Aaron Yi Rui Low
When the characteristic of the underlying field is at least 5, we prove that the adjustment matrix for blocks of $q$-Schur algebras of weights 3 and 4 is the identity matrix. Moreover, we show that the decomposition numbers for weight 3 blocks of $q$-Schur algebras are bounded above by one.
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A new construction of the asymptotic algebra associated to the $q$-Schur algebra
我们用A表示不定v中的洛朗多项式环，用K表示它的分数域。本文研究了A上“一般”q-Schur代数S_q（n，r）的表示论。本文将KS_q（n，r）上的每一个非退化对称迹形式τ都对应于KS_q（n，r）的一个子代数J_τ，它同构于J. Du定义的“渐近”代数\J（n，r）_A，从而给出了James猜想的一个新的判定准则.
Graded $q$-Schur algebras
推广了Brundan和Kleshchev的工作，在Dipper-James的$q$代数上引入了分次概念，并证明了当$q^2\neq1$和$q^3\neq1$分别满足时，关于$q$-Schur代数的分解数的Leclerc和Thibon猜想的分次类似.
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.
We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.
RoCK blocks of double covers of symmetric groups and generalized Schur algebras
本文研究对称群与交错群双覆盖的块，通过广义Schur超代数给出任意亏数RoCK块的局部描述，并在导出等价下推广至任意块。
We define and study a new class of bialgebras, which generalize certain Turner double algebras related to generic blocks of symmetric groups. Bases and generators of these algebras are given. We investigate when the algebras are symmetric, which is relevant to block theory of finite groups. We then establish a double centralizer property related to blocks of Schur algebras.
Real Subpairs and Frobenius-Schur Indicators of Characters in 2-Blocks
设B是有限群G的实2-块。那么B有一个真正的缺陷类。设g是这样一个类的元素。B的亏偶是（D，E），其中E是g的扩张中心化子C ^*（g）的Sylow 2-子群，D是E与中心化子C（g）的交集。已知（D，E）在G-共轭下是唯一确定的。我们证明了（D，E）决定了哪些B-子对是实数。我们还概述了（D，E）如何影响G-置换模的对应于G的共轭作用在其对合上的分量的顶点。我们应用这些方法来枚举的Frobenius-Schur指标的不可约字符在一个真正的块，有一个二面角亏损群。我们还确定了这样的块中的对合模的分支的顶点。
Extended zigzag Schur algebras are quasi-hereditary algebras which are conjecturally Morita equivalent to RoCK blocks of classical Schur algebras. We prove that extended zigzag Schur algebras are Ringel self-dual.
Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra
设$S$是与秩为$n$的Ariki-Koike代数$H_{n,r}$相关联的分圆$q$ -Schur代数，由Dipper-James-Mathas引入.对于每个$p = (r_1, ..., r_g)$使得$r_1 + ... + r_g = r$，定义了$S$的一个子代数$S^p$及其商代数$\bar S^p$ .证明了$S^p$是一个标准基代数，$\bar S^p$是一个胞腔代数.利用这些代数，我们证明了$S$的某些分解数可以表示为与较小的Ariki_koike代数$H_{n_k,r_k}$相关联的分圆$q$ -Schur代数的分解数的乘积.
On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras
设$\Sc(\vL)$是Dipper-James-Mathas引入的Ariki-Koike代数$\He_{n,r}$的分圆$q$代数.本文考虑$v$ -$\Sc(\vL)$的分解数，即关于Weyl模的Jantzen滤子的分解数.作为Shoji-Wada结果的$v$ -模拟，我们证明了$\Sc(\vL)$的$v$-分解数的乘积公式，它断言对于与秩较小的Ariki-koike代数$\He_{n_i,r_i}$相关联的各种分圆$q$ -Schur代数，某些$v$ -分解数表示为$v$-分解数的乘积.利用Schur函子证明了$\He_{n,r}$的$v$ -分解数的一个类似公式.
