Martin Hertweck
Let $G$ be a finite group having a normal $p$-subgroup $N$ that contains its centralizer $\text{C}_{G}(N)$, and let $R$ be a $p$-adic ring. It is shown that any finite $p$-group of units of augmentation one in $RG$ which normalizes $N$ is conjugate to a subgroup of $G$ by a unit of $RG$, and if it centralizes $N$ it is even contained in $N$.
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We prove \emph{the other half} of Brauer's Height Zero Conjecture in the case of principal blocks.
On the principal blocks of $S_{15}$ over a field of characteristic 5
该研究在特征为5的代数闭域$k$的语境下，完成了对称群$S_{15}$的群代数主块的Ext箭图构造，丰富了模表示论中相关群块结构的研究内容。
Characters of prime power degree in principal blocks
本文研究并描述了主块中仅包含素数幂次方阶特征的有限群。
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
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.
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
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.
We classify Sylow $p$-subgroups of finite groups whose principal $p$-blocks have precisely six ordinary irreducible characters.
If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support to the validity of these conjectures.
Let $G$ be a finite group with a Sylow $p$-subgroup $P$. We prove that the principal $p$-blocks of $G$ and $N_G(P)$ are perfectly isometric under the assumption $G$ has a cyclic $p$-hyperfocal subgroup.
James's Conjecture predicts that the adjustment matrix for blocks of the Iwahori-Hecke algebra of the symmetric group is the identity matrix when the weight of the block is strictly less than the characteristic of the field. In this paper, we consider the case when the characteristic of the field is greater than or equal to 5, and prove that the adjustment matrix for the principal block of $\mathcal{H}_{5e}$ is the identity matrix whenever $e\neq4$. When $e=4$, we are able to calculate all but two entries of the adjustment matrix.
We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.
Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$. Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal $2$-block of $G$ are fixed by a certain Galois automorphism $σ\in \mathrm{Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$. Recent work of Navarro-Vallejo has reduced this conjecture to a problem about finite simple groups. We show that their conjecture holds for all finite simple groups, thus establishing the conjecture for all finite g…
We have found a list of finite simple groups with cyclic Sylow $p$-subgroup whose principal $p$-blocks have Brauer trees in the shape of a star, that is a tree of diameter at most $2$. Moreover, for an arbitrary finite group $G$ with cyclic Sylow $p$-subgroup, we have obtained a necessary condition when the Brauer tree of the principal $p$-block of $G$ is a star.
Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a theorem of Isaacs-Smith, as well as a recent result of three of the present authors.
Principal blocks, irreducible restriction, fields and degrees
针对有限群表示论中确定几乎单群特征标是否属主块的问题，采用群论方法处理“下降”情形，将已知结果的块版本归约为几乎单群问题，并提出Malle和Navarro高度零等次数猜想的伽罗瓦类似物，“上升”情形仍未解决。
We compute the decomposition numbers of the unipotent characters lying in the principal $\ell$-block of a finite group of Lie type $B_{2n}(q)$ or $C_{2n}(q)$ when $q$ is an odd prime power and $\ell$ is an odd prime number such that the order of $q$ mod $\ell$ is $2n$. Along the way, we extend to these finite groups the results of \cite{DVV19} on the branching graph for Harish-Chandra induction and restriction.
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.
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.
Base change fundamental lemma for Bernstein centers of principal series blocks
本文研究了主系列块伯恩斯坦中心的基础变换同态，证明了相关函数的关联性，并为稳定伯恩斯坦中心扭曲摄动转移猜想提供了证据。
Decomposition numbers in the principal block and Sylow normalisers
本文研究有限群主块中高度为零的特征的 p-模分解数与群的 p-局部结构之间的关系。
Let $G$ be an unramified group over a $p$-adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for $G$ and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with $Γ_1(p)$-level structure initiated by M. Rapoport and the author in [HR2].
