Resolutions by permutation modules
Paul Balmer, Dave Benson
本文研究有限群模表示的消解问题，证明任意模表示在添加补模后都具有由置换模构成的有限消解。
我们证明，在相差添加一个补模的意义下，有限群的每个模表示都容许由置换模构成的有限消解。
We prove that, up to adding a complement, every modular representation of a finite group admits a finite resolution by permutation modules.
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We prove that all endo-$p$-permutation modules for a finite group are liftable from characteristic $p>0$ to characteristic $0$.
Young module multiplicities and classifying the indecomposable Young permutation modules
研究了作为置换模的直和项的Young模在Young子群陪集上的重数。这种多重性被称为p-Kostka数。对不可分解的Young置换模进行了分类，并利用p-置换模的Brauer构造，给出了p-Kostka数的一些新的约化.特别是，我们证明了p-Kostka数下保持乘分区由p，并加强了已知的减少亨克，对应于添加倍数的p-权力的第一行的分区。
It is shown that the endomorphism algebra of an arbitrary Young permutation module is cellular. Those are are quasi-hereditary are then determined.
A Specht Filtration of Permutation Modules Over KLR Algebras
本文在仿射A型中构造了由钩子分区索引的置换模的Specht滤过，并在线性箭图情形下推广到任意分区。
Permutation modules play an important role in the representation theory of the symmetric group. Hartmann and Paget defined permutation modules for non-degenerate Brauer algebras. We generalise their construction to a wider class of algebras, namely cellularly stratified algebras, satisfying certain conditions. Partition algebras are shown to satisfy these conditions, provided the characteristic of the underlying field is large enough. Thus we obtain a definition of permutation modules for partition algebras.
We study the double centraliser property and the annihilators ideals of certain permutation modules for symmetric groups and their quantum analogues. In version 2 some remarks have been added on cell ideals and annihilators.
We prove the existence and main properties of signed Young modules for the symmetric group, using only basic facts about symmetric group representations and the Brou{é} correspondence. We then prove new reduction theorems for the signed $p$-Kostka numbers, defined to be the multiplicities of signed Young modules as direct summands of signed Young permutation modules. We end by classifying the indecomposable signed Young permutation modules and determining their endomorphism algebras.
We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of permutation modules, and it uses the multiplicities of irreducible submodules within individual orbital spaces to express the resulting computational bounds. We conclude by showing that these bounds are surprisingly small when dealing with certain permutation modules arising from the action of the symmetric group on tabloids.
The 2-modular permutation modules on fixed point free involutions of symmetric groups
本文在偶特征上列举了偶次对称群的置换模中作用于其不动点自由对合集的分量。我们找到每个组件的顶点和Brauer商，以及与每个组件相关联的普通字符。
Let $G$ be a finite $p$-group with normal subgroup $N$ of order $p$. The first author and Zalesskii have previously given a characterization of permutation modules for $\mathbb{Z}_pG$ in terms of modules for $G/N$, but the necessity of their conditions was not known. We apply a correspondence due to Butler to demonstrate the necessity of the conditions, by exhibiting highly non-trivial counterexamples to the claim that if both the $N$-invariants and the $N$-coinvariants of a given lattice $U$ are permutation modules, then so is $U$.
A recognition theorem for permutation modules over $p$-groups extending Weiss' Theorem
本文提出一个关于 $p$-群上排列模的识别定理，扩展了 Weiss 定理，并在混合特征下对排列模块进行了更一般的刻画。
Applying Robert Boltje's theory of canonical induction, we give a restriction-preserving formula expressing any $p$-permutation module as a $\mathbb{Z}[1/p]$-linear combination of modules induced and inflated from projective modules associated with subquotient groups. The underlying constructions include, for any given finite group, a ring with a $\mathbb{Z}$-basis indexed by conjugacy classes of triples $(U, K, E)$ where $U$ is a subgroup, $K$ is a $p'$-residue-free normal subgroup of $U$ and $E$ is an indecomposable projective module of the group algebra of $U/K$.
