On Picent for blocks with normal defect group

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.

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. …

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.

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.

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.

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.

On the blockwise modular isomorphism problem

As a generalization of the modular isomorphism problem we study the behavior of defect groups under Morita equivalence of blocks of finite groups over algebraically closed fields of positive characteristic. We prove that the Morita equivalence class of a block B of defect at most 3 determines the defect groups of B up to isomorphism. In characteristic 0 we prove similar results for metacyclic defect groups and 2-blocks of defect 4. In the second part of the paper we investigate the situation for p-solvable groups G. Among other results we show that the group algebra of G itself determines if G…

Simple modules in the Auslander-Reiten quiver of principal blocks with abelian defect groups

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

Block theory and Brauer's first main theorem for profinite groups

We develop the local-global theory of blocks for profinite groups. Given a field $k$ of characteristic $p$ and a profinite group $G$, one may express the completed group algebra $k[[G]]$ as a product $\prod_{i\in I}B_i$ of closed indecomposable algebras, called the blocks of $G$. To each block $B$ of $G$ we associate a pro-$p$ subgroup of $G$, called the defect group of $B$, unique up to conjugacy in $G$. We give several characterizations of the defect group in analogy with defect groups of blocks of finite groups. Our main theorem is a Brauer correspondence between the blocks of $G$ with defe…

Inverse images of block varieties

We extend a result due to Kawai on block varieties for blocks with abelian defect groups to blocks with arbitrary defect groups. This partially answers a question by J. Rickard.

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

We determine the numerical invariants of blocks with defect group D_{2^n}\times C_{2^m}, where D_{2^n} denotes a dihedral group of order 2^n and C_{2^m} denotes a cyclic group of order 2^m. This generalizes Brauer's results for m=0. 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.

Cartan matrices and Brauer's k(B)-conjecture

We show (among other things) that Brauer's k(B)-conjecture holds for defect groups with are central extensions of metacyclic 2-groups by cyclic groups. The same holds for defect groups which contain a central cyclic subgroup of index at most 9. In particular the k(B)-conjecture holds for 2-blocks of defect at most 4 and 3-blocks of defect at most 3. As a byproduct we obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups.

Donovan's conjecture and extensions by the centralizer of a defect group

We consider Donovan's conjecture in the context of blocks of groups $G$ with defect group $D$ and normal subgroups $N \lhd G$ such that $G=C_D(D\cap N)N$, extending similar results for blocks with abelian defect groups. As an application we show that Donovan's conjecture holds for blocks with defect groups of the form $Q_8 \times C_{2^n}$ or $Q_8 \times Q_8$ defined over a discrete valuation ring.

Loewy lengths of blocks with abelian defect groups

We consider $p$-blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index $p$. Using this, we show that if $B$ is a $2$-block of a finite group with abelian defect group $D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s$, where $a_i > 1$ for all $i$ and $r \geq 0$, then $d < LL(B) \leq 2^{a_1}+\cdots+2^{a_r}+2s-r+1$, where $|D|=2^d$. When $s=1$ the upper bound can be improved to $2^{a_1}+\cdots+2^{a_r}+2-r$. Together these give sharp upper bounds for every isomorphism type of $D$. A …

分圆Hecke代数中的亏格

Defect in cyclotomic Hecke algebras

本文证明了亏格在无限系列复反射群及分圆Yokonuma-Hecke代数中是块不变量，并揭示了其与Fayers权概念及钩长推广的对应关系。

Broué's abelian defect group conjecture holds for the Harada-Norton sporadic simple group $HN$

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime $p$, if a $p$-block $A$ of a finite group $G$ has an abelian defect group $P$, then $A$ and its Brauer corresponding block $B$ of the normaliser $N_G(P)$ of $P$ in $G$ are derived equivalent (Rickard equivalent). This conjecture is called Broué's abelian defect group conjecture. We prove in this paper that Broué's abelian defect group conjecture is true for a non-principal 3-block $A$ with an elementary abelian defect group $P$ of order 9 of the Harada-N…

Broué's abelian defect group conjecture holds for the sporadic simple Conway group Co_3

In the representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block A_N of the normaliser N_G(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Strong Version of Broué's Abelian Defect Group Conjecture. In this paper, we prove that the strong version of Broué's abelian defect group conjecture is true for the non-principal 2-block A with an elementary abelian defect group P o…

On k(D)-blocks

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.

对称群二重覆盖的阿贝尔亏群RoCK块的分解数

Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

本文计算了对称群二重覆盖的阿贝尔亏群RoCK块的(超)分解矩阵，通过利用Morita超等价定理和圈超积的表示理论，验证了第一作者的猜想。

Broué's abelian defect group conjecture holds for the double cover of the Higman-Sims sporadic simple group

In the representation theory of finite groups, there is a well-known and important conjecture, due to Broué saying that for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the normaliser N_G(P) of P in G are derived equivalent. We prove in this paper, that Broué's abelian defect group conjecture, and even Rickard's splendid equivalence conjecture are true for the faithful 3-block A with an elementary abelian defect group P of order 9 of the double cover 2.HS of the Higman-Sims sporadic simple group. It then turns out…

Broué's abelian defect group conjecture and 3-decomposition numbers of the sporadic simple Conway group Co_1

In the representation theory of finite groups, Broué's abelian defect group conjecture says that for any prime p if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the normaliser N_G(P) of P in G are derived equivalent. We prove that Broué's conjecture, and even Rickard's splendid equivalence conjecture, are true for the unique 3-block A of defect 2 of the sporadic simple Conway group Co_1, implying that both conjectures hold for all 3-blocks of Co_1. To do so, we determine the 3-decomposition numbers of A, and we actually show that…

The inductive Alperin-McKay and blockwise Alperin weight conditions for blocks with cyclic defect groups

We verify the inductive blockwise Alperin weight (BAW) and the inductive Alperin-McKay (AM) conditions introduced by the second author for blocks of finite quasisimple groups with cyclic defect groups. Furthermore we establish a criterion that describes conditions under which the inductive AM condition for blocks with abelian defect groups implies the inductive BAW condition for those blocks.