A complete classification of $τ$-tilting finite Schur algebras
本文确定经典Schur代数部分块的τ-倾斜有限性，结合相关文献结果完成τ-倾斜有限Schur代数的完全分类，还得到Schur代数S(2,r)对应块的分类及严格野生Schur代数的分类，仅余三个未解决情况。
Blocks of affine quantum Schur algebras
仿射量子Schur代数是一类重要的无穷维代数，它的表示论与量子仿射$\frak{gl}_n$的表示理论有着密切的联系.将仿射量子Schur代数${\mathcal S}_{\vartriangle}(n,r)_{v}$的有限维不可约模分类到\cite{DDF}中，其中$v\in{\mathbb C}^*$不是单位根.本文将对仿射量子Schur代数${\mathcal S}_{\vartriangle}(n,r)_{v}$的块进行分类。
Schur-Weyl duality for higher levels
我们将Schur-Weyl对偶推广到任意水平的$l \geq 1$，这种情形$l=1$恢复了对称群和一般线性群之间的经典对偶.对于$A_\infty$型根系，一般用$\C$上的退化分圆Hecke代数代替对称群，并以$l$的水平权为参数.作为应用，我们证明了Dipper，James和Mathas构造的分圆Hecke代数的拟遗传覆盖的退化模拟与一般线性李代数的抛物范畴$\mathcal{O}$的某些块是Morita等价的.
For any algebra $A$ over an algebraically closed field $\mathbb{F}$, we say that an $A$-module $M$ is Schurian if $\mathrm{End}_A(M) \cong \mathbb{F}$. We say that $A$ is Schurian-finite if there are only finitely many isomorphism classes of Schurian $A$-modules, and Schurian-infinite otherwise. In this paper, we build on the work of Ariki and the second author to show that all blocks of type $A$ Hecke algebras of weight at least $2$ in quantum characteristic $e \geq 3$ are Schurian-infinite. This proves that if $e \geq 3$ then blocks of type $A$ Hecke algebras are Schurian-finite if and only …
Schurian-finiteness of blocks of type B Hecke algebras
本文研究了类型B Hecke代数块的Schurian有限性，确定了非积分情况下的所有块和积分情况下的大部分块的有限性，剩余情况通过分解数和表示论方法进行分析。
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…
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。
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.
Blocks of the truncated $q$-Schur algebras of type A
本文对以任意共饱和划分集为权偏序集的$A$型截断$q$ -Schur代数的块进行了分类.
Affine and cyclotomic $q$-Schur categories via webs
本文通过构造仿射 q-web 范畴、仿射 q-Schur 范畴及其分圆商范畴并建立图解整基，首次给出仿射与分圆 q-Schur 代数的图解表示。
Schurian-finiteness of blocks of type $A$ Hecke algebras
本研究借助$\tau$-倾斜有限性与Schurian有限性的等价关系，证明$e\geq3$的$A$型Hecke代数中权重≥2的块为Schurian-无限的，还得到分次Scopes等价的结论。
The Brauer algebra and the symplectic Schur algebra
设k是特征p>0的代数闭域，m，r是整数，m \ge 1，r \ge 0，m \ge r，S_0（2 m，r）是第一作者引入的k上的辛Schur代数.我们介绍了辛舒尔函子，推导出它的一些基本性质，并将其与哈特曼和佩吉特的工作。我们对正交Schur代数做同样的事情。我们给出了辛Schur代数的一个修正的Jantzen和公式和一个分块结果，在此基础上，我们推出了正交Schur代数的一个分块结果.最后，我们从前面的结果推出了一个新的证明的几何描述的块的Brauer代数的特征为0所获得的考克斯，德Visscher和马丁。
Let $A$ be a finite-dimensional algebra over a field of characteristic $p>0$. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective $A$--modules $P$ into those of the torsion submodules of $P$. As an application, we show that blocks of both the classical and quantum Schur algebras $S(2,r)$ and $S_q(2,r)$ are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain $2p^k$ simple modules for some $k$.
