Gerhard Roehrle, Raphael Rouquier
We prove analogues for reductive algebraic groups of some results for finite groups due to Knoerr and Robinson which play a central role in their reformulation of Alperin's conjecture for finite groups.
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In this paper we consider the inductive Alperin-McKay condition for quasi-isolated 2-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin-McKay conjecture for the prime 2.
We show that Dade's ordinary conjecture implies the Alperin-McKay conjecture. We remark that some of the methods can be used to identify a canonical height zero character in a nilpotent block.
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.
The Blockwise Navarro Alperin Weight Conjecture for Double Covers of Symmetric and Alternating Groups
本文证明了对称群与交错群双重覆盖的分块 Navarro Alperin 权猜想。
The Alperin weight conjecture was reduced to simple groups by the work of Navarro, Tiep and Späth. To prove Alperin weight conjecture, it suffices to show that all finite non-abelian simple groups are BAW-good. We reduce the verification of the inductive conditons for groups of Lie type in non-defining characteristic to quasi-isolated blocks.
The Alperin-McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Späth showed that the Alperin-McKay conjecture holds if the so-called inductive Alperin-McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this we obtain a proof of the Alperin-McKay conjecture for…
The so-called inductive McKay condition on finite simple groups, due to Isaacs-Malle-Navarro (2007), has been recently reformulated by Späth. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro-Tiep (2011). This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.
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.
In this paper characters of the normaliser of $d$-split Levi subgroups in $\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$ are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer.These results are applied to verify the Alperin-McKay conjecture for primes $\ell$ with $\ell\nmid 6(q^2-1)$ and the Alperin weight conjecture for $\ell$-blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of $\mathrm {SL}_n(q)$ and $\mat…
Equivariant chararacter bijections and the Alperin-McKay conjecture
本文研究了Lie型B和C群孤立块的归纳Alperin-McKay条件，完成了该类群归纳条件的验证。
Recently, there has been substantial progress on the Alperin weight conjecture. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for simple groups of classical type under some additional assumption.
The Alperin Weight Conjecture and the Glauberman correspondence via character triples
本文通过特征标三元组验证了纳瓦罗猜想，并将其简化为单列群问题，证明了若干类群和块的结论。
On Alperin's conjecture and functorial equivalence of blocks
本文在特征 $p$ 与特征 0 域上利用块的稳定函子等价性，提出并证明 Alperin 分块权猜想的函子版本等价于原猜想，且其在稳定对角 $p$-置换函子范畴中成立。
In this paper, we establish the inductive Alperin weight condition for the finite simple groups of Lie type $\mathsf A$, contributing to the program to prove the Alperin weight conjecture by checking the inductive condition for all finite simple groups.
Alperin's weight conjecture, Galois automorphisms, alternating sums, and functorial equivalences
本文通过函子等价方法，证明了块级伽罗瓦阿尔佩林重量猜想的等价性，并探讨了伽罗瓦共轭块的函子等价性。
The Galois Alperin weight conjecture for finite category algebras
本文提出有限范畴代数的伽罗瓦阿尔佩林权重猜想，将其转化为有限群问题，并针对EI范畴提出块级猜想。
By the reduction theorems of Navarro--Tiep and Späth, a way to prove the Alperin weight conjecture and its blockwise version is to verify the co-called inductive Alperin weight condition and inductive blockwise Alperin weight condition for all finie simple groups respectively. In this paper, we establish the inductive blockwise Alperin weight condition for simple groups of type $\mathsf B$ and odd primes, using a criterion given by Brough and Späth recently.
In this paper we prove the blockwise Alperin weight conjecture for finite special linear and unitary groups, for finite groups with abelian Sylow $3$-subgroups, and verify the inductive blockwise Alperin weight condition for certain cases of groups of type $\mathsf A$. We also give a classification for the 2-blocks of special linear and unitary groups.
The Alperin-McKay conjecture relates height zero characters of an $\ell$-block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin-McKay conditions about quasi-simple groups by the third author. Those conditions are still open for groups of Lie type. The present paper describes characters of height zero in $\ell$-blocks of groups of Lie type over a field with $q$ elements when $\ell$ divides $q-1$. We also give information about $\ell$-blocks and Brauer correspondents. As an application we show that quasi-simple groups of type $C$ over…
A reduction theorem for the blockwise Navarro Alperin weight conjecture via H-triples
本文通过H-三元组的块状导航阿尔佩林重量猜想提出简化定理，基于GAW猜想的归纳条件，建立更强的GAW猜想与块同构性关系。
In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions. Our purpose here is to show to the interested reader that the results in our book "Frobenius categories versus Brauer blocks", Progress in Math. 274(2009), and the reduction arguments in "On the reduction of Alperin's Conjecture to the quasi-simple groups", J. of Algebra 328(2011), suggest a numerical statement - implying Alperin's Conjecture block by block…
The so-called "local-global" conjectures in the representation theory of finite groups relate the representation theory of $G$ to that of certain proper subgroups, such as the normalizers of particular $p$-groups. Recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups. Here, we show that $G=Sp_6(2^a)$ is "good" for these reductions for the McKay conjecture, the Alperin weight conjecture, and their blockwise versions.