Structure of blocks with normal defect and abelian $p'$ inertial quotient

Let $k$ be an algebraically closed field of prime characteristic $p$. Let $kGe$ be a block of a group algebra of a finite group $G$, with normal defect group $P$ and abelian $p'$ inertial quotient $L$. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of $P$ with a certain subgroup of $L$. To do this, we first examine the associated graded algebra, using a Jennings--Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the non-principal block in the case of a semidirect product of an extr…

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.

Broué's abelian defect group conjecture for the sporadic simple Janko group J_4 revisited

We show that the 3-block of the sporadic simple Janko group J_4 with defect group C_3 x C_3, and the principal 3-block of the alternating group A_8 are Puig equivalent, answering a question posed in earlier work of Koshitani-Kunugi-Waki. To accomplish this, we apply computational techniques, in particular an explicit version of the Brauer construction.

Real characters in nilpotent blocks

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.

Splendid Morita equivalences for principal blocks with semidihedral defect groups

We classify principal blocks of finite groups with semidihedral defect groups up to splendid Morita equivalence. This completes the classification of all principal $2$-blocks of tame representation type up to splendid Morita equivalence and shows that Puig's Finiteness Conjecture holds for such blocks.

Profinite群块的块Pro-fusion系统与具有无限二面体亏群的块

Block pro-fusion systems for profinite groups and blocks with infinite dihedral defect groups

本文引入profinite群块的块pro-fusion系统，证明Puig幂零块结构定理的profinite版本，进而得出以无限二面体pro-2群为亏群的块仅一个Morita等价类的结论。

Descent of equivalences for blocks with Klein four defect groups

We show that the splendid Rickard complexes for blocks with Klein four defect groups constructed by Rickard and Linckelmann descend to non-split fields. As a corollary, Navarro's refinement of the Alperin-McKay conjecture holds for blocks with a Klein four defect group. We also prove that splendid Morita equivalences between blocks and their Brauer correspondents (if exist) descend to non-split situations.

On centers of blocks with one simple module

Let G be a finite group, and let B be a non-nilpotent block of G with respect to an algebraically closed field of characteristic 2. Suppose that B has an elementary abelian defect group of order 16 and only one simple module. The main result of this paper describes the algebra structure of the center of B. This is motivated by a similar analysis of a certain 3-block of defect 2 in [Kessar, 2012].

On the Alperin-McKay conjecture for 2-blocks of maximal defect

In this paper, we show that the Alperin-McKay conjecture holds for 2-blocks of maximal defect. A major step in the proof is the verification of the inductive Alperin-McKay condition for the principal 2-block of groups of Lie type in odd characteristic.

Blocks of profinite groups with cyclic defect group

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.

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.

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.

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.

Cartan matrices and Brauer's k(B)-Conjecture V

We prove Brauer's k(B)-Conjecture for the 3-blocks with abelian defect groups of rank at most 5 and for all 3-blocks of defect at most 4. For this purpose we develop a computer algorithm to construct isotypies based on a method of Usami and Puig. This leads further to some previously unknown perfect isometries for the 5-blocks of defect 2. We also investigate basic sets which are compatible under the action of the inertial group.

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.

Real Subpairs and Frobenius-Schur Indicators of Characters in 2-Blocks

Let B be a real 2-block of a finite group G. Then B has a real defect class. Let g be an element of such a class. A defect couple of B is (D,E), where E is a Sylow 2-subgroup of the extended centralizer C^*(g) of g, and D is the intersection of E with the centralizer C(g). It is known that (D,E) is uniquely determined up to G-conjugacy. We show that (D,E) determines which B-subpairs are real. We also outline how (D,E) influences the vertices of components of the G-permutation module corresponding to the conjugation action of G on its involutions. We apply these methods to enumerate the Frobeni…

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.

驯顺表示型 $2$-块的函子等价类

Functorial equivalence classes of $2$-blocks of tame representation type

本文显式确定了特征2代数闭域上具有二面体、半二面体或广义四元数亏群的有限群块在特征0代数闭域上的对角p-置换函子分解，并证明此类块函子等价当且仅当其融合系统同构。

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。

Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.

Brauer's Height Zero Conjecture for metacyclic defect groups

We prove that Brauer's Height Zero Conjecture holds for p-blocks of finite groups with metacyclic defect groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate and p-rational irreducible characters. Then the Alperin-McKay Conjecture follows provided p=3. Along the way we verify a few other conjectures. Finally we consider the extraspecial defect group of order p^3 and exponent p^2 for an odd prime more closely. Here for blocks with inertial index 2 we prove the Galois-Alperin-McKay Conjecture by computing k_0(B). Then for p…

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.