Permutation modules of the walled Brauer algebras
本文研究了对称群直积群代数与有墙Brauer代数的置换模和Young模，证明在特征不为2、3时置换模具有对偶Specht滤过结构，并给出其Young模直和分解。
On permutation modules and decomposition numbers for symmetric groups
研究了对任意奇素特征$p$的域$\mathbb{F}$，将平凡$\mathbb{F}(S_a\wr S_n)$ -模诱导为全对称群$S_{an}$，得到置换模的不可分解和数，使得$a<p\leq n$ .特别是我们的特点，这样的不可分解的和项的顶点。作为一个推论，我们将证伪一个模形式的福克斯猜想。在文章的第二部分中，我们将利用这些信息，根据Foulkes模$ϕ^{(a^n)}$的普通特征标，给出对称群分解矩阵的某些列的一种新的描述。
Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}_q$, the finite field with $q$ elements. Let ${\bf B}$ be an Borel subgroup defined over $\mathbb{F}_q$. In this paper, we completely determine the composition factors of the induced module $\mathbb{M}(\op{tr})=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}\op{tr}$ ($\op{tr}$ is the trivial ${\bf B}$-module) for any field $\Bbbk$.
Given $r \in \mathbf{N},$ let $λ$ be a partition of $r$ with at most two parts. Let $\mathbf{F}$ be a field of characteristic 3. Write $M^λ$ for the $\mathbf{F}S_r$-permutation module corresponding to the action of the symmetric group $S_r$ on the cosets of the maximal Young subgroup $S_λ.$ We construct a full set of central primitive idempotents in $\text{End}_{\mathbf{F} S_r}(M^λ)$ in this case. We also determine the Young module corresponding to each primitive idempotent that we construct.
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。
The decomposition of permutation module for infinite Chevalley groups, II
本文研究了代数闭域上连通约化代数群的置换模分解问题，完全确定了任意域上由Borel子群导出的置换模的所有合成因子。
We construct a class $Θ_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^λ$ to a signed permutation module $M_{\mathbb{Z}}(α|β)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $ϕ\in \operatorname{Hom}_{\mathbb{Z}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^λ, M_{\mathbb{Z}}(α|β)\big)$ lies in the $\mathbb{Q}$-span of $Θ_{\text{sstd}}$, a subset of $Θ_{\mathscr{R}}$ corresponding to semistandard $λ$-tableaux of type $(α|β)$. We also study the conditions for which $Θ^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\…
Is the projective cover of the trivial module in characteristic $11$ for the sporadic simple Janko group $J_4$ a permutation module?
本文确定了特征 $11$ 下散在单 Janko 群 $J_4$ 的平凡模的射影覆盖的常特征标，从而回答了其是否为置换模的问题。
Permutation dimensions of prime cyclic groups
受 Balmer 与 Gallauer 的置换分解工作启发，本文定义有限群的 p-置换维数，并计算素数阶循环群的该不变量。
Let $n$ be a positive integer and $λ$ be a partition of $n$. Let $M^λ$ be the Young permutation module labelled by $λ$. In this paper, we study symmetric and exterior powers of $M^λ$ in positive characteristic case. We determine the symmetric and exterior powers of $M^λ$ that are projective. All the indecomposable exterior powers of $M^λ$ are also classified. We then prove some results for indecomposable direct summands that have the largest complexity in direct sum decompositions of some symmetric and exterior powers of $M^λ$. We end by parameterizing all the Scott modules that are isomorphic…
Descent and finite permutation resolutions for discrete groups
对于具有有限维proper作用模型空间的离散群和有限整体维数的交换Noether环，证明了投射kG-模的同伦范畴、Mazza-Symonds定义的kG-模的稳定模范畴以及具有有限迷向的置换kG-模的导出范畴可以下降至有限子群，并应用该结果推广了Balmer-Gallauer的定理。
Fixed point sets for permutation modules
本文研究了有限特征p的代数闭域上作用在其共轭类的不动点自由对合上的对称群的表示，讨论了作用所给模的不可约分支的任意顶点所固定的对合集的一般形式，并展示了这样一个集合是如何由某些基本不动点集合组成的，我们可以在p = 2的情况下枚举这些集合。Broué对应允许我们从不动点集传递到顶点。
We investigate finite non-Abelian simple groups $G$ for which the projective cover of the trivial module coincides with the permutation module on a subgroup and classify all cases unless $G$ is of Lie type in defining characteristic.
