathcal O$ with three simple modules and the same generalized quaternion defect group are derived equivalent.

2-blocks with an abelian defect group and a freely acting cyclic inertial quotient

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.

Blocks with quaternion defect group over a 2-adic ring: the case \tilde{A}_4

Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a p-adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed field k of characteristic 2 is Morita equivalent to $k\tilde A_4$. The main ingredients are Erdmann's classification of tame blocks and work of Cabanes and Picaronny on perfect isometries between tame blocks.

关于具有四元数亏格群的2-块

On $2$-blocks with quaternion defect groups

该研究确定了任意2幂阶且具有四元数亏格群的2-块的Morita等价类，完成了驯服表示型块的Donovan猜想的证明。

2-blocks with abelian defect groups

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan matrices are bounded in terms of the defect for arbitrary abelian 2-groups. We also show that a block with defect groups of the form $C_{2^m} \times C_{2^m}$ for $m \geq 2$ has one of two Morita equivalence types and hence is Morita equivalent to the Brauer correspondent block of the normaliser of a defect group. This completes the analysis of the Morita eq…

Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32

The first author has recently classified the Morita equivalence classes of 2-blocks B of finite groups with elementary abelian defect group of order 32. In all but three cases he proved that the Morita equivalence class determines the inertial quotient of B. We finish the remaining cases by utilizing the theory of lower defect groups. As a corollary, we verify Broué's Abelian Defect Group Conjecture in this situation.

The classification of the trivial source modules in blocks with cyclic defect groups

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.

惯性2块与阿贝尔缺陷群

Inertial 2-blocks with abelian defect groups

本文证明了具有阿贝尔缺陷群的2块为惯性块，方法基于缺陷群的结构分析。

Morita equivalence classes of blocks with elementary abelian defect groups of order 32

We describe a general technique to classify blocks of finite groups, and we apply it to determine Morita equivalence classes of blocks with elementary abelian defect groups of order 32 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two. As a consequence we verify that a conjecture of Harada holds on these blocks.

Splendid Morita equivalences for principal blocks with generalised quaternion defect groups

We prove that splendid Morita equivalences between principal blocks of finite groups with dihedral Sylow $2$-subgroups realised by Scott modules can be lifted to splendid Morita equivalences between principal blocks of finite groups with generalised quaternion Sylow $2$-subgroups realised by Scott modules.

Picard groups for some blocks with TI defect groups

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.

Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64

We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.

Donovan's conjecture and blocks with abelian defect groups

We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian $2$-groups.

On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the abelian defect group conjecture

Using a stable equivalence due to Rouquier, we prove that Broue's abelian defect group conjecture holds for 3-blocks of defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules. The proof makes use of the fact, also due to Rouquier, that a stable equivalence of Morita type between self-injective algebras induces an isomorphism between the connected components of the outer automorphism groups of the algebras.

$2$-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle II

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.

具有循环亏群的块中的自对偶不可分解模

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。

Lower defect groups and vertices of simple modules

We compare lower defect groups associated with $p$-regular classes and vertices of simple modules for a block of a finite group algebra. We show that lower defect groups are contained in vertices of simple modules after suitable reordering. Moreover, for a block of principal type, we show that a $p$-regular lower defect group which contains a vertex of simple module is a defect group of the block.

具有超特殊亏格群 $p_+^{1+2}$ 的块的Morita等价类

Morita equivalence classes of blocks with extraspecial defect groups $p_+^{1+2}$

本研究刻画了 $p\geq5$ 时具有超特殊亏格群 $p_+^{1+2}$ 的块的Morita等价类，验证了相关猜想对这类 $p$-群成立，还将 $p=3$ 时的Donovan猜想约化，并以 $p=5$ 为例列出对应等价类。

Splendid Morita equivalences for principal 2-blocks with dihedral defect groups

Given a dihedral $2$-group $P$ of order at least~8, we classify the splendid Morita equivalence classes of principal $2$-blocks with defect groups isomorphic to $P$. To this end we construct explicit stable equivalences of Morita type induced by specific Scott modules using Brauer indecomposability and gluing methods; we then determine when these stable equivalences are actually Morita equivalences, and hence automatically splendid Morita equivalences. Finally, we compute the generalised decomposition numbers in each case.

