Shigeo Koshitani, Caroline Lassueur
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.
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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.
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.
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。
We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined \linebreak by R.~Kessar, N.~Kunugi and N.~Mitsuhashi. The motivation of \linebreak this paper is a fact that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of t…
Pointed fusion systems of blocks
本文研究块的尖融合系统，证明其在华丽Morita等价下完全保持，在Morita型华丽稳定等价下部分保持。
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.
By results of the second author, a source algebra equivalence between two $p$-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equi…
Morita equivalent blocks with an endotrivial source that are not globally isotypic
本文在2-模系统下构造了$\mathcal{O}{\rm SU}_3(2)$的块代数$B$，其与$\mathcal{O} Q_8$森田等价但非整体同型，进而得出二者非$p$-置换等价与精彩Rickard等价的结论。
Categorical equivalences between block algebras of finite groups - such as Morita and derived equivalences - are well-known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non splitting fields. This article presents various result on the theme of descent. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centers in non-split situations, and explain connections with Navarro's generalisation of the …
The refined Broué conjecture for RoCK blocks of double covers of symmetric and alternating groups
本文针对对称群与交错群双覆盖的RoCK $p$-块，证明Kleshchev和Livesey构造的Morita等价与精彩Rickard等价可下降到$p$-进整数环，验证了Kessar与Linckelmann提出的精细Broué阿贝尔亏群猜想。
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.
We give a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under taking the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a wreathed $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing met…
We give a handy way to have a situation that the $kG$-Scott module with vertex $P$ remains indecomposable under taking the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $p>0$. The motivation is that the Brauer indecomposability of a $p$-permutation bimodule is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method, that then can possibly lift to a splendid derived equivalence. Further our result explains a hidden reason why the Brauer indecomposability of the Scot…
Rickard's Derived Morita Theory: Review and Outlook
本文回顾里卡德在导出范畴莫里塔理论中的核心成果，并探讨其对后续发展的影响，包括在代数和拓扑领域中的应用及对布鲁埃猜想的贡献。
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.
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.
By results of Rognerud, a source algebra equivalence between two $p$-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above me…
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。
We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of the class of Erdmann's algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander-Reiten conjecture for these classes of algebras.
We use the dimension and the Lie algebra structure of the first Hochschild cohomology group to distinguish some algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type. In particular, we complete the classification of algebras of dihedral type that was mostly determined by Zhou and Zimmermann.
Trivial source characters in blocks of domestic representation type
研究国内表示类型块中平凡源模块的普通特征，确定其在不同块情况下的表现。
We introduce and study the algebras of generalized quaternion type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups. We prove that all these algebras, with 2-regular Gabriel quivers, are periodic algebras of period 4 and very specific deformations of the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles. The main result o f the paper forms an important step towards the Morita equivalence classification of all periodic symmetric tame algebras of non-polynomial gro…
This paper is a continuation and a completion of [BoRo1]. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in non-describing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic subgroups. We also show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of [BoRo1], …
A generalization of Dugas' construction on stable auto-equivalences for symmetric algebras
本文将Dugas关于Morita型稳定自同构的构造从局部对称代数推广到任意对称代数，恢复了p-群群代数上对应内平凡模的自同构，并给出了非局部对称代数的多个实例。
Fusion-stable endosplit $p$-permutation resolutions
本文证明饱和融合系下满足特定 Dade 类条件的稳定内置换模具有内分裂 $p$-置换分解，并由此推出 Morita 等价的两个块是华丽 Rickard 等价的。
For a symmetric algebra A over a field K of characteristic p > 0 K{ü}lshammer constructed a descending sequence of ideals of the centre of A. If K is perfect this sequence was shown to be an invariant under derived equivalence and for algebraically closed K under stable equivalence of Morita type. Erdmann classified algebras of tame representation type which may be blocks of group algebras, and Holm classified Erdmann's list up to derived equivalence. In both classifications certain parameters occur in the classification, and it was unclear if different parameters lead to different algebras. E…
The source permutation module of a block of a finite group algebra
本文引入有限群代数块分量的源置换模，证明其在 splendid Morita 等价下不变，并将 Sylow 置换模自内射性与 Alperin 权猜想的相关结果推广到该模，同时计算了若干具体情形。
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