V. Heiermann, G. Muic
Let G be a quasi-split p-adic group. Under the assumption that the local coefficients $C_ψ$ defined with respect to $ψ$-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber, we show that the standard module conjecture is true, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Twisted Jacquet modules: a conjecture of D. Prasad
本文研究非阿基米德局部域上可除代数对应的GL₂(D)主系列表示子商的扭化Jacquet模，证明了四元数情形下D. Prasad关于Speh表示扭化Jacquet模的猜想，并计算了任意可除代数下深度零主系列中广义Speh表示的扭化Jacquet模维数、明确其结构。
We verify a finiteness conjecture of Feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups.
In this article we prove a conjecture of Braverman-Kazhdan in \cite{BK} on the acyclicity of gamma sheaves in the de Rham setting. The proof relies on the techniques developed in \cite{BFO} on Drinfeld center of Harish-Chandra bimodules and character D-modules. As an application, we show that the functors of convolution with gamma D-modules, which can be viewed as a version of non-linear Fourier transforms, commute with induction functors and are exact on the category of admissible D-modules on a reductive group.
We give a general proof of Shahidi's tempered L-function conjecture, which has previously been known in all but one case. One of the consequences is the standard modules conjecture for p-adic groups, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible and the inducing data generic. We have also included the result that every generic tempered representation of a p-adic group is a sub-representation of a representation parabolically induced from a generic supercuspidal representation with a non-negative real central character.
We describe a general conjecture on how one may derive from the generic bosonic case all structural properties of multivariate diagonal coinvariant modules in $k$ sets of $n$ commuting variables (bosons), and $j$ sets of $n$ anticommuting variables (fermions).
Let $E/F$ be a quadratic extension of p-adic fields. We prove that every smooth irreducible ladder representation of the group $GL_n(E)$ which is contragredient to its own Galois conjugate, possesses the expected distinction properties relative to the subgroup $GL_n(F)$. This affirms a conjecture attributed to Jacquet for a large class of representations. Along the way, we prove a reformulation of the conjecture which concerns standard modules in place of irreducible representations.
A conjecture on the tensor ideal for an elementary p-group generated by the restriction of a Steinberg module
本文针对初等阿贝尔p-群的模表示与最大非投射受限Steinberg SL_2-模的张量积是否为受限倾斜模这一猜想，提供了支持性证据，并指出其成立将对正特征下不可压缩对称张量范畴的分类产生重要影响。
We provide a self-contained proof of the main properties of Brauer quotients of Young modules. We then use these results to give a new inductive proof of Nakayama's Conjecture on the blocks of the symmetric group.
For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan-Lusztig cells in the affine Weyl group. In this paper, we propose a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys.
The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. Here we establish the existence of the umbral moonshine modules in the remaining 22 cases.
We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.
In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic $p$ that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin's longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a $p$-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the $(p,r)$-Filtration Conjecture.
Two proofs of a Jantzen Conjecture for Whittaker Modules
本文定义了复半单李代数上标准 Whittaker 模的滤过并证明其基本性质，将 Jantzen 猜想推广至 Whittaker 模，并分别通过代数和几何方法给出了两个证明。
This paper proves Lee-Lee's conjecture that establishes a coincidence between the set of associated roots of non-self-intersecting curves in a $n$-punctured disc and the set of real Schur roots of acyclic (valued) quivers with $n$ vertices.
Is a Verma module transformed into another Verma module by a selfequivalence? The answer is affirmative and the proof suggests a notion of standard object in the category of Harish-Chandra modules that coincides often, but not always, with the usual one.
In this paper the authors provide a complete answer to Donkin's Tilting Module Conjecture for all rank $2$ semisimple algebraic groups and $\text{SL}_{4}(k)$ where $k$ is an algebraically closed field of characteristic $p>0$. In the process, new techniques are introduced involving the existence of $(p,r)$-filtrations, Lusztig's character formula, and the $G_{r}$T-radical series for baby Verma modules.
Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a til…
On Donkin's Tilting Module Conjecture IV: New Frontiers
本文回顾Donkin倾斜模猜想，基于2019年发现的反例提出新版本猜想，并通过新计算为A_n型根系约化群在正特征下支持该猜想提供证据。
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the r…
In this paper we give a sufficient condition of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations. We also introduce property (W$^{k}$). As an application of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations we give a connection between the finitistic dimension conjecture, the Auslander-Reiten conjecture and property (W$^{k}$).
We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: Whenever a direct product $\prod_{n \in \Bbb N} M_n$ of finitely generated indecomposable modules $M_n$ is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the $M_n$. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.