亏群为Suzuki 2-群的块

Blocks whose defect groups are Suzuki $2$-groups

本研究完成了亏群为Suzuki 2-群的块的基本Morita等价分类，验证了Donovan猜想对这类群成立，并推导得到无非平凡奇阶正规子群的有限群的Suzuki Sylow 2-子群为平凡交的推论。

profinite 块与亏群为 $\mathbb Z_p^n$ 的块的有限性性质

Finiteness properties of profinite blocks and blocks with defect group $\mathbb Z_p^n$

研究profinite群块的有限性性质，证明拓扑有限生成亏群的块同构于virtually pro-p群的块，且亏群为Z_p^n的块Morita等价于某扭曲群环，从而验证Donovan猜想。

Towards Donovan's conjecture for abelian defect groups

We define a new invariant for a $p$-block, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete the proof of Donovan's conjecture for 2-blocks with abelian defect groups of rank at most 4 and for 2-blocks with abelian defect groups of order at most 64.

Frobenius-Schur indicators of characters in blocks with cyclic defect

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…

On blocks of defect two and one simple module, and Lie algebra structure of $HH^1$

Let $k$ be a field of odd prime characteristic $p$. We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over $k$. As a consequence, we prove that if $B$ is a defect $2$-block of a finite group algebra $kG$ whose Brauer correspondent $C$ has a unique isomorphism class of simple modules, then a basic algebra of $B$ is a local algebra which can be generated by at most $2\sqrt I$ elements, where $I$ is the inertial index of $B$, and where we assume that $k$ is a splitting field for $B$ and $C$.

Kuelshammer ideals and the scalar problem for blocks with dihedral defect groups

In by now classical work, K. Erdmann classified blocks of finite groups with dihedral defect groups (and more generally algebras of dihedral type) up to Morita equivalence. In the explicit description by quivers and relations of such algebras with two simple modules, several subtle problems about scalars occurring in relations remained unresolved. In particular, for the dihedral case it is a longstanding open question whether blocks of finite groups can occur for both possible scalars 0 and 1. In this article, using Kuelshammer ideals (a.k.a. generalized Reynolds ideals), we provide the first …

关于具有循环缺陷群的块的源代数等价类，第三部分

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。

Blocks with normal abelian defect and abelian p' inertial quotient

Let $k$ be an algebraically closed field of characteristic $p$, and let $\mathcal{O}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ a finite group and $B$ a block of $\mathcal{O}G$ with normal abelian defect group and abelian $p'$ inertial quotient. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan's conjecture. For $\mathcal{O}=k$, we give an explicit description of the basic algebra of $B$ as a quiver with relations. It is a quantised version of the group algebra of t…

Morita equivalence classes of 2-blocks of defect three

We give a complete description of the Morita equivalence classes of blocks with elementary abelian defect groups of order 8 and of the derived equivalences between them. A consequence is the verification of Broué's abelian defect group conjecture for these blocks. It also completes the classification of Morita and derived equivalence classes of 2-blocks of defect at most three defined over a suitable field.

On 2-blocks with minimal nonabelian defect groups II

We determine the structure of 2-blocks with minimal nonabelian defect groups, by making use of the classification of finite simple groups.

On blocks with cyclic defect group and their head orders

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.

2-Blocks with minimal nonabelian defect groups

We study numerical invariants of 2-blocks with minimal nonabelian defect groups. These groups were classified by Rédei. If the defect group is also metacyclic, then the block invariants are known. In the remaining cases there are only two (infinite) families of "interesting" defect groups. In all other cases the blocks are nilpotent. We prove Brauer's k(B)-conjecture and the Olsson-conjecture for all 2-blocks with minimal nonabelian defect groups. For one of the two families we also show that Alperin's weight conjecture and Dade's conjecture is satisfied. This paper is a part of the author's P…

On Picent for blocks with normal defect group

We prove that if $b$ is a block of a finite group with normal abelian defect group and inertial quotient a direct product of elementary abelian groups, then $\operatorname{Picent}(b)$ is trivial. We also provide examples of blocks $b$ of finite groups with non-trivial $\operatorname{Picent}(b)$. We even have examples with normal abelian defect group and abelian inertial quotient.

循环与 Klein 四缺陷块的特征标次数

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。

On Picard groups of blocks with normal defect groups

Let $b$ be a block with normal abelian defect group and abelian inertial quotient. We prove that every Morita auto-equivalence of $b$ has linear source. We note that this improves upon results of Zhou and also Boltje, Kessar and Linckelmann.

Morita Equivalence Classes of Tame Blocks of Finite Groups

We show that several Morita equivalence classes of tame algebras do not occur as blocks of finite groups. This refines classifications by Erdmann of classes of blocks with dihedral, semidihedral, and generalised quaternion defect groups. In particular we now have a complete classification of the Morita equivalence classes of blocks of finite groups with dihedral defect groups.

Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings

We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring $\mathcal{O}$. Consequences are that Donovan's conjecture holds for $\mathcal{O}$-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan's conjecture for $\mathcal{O}$-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. …

$2$-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle

We consider a block $B$ of a finite group with defect group $D \cong (C_{2^m})^n$ and inertial quotient $\mathbb{E}$ containing a Singer cycle (an element of order $2^n-1$). This implies $\mathbb{E} = E \rtimes F$, where $E \cong C_{2^n-1}$, $F \leq C_n$, and $E$ acts transitively on the elements in $D$ of order $2$, and freely on $D \backslash \{1\}$. We classify the basic Morita equivalence classes of $B$ over a complete discrete valuation ring $\mathcal{O}$: when $m=1$, $B$ is basic Morita equivalent to the principal block of one of $SL_2(2^n) \rtimes F$, $D \rtimes \mathbb{E}$, or $J_1$ (w…

On the number of simple modules in a block of a finite group

We prove that if $B$ is a $p$-block with non-trivial defect group $D$ of a finite $p$-solvable group $G$, then $\ell(B) < p^r$, where $r$ is the sectional rank of $D$. We remark that there are infinitely many $p$-blocks $B$ with non-Abelian defect groups and $\ell(B) = p^r - 1$. We conjecture that the inequality $\ell(B) \leq p^r$ holds for an arbitrary $p$-block with defect group of sectional rank $r$. We show this to hold for a large class of $p$-blocks of various families of quasi-simple and nearly simple groups.

具有圈积亏群的主2-块在splendid Morita等价下的分类

Principal 2-blocks with wreathed defect groups up to splendid Morita equivalence

本文在Morita等价和splendid Morita等价下分类了Sylow 2-子群同构于圈积2-群的主2-块，证明了Puig的Finiteness Conjecture对此成立，并给出了相关群的纯群论分类。

Picard groups for blocks with normal defect groups and linear source bimodules

It is an open problem as to whether any bimodule inducing a Morita auto-equivalence of a block must have endopermutation source. We prove that, for blocks $b$ with normal defect groups in odd characteristic, a stronger result holds, namely that all such bimodules have linear source. We also prove the analogous result in characteristic $2$, provided that the defect group is of a specific, slightly restrictive, form.

On Picard groups of blocks of finite groups

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 …

Real characters in blocks

We consider real versions of Brauer's k(B) conjecture, Olsson's conjecture and Eaton's conjecture. We prove the real version of Eaton's conjecture for 2-blocks of groups with cyclic defect group and for the principal 2-blocks of groups with trivial real core. We also characterize G-classes, real and rational G-classes of the defect group of a block B.

The strong Frobenius numbers for cyclic defect blocks are equal to one

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.

关于具有循环缺陷群的块的源代数等价类，II

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。

On centers of blocks with non-cyclic defect groups

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.

Trivial source characters in blocks with 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.

Basic Orders for Defect Two Blocks of $\Z_p\Sym_n$

We show how basic orders for defect two blocks of symmetric groups over the ring of $p$-adic integers can be constructed by purely combinatorial means.

重访具有二面体亏格群的实块

Real blocks with dihedral defect groups revisited

本研究针对具有二面体亏格群的实2-块特征标Frobenius–Schur指示子的已有结论，证明Murray提出的两个无限族不存在，构造了其余族的实例，并给出其他tame块相关指示子的部分结果。

Classifying blocks with abelian defect groups of rank $3$ for the prime $2$

In this paper we classify all blocks with defect group $C_{2^n}\times C_2\times C_2$ up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of $2$-blocks with abelian defect groups of rank at most $3$. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index $2$, so requires nove…

Loewy lengths of centers of blocks and exponents of defect groups

In this paper we study the Loewy structure of the center $ZB$ of a block of a finite group with respect to an algebraically closed field of prime characteristic. We first state a new method for calculating the Loewy length $LL(ZB)$ of $ZB$ in terms of subsections and lower defect groups. By applying this result we give an upper bound of $LL(ZB)$ which depends on the exponent of a defect group of $B$ for some cases.

Blocks with defect group D_{2^n} * C_{2^m}

We determine the numerical invariants of blocks with defect group D_{2^n} * C_{2^m} = Q_{2^n} * C_{2^m} (central product), where n > 2 and m > 1. As a consequence, we prove Brauer's k(B)-conjecture, Olsson's conjecture (and more generally Eaton's conjecture), Brauer's height zero conjecture, the Alperin-McKay conjecture, Alperin's weight conjecture and Robinson's ordinary weight conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case.

Symmetric group blocks of small defect

This paper is an attempt to compute the decomposition numbers of the blocks of the symmetric group which have "small defect"; that is, blocks of weight smaller than the characteristic. We present various methods for computing such decomposition numbers and use these as support for a conjecture which relates decomposition numbers in different characteristics. Finally, we apply our techniques to blocks of weight three.