We show that the refinement of Alperin's Conjecture proposed in "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, can be proved by checking that this refinement holds on any central k*-extension of a finite group H containing a normal simple group S with trivial centralizer in H and p'-cyclic quotient H/S. This paper improves our result in [ibidem, Theorem 16.45] and repairs some bad arguments there.
In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions. Our purpose here is to show that the equivariant refinement of the Alperin's Conjecture for blocks formulated by Geoffrey Robinson in the eighties can be reduced to checking the same statement on any central k*-extension of any finite almost-simple group, or of any finite simple group up to verifying an "almost necessary" condition. In an Appendix we develo…
As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro--Tiep and Späth for simple groups of Lie type $\mathsf A$, split or twisted. Key to the proofs is to reduce the verification of the inductive condition to the isolated (that means unipotent) blocks, using the Jordan decomposition for blocks of finite reductive groups given by Bonnafé, Dat and Rouquier.
A Reduction Theorem for the Alperin-McKay-Navarro Conjecture
本文通过将 Alperin-McKay-Navarro 猜想约化到非阿贝尔有限单群的万有覆盖群上的归纳条件，并验证若干零星群和 Suzuki 群、小 Ree 群的情形，证明了该猜想在这些情形下成立。
Alperin's Main Problem of Block Theory
本文提出一个猜想框架，通过非消失字符集和子正规化子群重新表述Alperin的块理论主要问题，并在多个族（包括具有TI Sylow p-子群的单群）中验证主要猜想。
Generic weights for finite reductive groups
本文受 Alperin 权猜想启发，推广了有限简约群 e-Harish-Chandra 理论中的 e-cuspidality 概念，定义了非定义特征下的通用权，并证明其在研究 Lie 型单群的归纳 Alperin 权条件时发挥类似作用，为最终证明该猜想提供新思路。
Let $p$ be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd $p$-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture and the Alperin-McKay conjecture hold for such group.
Alperin's bound and normal Sylow subgroups
本文将关于 $p$-Brauer 特征标数量与 Sylow $p$-子群正规性的猜想归约到有限单群并证明 $p=2$ 情形，同时建立 Alperin 下界分块版本的归约定理。
This paper has two main parts. Firstly, we give a classification of the $\ell$-blocks of finite special linear and unitary groups $SL_n(εq)$ in the non-defining characteristic $\ell\ge 3$. Secondly, we describe how the $\ell$-weights of $SL_n(εq)$ can be obtained from the $\ell$-weights of $GL_n(εq)$ when $\ell\nmid\mathrm{gcd}(n,q-ε)$, and verify the Alperin weight conjecture for $SL_n(εq)$ under the condition $\ell\nmid\mathrm{gcd}(n,q-ε)$. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent $…
We show that Brauer's height zero conjecture holds for blocks of finite quasi-simple groups. This result is used in Navarro-Späth's reduction of this conjecture for general groups to the inductive Alperin-McKay condition for simple groups.
The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Spaeth and Koshitani. Thus, to prove this conjecture, it suffices to verify the corresponding inductive condition for all finite simple groups. The first is to establish an equivariant bijection between irreducible Brauer characters and weights for the universal covering groups of simple groups. Assume q is a power of some odd prime p. We first prove the blockwise Alperin weight conjecture for Sp2n(q) and odd non-defining characteristics. If the decomposition matri…
Radical subgroups of finite reductive groups
本文提出统一方法分类有限约化群的根$p$-子群，并验证其归纳分块Alperin权条件，应用于经典群与$F_4$型群，推进Alperin权猜想的证明。
We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups.
Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace $p$ by a set of primes pi and prove a pi-version of Dade's conjecture for pi-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a pi-version of Alperin's weight conjecture previously established by the authors.