We define and study new classes of quasi-hereditary and cellular algebras which generalize Turner's double algebras. Turner's algebras provide a local description of blocks of symmetric groups up to derived equivalence. Our general construction allows one to `schurify' any quasi-hereditary algebra $A$ to obtain a generalized Schur algebra $S^A(n,d)$ which we prove is again quasi-hereditary if $d\leq n$. We describe decomposition numbers of $S^A(n,d)$ in terms of those of $A$ and the classical Schur algebra $S(n,d)$. In fact, it is essential to work with quasi-hereditary superalgebras $A$, in w…
Given two m x n matrices A = (a_{ij}) and B=(b_{ij}) with entries in B(H), the Schur block product is the m x n matrix A \square B := (a_{ij}b_{ij}). There exists an m x n contraction matrix S = (s_{ij}), such that A \square B = diag(AA*)^(1/2) S diag(B*B)^(1/2). This decomposition is also valid for the block Schur tensor product. It is shown, via the theory of random matrices, that the set of contractions S, which may appear in such a decomposition, is a very thin subset of the unit ball of M_n(B(H)).
The weighted fusion category algebra and the q-Schur algebra for \mathrm{GL}_2(q)
证明了$\mathrm{GL}_2(q)$的主2-块$b_0$的加权融合范畴代数是$q$-Schur代数$\mathcal{S}_2(q)$乘其基柱的商，其中$q$是奇素幂.因此，在这种情况下，我们得到了简单$k\mathrm{GL}_2(q)b_0$ -模的同构类集与$b_0$ -权的共轭类集之间的一个典范双射。
The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra
本文在奇特征非平凡情形下，建立了 $p$-adic $\mathrm{GL}_2$ 的幺幂表示导出范畴与 dg Schur 代数完美复形间的三角等价，并构造经典生成元。
Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by $p$-restricted partitions. We prove an analogue of this result in the very general setting of "Schur pairs". As an application we show that the socle of a Weyl module of a cyclotomic $q$-Schur algebra is a sum of simple modules labelled by Kleshchev multipartitions and we use this result to prove a conjecture of Fayers that leads to an efficient LLT algorithm for the higher level cyclotomic Hecke algebras of type $A$. Finally, we prove a cyclotomic analogue of the Carter-Lusz…
Representations of little $q$-Schur algebras
在\cite{DFW}和\cite{Fu07}中，小$q$ -Schur代数作为无穷小量子群的同态像被引入.在本文中，我们将研究这些代数的表示。我们将对小$q$ -Schur代数的单模进行分类，并对半单模的小$q$ -Schur代数进行分类.此外，通过对$n=2$的小$q$-Schur代数的块的分类，确定了在单位根为奇数的情况下，有限表示型的小$q$代数.
Turner's Conjecture describes all blocks of symmetric groups and Hecke algebras up to derived equivalence in terms of certain double algebras. With a view towards a proof of this conjecture, we develop a general theory of Turner doubles. In particular, we describe doubles as explicit maximal symmetric subalgebras of certain generalized Schur algebras and establish a Schur-Weyl duality with wreath product algebras.
We compute the Hochschild cohomology of any block of $q$-Schur algebras. We focus the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of $q$-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all $q$-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derived equivalence.
Rock blocks
考虑表示论与对称群、A型Hecke代数、q-Schur代数、非描述特征的有限一般线性群的关系。岩石块是出现在这种表示论中的某些组合定义的块，首先由R.鲁基耶岩石块比一般块更对称，并且每个块都等效于岩石块。受J. Chuang和R. Kessar在交换亏损的对称群块的情况下，我们寻求这些块的结构定理。
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 compute explicitly the submodule structure of the Young modules for symmetric groups $S_n$ over fields of characteristic 2, when $n \le 7$. We use this information to compute the submodule structure of indecomposable projectives for the corresponding Schur algebras when $n \le 5$, and we give give partial information when n=6,7, including the Gabriel quiver and the structure of the Weyl modules. We resolve all Morita equivalences between blocks of these algebras.