Let ${\bf G}$ be a connected reductive group over $\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let ${\bf B}$ be an $F$-stable Borel subgroup. Let $\Bbbk$ be a field of characteristic $r\neq p$. In this paper, we completely determine the composition factors of the induced module $Ind_{B}^{G}{tr}=\Bbbk{G}\otimes_{\Bbbk{\bf B}}$ tr (here $\Bbbk{H}$ is the group algebra of the group ${H}$, and tr is the trivial $B$-module). In particular, we find a new family of infinite dimensional irreducible abst…
Generalized permutation matrices and non-weight modules over $\mathfrak{sl}(m|1)$
本文研究$\frak{sl}(m|1)$上秩$(k|k)$的非权模，构造显式模族，给出同构定理与不可分解性判据。
The structure of rank 3 permutation modules for O_{2n}(2) and U_m(2) acting on nonsingular points
研究了$O_{2n}^{\pm}(2)$（$n\geq3$）和$U_{m}(2)$（$m\geq4$）的秩为3的自然作用在其标准模的非奇点上的置换模的奇特征结构.
On the permutation modules for orthogonal groups $O_{m}^{\pm}(3)$ acting on nonsingular points of their standard modules
我们描述了正交群$O_{m}^{\pm}(3)$与$m\geq6$在其标准模的非奇点上的自然作用的交叉特征置换模的结构，包括合成因子和子模格.这些行动连同那些在\cite{HN}研究的都是原始秩3行动的有限典型群的非奇异点的例子。
Annihilators of permutation modules
置换模是对称群$\Sym_n$及其对应的Iwahori-Hecke代数$\He = \He(\Sym_n)$的表示论的基础。我们找到了置换模的零化子在“积分”情形下的一个显式组合基--表明它是G. E. $\He$的Murphy细胞结构。只要$\He$是半单的，同样的结果也成立，但在非半单的情况下可能失败。
In a recent paper, Dave Benson and Peter Symonds defined a new invariant $γ_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for permutation modules of the symmetric group corresponding to two-part partitions using tools from representation theory and combinatorics.
Permutation modules over cyclic $p$-groups
本文推广了循环 $p$-群上 $RG$-置换模的刻画结果，将 Torrecillas 和 Weigel 的结论从 $p$ 不分歧扩展到可能分歧的情形，证明仅用群上同调与 Weiss 定理。
The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $γ_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.
Deligne's category $\underline{\rm Rep}(S_t)$ is a tensor category depending on a parameter $t$ "interpolating" the categories of representations of the symmetric groups $S_n$. We construct a family of categories $\mathcal{C}_λ$ (depending on a vector of variables $λ= (λ_1, λ_2, \ldots, λ_l)$, that may be specialised to values in the ground ring) which are module categories over $\underline{\rm Rep}(S_t)$. The categories $\mathcal{C}_λ$ are defined over any ring and are constructed by interpolating permutation representations. Further, they admit specialisation functors to $S_n$-mod which are …
The tt-geometry of permutation modules. Part I: Stratification
本文在正特征情形下研究有限群置换模导出范畴的张量三角几何，利用Brauer商实现分层，刻画紧对象tt-谱的底层集合并给出实例。
Let $k$ be a field of characteristic $p > 0$. For $G$ an elementary abelian $p$-group, there exist collections of permutation module such that if $C^*$ is any exact bounded complex whose terms are sums of copies of modules from the collection, then $C^*$ is contractible. A consequence is that if $G$ is any finite group whose Sylow $p$-subgroups are not cyclic or quaternion, and if $C^*$ is a bounded exact complex such that each $C^i$ is direct sum of one dimensional modules and projective modules, then $C^*$ is contractible.