The Character Triple Conjecture for maximal defect characters and the prime 2
本文将极大亏格情形归约为归纳 Alperin–McKay 条件，证明 Späth 特征三元组猜想在素数2对所有有限群成立，并推出阿贝尔亏群2-块及任意素数无块版本的结果。
On the inductive blockwise Alperin weight condition for type $\mathsf B$ and type $\mathsf C$
本文证明类型 B 和 C 的有限单群满足归纳分块 Alperin 权条件，通过置换格、Jordan 约化等方法去除单位三角假设，并补全散在群情形，从而推出相关群的块权猜想成立。
The weights for a finite group G with respect to a prime number p where introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulates a refinement of Alperin's conjecture involving ordinary irreducible characters - with their defect - and, in 2000, Geoffrey Robinson proves that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in…
Galois action on the principal block and generation of Sylow 3-subgroups
本文证明了关于有限群Sylow 3-子群生成的猜想，并验证了块级Galois-McKay猜想的另一个版本。
Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgroup have equal numbers of irreducible characters with degrees not divisible by $p$. The Alperin-McKay conjecture is a version of this as applied to individual Brauer $p$-blocks of $G$. We offer evidence that perhaps much stronger forms of both of these conjectures are true.
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。
On Héthelyi-Külshammer's conjecture for principal blocks
本文证明了有限群主 $p$-块中不可约常特征标数至少为 $2\sqrt{p-1}$，从而证实 Héthelyi-Külshammer 猜想并肯定回答有界亏群主块的 Brauer 问题 21。
Blocks with a single automorphism orbit of irreducible Brauer characters
本文证明了当块覆盖一个 p-幂指数的正规子群的块且仅有一个不可约 Brauer 特征标轨道时，具有阿贝尔缺陷群的块是惰性的。
The blocks with five irreducible characters
在 Alperin–McKay 猜想成立的前提下，本文证明含五个不可约常特征标的 Brauer $p$-块的亏群只能是 $C_5$、$C_7$、$D_8$ 或 $Q_8$。
Galois automorphisms and blocks covering unipotent blocks
本文证明了有限简单群中关于特征值延伸的条件成立，从而为有限群具有正常ℓ-补提供了新特征化，并研究了单幂块的特征分布及块级Galois-McKay猜想。
An equivariant bijection of irreducible Brauer characters above the Dade-Glauberman-Nagao correspondence
本文利用$\hH$-三元组框架，证明DGN对应之上存在保持顶点且对Galois与群自同构等变的不可约Brauer特征标双射，从而建立Galois Alperin权猜想的约化定理。
Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and $k$ an algebraically closed field of characteristic $p$. We study the relationship between the category $\Ff_P(G)$ and the behavior of $p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We give a sufficient condition for $\Ff_P(G)$ to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of $(b, G)$-Brauer pairs in the sense of Alperin-Brou…
Weights for $\ell$-local compact groups
本文研究了$\ell$-局部紧群的$\mathcal{F}$-权重，提出了关于权重数量上界的猜想，并在单群且$|S:T|=\ell$的情况下结合结构理论与块理论给出了证明，同时提出了Robinson普通权重猜想零高度情形的类似猜想。
This survey is about old and new results about the modular representation theory of finite reductive groups with a strong emphasis on local methods. This includes subpairs, Brauer's Main Theorems, fusion, Rickard equivalences. In the defining characteristic we describe the relation between $p$-local subgroups and parabolic subgroups, then give classical consequences on simple modules and blocks, including the Alperin weight conjecture in that case. In the non-defining characteristics, we sketch a picture of the local methods pioneered by Fong-Srinivasan in the determination of blocks and their…
Let $G$ be a connected reductive algebraic group defined over the finite field $\FF_q$, where $q$ is a power of a good prime for $G$. We write $F$ for the Frobenius morphism of $G$ corresponding to the $\FF_q$-structure, so that $G^F$ is a finite group of Lie type. Let $P$ be an $F$-stable parabolic subgroup of $G$ and $U$ the unipotent radical of $P$. In this paper, we prove that the number of $U^F$-conjugacy classes in $G^F$ is given by a polynomial in $q$, under the assumption that the centre of $G$ is connected. This answers a question of J. Alperin in \cite{alperin}. In order to prove the…
In this paper, we prove the algebraic K-theory Novikov conjecture for group algebras over the ring of Schatten class operators. The main technical tool in the proof is an explicit construction of the Connes-Chern character.
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.
In the present paper we investigate the question about the injectivity of the map F(R) --> F(K) induced by the canonical inclusion of a local regular ring of geometric type R to its field of fractions K for a homotopy invariant functor F with transfers satisfying some additional properties. As an application we get the original proof of Special Unitary Case of Grothendieck's conjecture about principal homogeneous spaces and some other interesting examples.
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