An enhanced algebraic group $\uG$ of $G=\GL(V)$ over $\bbc$ is a product variety $\GL(V)\times V$, endowed with an enhanced cross product. Associated with a natural tensor representation of $\uG$, there are naturally Levi and parabolic Schur algebras $\mathcal{L}$ and $\mathcal{P}$ respectively. We precisely investigate their structures, and study the dualities on the enhanced tensor representations for variant groups and algebras. In this course, an algebraic model of so-called degenerate double Hecke algebras (DDHA) is produced, and becomes a powerful implement. The connection between $\math…
Blocks of cyclotomic Hecke algebras
本文对任意域上G（r，1，n）型分圆Hecke代数的块进行了分类。而不是直接与Hecke代数，我们的工作，而不是与分圆舒尔代数。这些代数的优点是分圆Jantzen和公式给出了分圆Schur代数块的简单组合特征。我们通过分析“Jantzen等价”的组合数学得到了块的明确描述。我们注意到，在1999年宣布的分圆Hecke代数块的分类证明。不幸的是，考克斯发现这个先前的证明是不完整的。
In this paper we consider the (affine) Schur algebra introduced by Vignéras as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a $p$-adic field, a large part of the unipotent block over fields of characteristic different from $p$. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a bypr…
A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors
我们用基本（二维）sl（2）-模的n次张量幂幂来确定sl（n）的某个最高权范畴的奇异块直和的Grothendieck群. U（sl（2））的作用由投射函子给出，Temperley-Lieb代数的交换作用由Zuckerman函子给出.不可分解的投射函子对应于U（sl（2））中的Lusztig标准基。在对偶实现中，基本表示的n阶张量幂被确定为具有最高权类别的抛物块的直和。跨墙平移函子充当Temperley-Lieb代数的生成元，而Zuckerman函子充当U（sl（2））的生成元。
Weight 2 blocks of general linear groups and modular Alvis-Curtis duality
给出了q-Schur代数的权2块和[2：1]-对的结构，并明确计算了有限一般线性群的权2块在非定义特征下的模Alvis-Curtis对偶.
The minimal counterexample to James's conjecture
本文计算了对称群Hecke代数中一个显式的反例，揭示了在特定参数下分解矩阵的结构特性。
Hook modules for general linear groups
对于特征p > 0的任意无限域k，我们描述了代数幺半群M_n（k）（k上的所有n × n矩阵）的块的结构，或者等价地描述了Schur代数S（n，p）的块的结构，它的单模由p-hook划分索引。结果是已知的;本文仅根据Peel和Donkin关于Schur代数块的描述的一个结果，给出了一个初等的、自包含的证明。这个结果给出了GL_n（k）-模的一个特征标公式，它对所有n和所有p都成立.这个特征标公式是Brundan，Kleshchev和Suprunenko以及Mathieu和Papadopoulo分别发现的一个特征标公式的特例.
Let $U_ζ$ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra $\mathfrak g$ and a root of unity $ζ$. When $L,L'$ are irreducible $U_ζ$-modules having regular highest weights, the dimension of $\operatorname{Ext}^n_{U_ζ}(L,L')$ can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of $U_ζ$. This paper shows for $L,L'$ irreducible modules in a singular block that $\dim\operatorname{Ext}^n_{U_ζ}(L,L')$ is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. …
On the representation theory of Schur algebras in type $B$
本文研究类型 B Schur 代数 $\mathcal{L}^n(m)$ 的表示论，通过与环割 $q$-Schur 代数比较构造细胞结构，索引简单模，并修正了 Lai-Nakano-Xiang 猜想，给出拟遗传性的参数条件及加强结果。
Cuspidal systems for affine Khovanov-Lauda-Rouquier algebras
仿射Khovanov-Lauda-Rouquier代数$R_\al$的尖点系统给出了标准模理论.这使得我们可以将$R_\al$上的不可约模分类到所谓的虚模。本文对不可约$R_\al$ -模的模$p$约化提出了一个猜想，推广了James猜想.我们还描述了极小虚模，为今后虚Schur-Weyl对偶的研究奠定了基础。我们引入了有色虚张量空间，并将虚模的分类归结为一种颜色。研究了尖点模的性质。我们证明了在Khovanov-Lauda-Rouquier范畴下，尖点模对应于对偶根向量.