On The Eaton-Moretó Conjecture for Principal Blocks of Finite Groups
该研究针对有限群主块的Eaton-Moretó猜想，通过构造主p-块中满足次数条件的不可约特征标，结合Dade射影猜想证明了该猜想成立。
We formulate conjectures concerning the dimension of the principal block of a ${\mathbb Z}_\ell$-spets (as defined in our earlier paper), motivated by analogous statements for finite groups. We show that these conjectures hold in certain situations. For this we introduce and study a Yokonuma type algebra for torus normalisers in $\ell$-compact groups which may be of independent interest.
Let $F$ be a field of characteristic $p$ at least 5. We study the Loewy structures of Specht modules in the principal block of $FΣ_{3p}$. We show that a Specht module in the block has Loewy length at most 4 and composition length at most 14. Furthermore, we classify which Specht modules have Loewy length 1, 2, 3, or 4, produce a Specht module having 14 composition factors, describe the second radical layer and the socle of the reducible Specht modules, and prove that if a Specht module corresponds to a partition that is $p$-regular and $p$-restricted then the head of the Specht module does not…
For a prime $p$, we determine a Sylow $p$-subgroup $D$ of a finite group $G$ such that the principal $p$-block $B$ of $G$ has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three by work by R. Brauer, J. Brandt, and V.A. Belonogov thirty years ago. Our proof relies on the classification of finite simple groups.
A group-theoretic condition equivalent to a condition on principal blocks
本文提出一个群论条件，等价于有限群在指定素数集的主块中仅含平凡复不可约特征的性质。
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.
In this paper we construct an action of the affine Hecke category (in its "Soergel bimodules" incarnation) on the principal block of representations of a simply-connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical $p$-Kazhdan-Lusztig polynomials.
The continuity of $p$-rationality of characters and the principal block
本文研究有限群不可约特征的有理性，证明$2$-有理性连续性可在主$2$-块中探测以改进前人结果，并为所有素数提出$1$级连续性间隙的群论准则猜想。
We characterise finite groups such that for an odd prime $p$ all the irreducible characters in its principal $p$-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by $p$ unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\neq 7$ or if $M_{22}$ is not a composition factor of a group $G$, then the condition above is equivalent to $G/O_{p'}(G)$ having odd order.
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。
On almost $p$-rational characters in principal blocks
该研究针对素数$p$，给出了阶含$p$因子的有限群主$p$-块中次数与$p$互素的几乎$p$-有理特征标数的下界，并刻画了取等时群的$p$-局部结构。
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。
The block decomposition of the principal representation category of reductive algebraic groups with Frobenius maps
本文研究了具有 Frobenius 映射的约化代数群的主表示范畴中简单模的扩张，并得到了由群中心特征标参数化的范畴块分解。
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。
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。
We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The exemplary case of $\mathfrak{sl}_3$ is computed explicitly, and further evidence of $\mathfrak{sl}_4$ is sketched. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum $\mathfrak{sl}_m$ at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group $S_{m}$.
A depth-zero principal-series block whose Hecke algebra has a non-trivial two-cocycle
本文证明文献中一个被提议的反例并非Kim-Yu类型赫克代数同构定理的反例，并展示该深度为零主序列类型的赫克代数具有非平凡二上循环的有趣性质。
This paper studies intersections of principal blocks of a finite group with respect to different primes. We first define the block graph of a finite group $G$, whose vertices are the prime divisors of $|G|$ and there is an edge between two vertices $p\neq q$ if and only if the principal $p$- and $q$-blocks of $G$ have a nontrivial common complex irreducible character of $G$. Then we determine the block graphs of finite simple groups, which turn out to be complete except those of $J_1$ and $J_4$. Also, we determine exactly when the Steinberg character of a finite simple group of Lie type lies i…
Weight conjectures for Parker--Semeraro fusion systems
该研究证明Parker--Semeraro系统满足饱和融合系统的9个Kessar--Linckelmann--Lynd--Semeraro权猜想中的6个，并推导出Robinson常权猜想对多类有限群的主p-块成立。
Let $\mathcal{G}$ be a split reductive $p$-adic group with connected centre. We show that each Bernstein block in the principal series of $\mathcal{G}$ admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form $T//W$ where $T$ is a maximal torus in the Langlands dual group of $\mathcal{G}$ and $W$ is the Weyl group of $\mathcal{G}$.