We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type $A$. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $w\in S_n$ into the dominant Verma module are shown to be determined by the essential set of $w$ and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from…
Permutation modules and cohomological singularity
本文利用群上同调与系数环的奇点范畴，为有限群在交换 Noether 环上的有限生成表示构造新不变量，以刻画表示是否由置换模控制。
Permutation modules, Mackey functors, and Artin motives
本文详细阐述了置换模、Mackey函子与Artin动机三个概念间的联系，并探讨了arXiv:2009.14093引入的置换模“大”导出范畴如何融入该理论框架。
The permutation dimension of the Klein 4-group
本文基于Balmer和Gallauer的近期结果，研究有限群的 $p$-置换维数这一新不变量，并计算了特征2下Klein 4-群的全局 $p$-置换维数及其所有不可分解模的维数。
Endomorphism rings of permutation modules over maximal Young subgroups
令$K$是特征为2的域，令$λ$是某个自然数$r$的两部分划分。将对应于$Σ_r$中的（最大）Young子群$Σ_λ$的置换模表示为$M^λ$。构造了Schur代数$S_K(2,r)$的中心化子代数$S_K(λ) = 1_λS_K(2,r) 1_λ= End_{KΣ_r}(M^λ)$的正交本原幂等元的一个全集.这些幂等元自然地与2-Kostka数一一对应。
Permutation, stabilization and decomposition
本文基于排列模块的tt-几何理解，定义了有限群的稳定排列范畴，并证明该范畴在循环群和广义四面体群中分解。
Hopf subalgebras and tensor powers of generalized permutation modules
利用有限张量范畴中的某个模V及其张量幂幂，研究了有限维霍普夫代数H的子代数R的深度是否有限的问题.模V是从R导出到H的可数表示，它是一个广义置换模，也是一个模余代数。证明了：如果在子代数对中，任一霍普夫代数具有有限表示型，或V是R* 指向的半单的，或V是射影的，或V的张量幂满足包含R的有限集上的Burnside环公式，则R在H中的深度是有限的.通过比较V在有限张量范畴中的截断张量代数，给V或任何其他H-模赋予一个非负整数深度，从而得到一个Hopf子代数的深度的上界和下界.例如，一个相对霍普夫限制模的深度为1，一个无核子群的置换模的深度小于其特征标所假定的值的个数。
One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation $R$-lattice for the finite $p$-group $G$ in terms of the restriction to a normal subgroup $N$ and the $N$-fixed points of the lattice, where $R$ is a finite extension of the $p$-adic integers. Using techniques from relative homological algebra, we generalize Weiss' Theorem to the class of infinitely generated pseudocompact lattices for a finite $p$-group, allowing $R$ to be any complete discrete valuation ring in mixed characteristic. A related theorem of…
Permutation resolutions for Specht modules of Hecke algebras
在[Boltje，Hartmann：Permutation resolutions for Specht modules，J. Algeorge Combin. 34（2011），141-162]，以组合方式构造链复形，其实质上是对称群的积分Specht模（的对偶）在置换模方面的分解。本文将链复形的定义推广到整Iwahori Hecke代数，并证明了与在对称群情形下相同的部分正合性结果.在对称群情况下的正合性猜想的完整证明最近由Santana和Yudin，Adv. in Math.229（2012），2578-2601给出。
The geometry of permutation modules
本文利用Brauer商对正特征下有限群置换模的导出范畴进行分层，并通过约化到初等阿贝尔群及扭曲上同调描述紧对象谱，最终给出thick理想与localizing理想的分类。
Finite permutation resolutions
本文证明特征 p 域上有限群的任意有限维表示均存在由 p-置换模构成的有限分解，并借助导出范畴的重新表述完成论证。
Consider a discrete valuation ring $R$ whose residue field is finite of cardinality at least $3$. For a finite torsion module, we consider transitive subsets $O$ under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space $C[O]$ is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of $O\times O$ under the diagonal action of the automorphism group.
We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime $p$ the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pairs of partitions. As a …
We show that the signed $p$-Kostka numbers depend just on $p$-Kostka numbers and the multiplicities of projective indecomposable modules in certain signed Young permutation modules. We then examine the signed $p$-Kostka number $k_{(α|β),(λ|pμ)}$ in the case when $|β|=p|μ|$. This allows us to explicitly describe the multiplicities of direct summands of a signed Young permutation module lying in the principal block of $F\mathfrak{S}_{mp}$ in terms of the $p$-Kostka numbers.