Imaginary Schur-Weyl duality for quiver Hecke superalgebras
本文研究箭图 Hecke 超代数的虚尖代数，通过构造非负分次的 Morita 等价代数 C(n,d)，将其表示论归结为经典 Schur 代数，从而用多重分拆典范分类不可约模。
For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category $\mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the cate…
We study the representation theory of the uniform block permutation algebra in the context of the representation theory of factorizable inverse monoids. The uniform block permutation algebra is a subalgebra of the partition algebra and is also known as the party algebra. We compute its characters and provide a Frobenius characteristic map to symmetric functions. This reveals connections of the characters of the uniform block permutation algebra and plethysms of Schur functions.
We study the homological algebra in the category $\mathcal{P}_p$ of strict polynomial functors of degree $p$ over a field of positive characteristic $p$. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras of simple functors and Schur functors. We observe that the category $\mathcal{P}_p$ has a Kazhdan-Lusztig theory and we show that the DG algebras computing the Ext-algebras for simple functors and Schur functors are formal. These l…
The non-projective part of the Lie module for the symmetric group
已知对称群的群代数$FS_n$的李模不是投射的当且仅当$F$的特征$p$整除$n$。我们表明，在这种情况下，它的非投射和属于主块的$FS_n$。令$V$是$F$上的维数为$m$的向量空间，并且令$L^n(V)$是$V$上的自由李代数的第$n$次齐次部分;这是次数为$n$的$GL_m(F)$的多项式表示，或者等价地，Schur代数$S(m,n)$的模。我们的结果意味着，当$m \geq n$时，$L^n(V)$的每一个非倾斜模的被加数都属于$S(m,n)$的主块，这里的主块指的是包含$V$的$n$次对称幂的块.
Moving vectors I: Representation type of blocks of Ariki-Koike algebras
本文提出Ariki-Koike代数块的新不变量“块移动向量”，据此分类表示有限块，并给出非导出等价与不同轨道的导出等价块实例，同时确定了cyclotomic $q$-Schur代数块的表示型。
Modular Representation Theory of Symmetric Groups
综述了对称群及相关Hecke代数的模表示论的最新进展。本文讨论了群代数$FΣ_n$与Khovanov-Lauda-Rouquier代数的联系及其所揭示的群代数块上的分次，分次范畴与量子群和晶体基的联系，分次胞腔结构与分次Specht模，分次格腔结构与分次格腔结构之间的关系。仿射KLR代数的尖点系统和虚Schur-Weyl对偶，将这些代数的表示论与通常的小秩Schur代数联系起来.
Outer derivations on blocks of group algebras
研究块群代数的外导出与Hochschild上同调群的关系，证明了多种情况下HH^1(B,B)非零。
Wild blocks of type $A$ Hecke algebras are strictly wild
本文证明了类型 A Hecke 代数中量子特征 e ≥ 3 的野块均为严格野，且 q-Schur 代数的野块也均为严格野。
On Noetherian algebras, Schur functors and Hemmer-Nakano dimensions
本文研究（投射）诺特代数B的模范畴与B上生成-相对上生成子的自同态代数的模范畴间联系的质量，通过舒尔函子分析Hemmer-Nakano维数，建立整数情形与有限维情形的计算关联，推导舒尔代数和BGG范畴O的结果并展示形变的结构性质。
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