Character degrees in principal blocks for distinct primes
本文证明阶被不同素数整除的有限群在特定条件下存在同时位于主块且次数不被素数整除的非平凡不可约特征标，推广了前人结果并完成相关猜想的证明。
Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability
针对具有公共中心 p-子群的有限群，本文提出构造 Morita 型相对 Z-稳定等价的方法，推广 Linckelmann 的相关结论，并引入相对 Brauer 不可分解性概念及其等价判定条件。
We determine the cohomology of Deligne-Lusztig varieties associated to some short-length regular elements for split groups of type F4 and En. As a byproduct, we obtain conjectural Brauer trees for the principal Phi_{14}-block of E7 and the principal Phi_{24}-block of E8.
The Mathieu group ${\sf M}_{12}$ vs ${\mathbf{SL_3(3)}}$ in characteristic 3
研究特征3下马蒂厄群${\sf M}_{12}$与${\mathbf{SL_3(3)}}$，运用里卡德定理证明其主3 - 块出色里卡德等价及导出等价，还回答了W. 墨菲关于主3 - 块一阶霍赫希尔德上同调的问题。
Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections, and are eigenvectors of an associated matrix pencil.
Let B be a p-block of a finite group G with abelian defect group D such that S\unlhd G, S'=S, G/Z(S)\le\Aut(S) and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N_G(D) in the sense of Broué. This has been done by [Rouquier, 1994] for principal blocks and it remains to deal with the non-principal blocks.
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 …
Category $\mathcal{O}$ for hybrid quantum groups and non-commutative Springer resolutions
本文研究了杂交量子群的Category $\mathcal{O}$，通过非交换Springer解析构建其主块模型，并证明主块与仿射Hecke范畴导出等价，同时揭示了简单模在泛化Kazhdan-Lusztig多项式中的分级乘数。
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 $\ell>71$.
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.
On local character expansions for principal series representations of general linear groups
本文给出了p-adic一般线性群主块中不可约表示的完整局部特征展开的两个显式公式，分别基于退化Whittaker模型维数和Kazhdan-Lusztig多项式值。
In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic $p$ bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application…
Let $G=SL(2,5)$ be the special linear group of $2 \times 2$-matrices with coefficients in the field with $5$ elements. We show that the principal block over a splitting field $K$ of characteristic two of the group algebra $KG$ has a $3$-cluster tilting module. This gives the first example of a representation-infinite block of a group algebra having a cluster tilting module and answers a question by Erdmann and Holm.
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.
We study the cohomology of parabolic Deligne-Lusztig varieties associated to unipotent blocks of GLn(q). We show that the geometric version of Broué's conjecture over Q_\ell, together with Craven's formula, holds for any unipotent block whenever it holds for the principal Phi_1-block, that is for the variety X(π).
For finite groups $G$ with non-abelian, trivial intersection Sylow $p$-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of $G$ and the centre of the Brauer correspondent. This was already known for the Suzuki groups; the result will be generalised to cover more groups with trivial intersection Sylow $p$-subgroups.
We calculate examples of Picard groups for 2-blocks with abelian defect groups with respect to a complete discrete valuation ring. These include all blocks with abelian 2-groups of 2-rank at most three with the exception of the principal block of J1. In particular this shows directly that all such Picard groups are finite and Picent, the group of Morita auto-equivalences fixing the centre, is trivial. These are amongst the first calculations of this kind. Further we prove some general results concerning Picard groups of blocks with normal defect groups as well as some other cases.
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.
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