Brauer-friendly modules and slash functors
本文引入Brauer-友好模的概念，它是endo-p-置换模的推广。块代数OGe上的模称为Brauer友好的，如果它是具有相容融合稳定内置换源的不可分解模的直和.我们得到，这些模块，一个函子版本的戴德的斜杠建设，也被称为紧缩限制。我们证明，我们的斜杠函子，定义在布劳尔友好的范畴，共享大多数非常有用的性质，满足布劳尔函子的p-置换OGe-模的范畴。特别是，我们给出了一个参数化的不可分解的布劳尔友好模块，这开辟了一个完整的分类时，融合稳定的来源进行分类的方式。这些工具已被用来证明存在一个稳定等价之间的非主块的上下文中的最小反例奇Z* p定理。
The Balmer spectrum of integral permutation modules
本文将有限群在域上置换模小导出范畴的tt-几何分析扩展到交换诺特基情形，描述tt-谱集合并简化其拓扑研究至初等阿贝尔情形，在基环温和假设下发展扭曲上同调理论，将初等阿贝尔p-群的tt-谱实现为狄拉克概形。
Almost all permutations and involutions are Kostant negative
该研究证明当$n$趋于无穷时，复李代数$\frak{sl}_n(\bbC)$的BGG范畴$\frak{O}$主块中几乎所有单最高权模的Kostant问题答案为负，解决了相关代数表示论问题。
Construction of Young modules and filtration multiplicities for Brauer algebras of type $C$
本文通过扩展超八面体群的表示理论构造了C型Brauer代数的置换模与Young模，建立分层系统以确定胞腔模滤过重数的良定义性，并证明了当域特征不为2和3时置换模可分解为不可分解Young模的直和。
The permutation action of finite symplectic groups of odd characteristic on their standard modules
受辛极空间中点与面的关联问题的启发，研究了奇特征有限辛群的标准模上函数空间的一大子模。这类子模的结构结果使我们能够确定辛极空间的点与平面之间的关联矩阵的$p$ -秩。特别地，我们给出了广义四边形${\rm W}(3,q)$的$p$ -秩的一个显式公式，其中$q$是奇素数幂.结合Sastry和Sin关于${\rm W}(3,2^t)$的2-秩的早期结果，完成了${\rm W}(3,q)$的$p$ -秩的确定.
Kostant's problem for permutations of shape $(n-2,1,1)$ and $(n-3,2,1)$
本文研究了形如(n-2,1,1)和(n-3,2,1)的排列对应的Kostant问题，并验证了相关猜想的成立。
The vertices and sources of the natural simple module for the alternating group in even characteristic
对于$n\geq 5$，交替群$\mathfrak{A}_n$的自然置换模具有唯一的非平凡合成因子，称为其自然单模。我们确定的顶点和源的自然简单的$\mathfrak{A}_n$ -模的特征2。
We construct a resolution of irreducible complex representations of the symmetric group $S_n$ by restrictions of representations of $GL_n(\mathbb{C})$ (where $S_n$ is the subgroup of permutation matrices). This categorifies a recent result of Assaf and Speyer. Our construction also gives minimal resolutions of simple $\mathcal{F}$-modules (here $\mathcal{F}$ is the category of finite sets).
On endosplit $p$-permutation resolutions and Broué's conjecture for $p$-solvable groups
本文通过端分裂 $p$-置换分解的新刻画和 Galois 下降的约化，证明了 Kessar-Linckelmann 提出的 Broué 猜想细化版本对满足 $G = O_{p',p,p'}(G)$ 且具有交换 Sylow $p$-子群的某些块成立，这是 Harris-Linckelmann 验证所有 $p$-可解群的 Broué 猜想的关键约化步骤。
We prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's a-function. Our arguments are combinatorial, and include a description of the intersection of two principal order ideals when at least one permutation is boolean. An important object in our work is a reduced word written as minimally many runs of consecutive integers, and one step of our argument shows that this minimal quantity is equal to the length of the second row in the permutation's shape under the Robinson…
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