In the theory of Lie groups, the irreducibility of a unitary representation is not preserved in general by restriction to a subgroup. Kirillov's conjecture says that it is preserved for the groups Gl(n,R) or Gl(n,C) when the subgroup is the subgroup of matrices leaving invariant a non zero vector. This conjecture was proved by Barush using a detailed study of nilpotent orbits. In fact, it is not difficult to see that the conjecture is equivalent to the fact that some system of partial differential equations has no singular distributions as solutions. This system of equations is a regular holon…
The Vogan Conjectures (sometimes called Kazhdan-Lusztig Conjectures) say that a certain algorithm works both on the category of BGG modules and on the category of Harish-Chandra modules. The Alexandru Conjecture tries to uncover the general property common to these two categories which makes Vogan's algorithm work.
In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded.
Let $A$ be an Iwanaga-Gorenstein ring. Enomoto conjectured that a self-orthogonal $A$-module has finite projective dimension. We prove this conjecture for $A$ having the property that every indecomposable non-projective maximal Cohen-Macaulay module is periodic. This answers a question of Enomoto and shows the conjecture for monomial quiver algebras and hypersurface rings.
We show that the Jantzen filtration of a Verma module (possibly singular) coincides with its radical filtration. It implies that the Jantzen Conjecture on Verma modules holds for all infinitesimal characters, while the regular case was settled by Beilinson and Bernstein by geometric methods and reproved by Williamson by an algebraic approach. The coincidence between Jantzen filtration and radical filtration is also generalized to the case of parabolic Verma modules by Shan's results.
It is shown that, if $R$ is either an Artin algebra or a commutative noetherian domain of Krull dimension $1$, then infinite direct products of $R$-modules resist direct sum decomposition as follows: If $(M_n)_{n \in \Bbb N}$ is a family of non-isomorphic, finitely generated, indecomposable $R$-modules, then $\prod_{n\in \Bbb N} M_n$ is not a direct sum of finitely generated modules. The bearing of this direct product condition on the pure semisimplicity problem is discussed.
Irreducible modules of modular Lie superalgebras and super version of the first Kac-Weisfeiler conjecture
本文针对特征p>2的代数闭域上有限维限制李超代数，提出第一Kac-Weisfeiler猜想的超推广猜想，证明其对基本典型李超代数和完全可解限制李超代数成立，并研究了可解李超代数的不可约表示。
In this article, we show in the ADE case that the fusion product of Kirillov-Reshetikhin modules for a current algebra, whose character is expressed in terms of fermionic forms, can be constructed from one-dimensional modules by using Joseph functors. As a consequence, we obtain some identity between fermionic forms and Demazure operators. Since the same identity is also known to hold for one-dimensional sums of nonexceptional type, we can show from these results the X = M conjecture for type $A_n^{(1)}$ and $D_n^{(1)}$.
We classify indecomposable pure injective modules over domestic string algebras, verifying Ringel's conjecture on the structure of such modules.
Let $G$ be a reductive algebraic group scheme defined over $\mathbb{F}_p$ and let $G_1$ denote the Frobenius kernel of $G$. To each finite-dimensional $G$-module $M$, one can define the support variety $V_{G_1}(M)$, which can be regarded as a $G$-stable closed subvariety of the nilpotent cone. A $G$-module is called a tilting module if it has both good and Weyl filtrations. In 1997, it was conjectured by J.E. Humphreys that when $p\geq h$, the support varieties of the indecomposable tilting modules coincide with the nilpotent orbits given by the Lusztig bijection. In this paper, we shall verif…
We address two problems regarding the structure and representation theory of finite W-algebras associated with the general linear Lie algebras. Finite W-algebras can be defined either via the Whittaker model of Kostant or, equivalently, by the quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of the finite W-algebras. The second main result is a parametrization of finite families of irreducible Harish-Chandra modules by the characters of the Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classif…
We present three examples of Green biset functors for which their simple modules can be parametrized. These are particular cases of a conjecture by Serge Bouc classifying the simple modules over a Green biset functor A, that generalizes the classification of simple biset functors. We also prove this conjecture under certain hypothesis for A.
The Brauer-Thrall Conjectures, now theorems, were originally stated for finitely generated modules over a finite-dimensional $\sk$-algebra. They say, roughly speaking, that infinite representation type implies the existence of lots of indecomposable modules of arbitrarily large $\sk$-dimension. These conjectures have natural interpretations in the context of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings. This is a survey of progress on these transplanted conjectures.
The Telescope Conjecture for Global Representations and FI-modules
针对特征零域\(k\)上有限群的无限族\(U\)，对全局表示导出范畴\(D(U)\)的局部化理想分类，证明其望远镜猜想成立，通过对偶性确立导出 VI - 模相关猜想，还证明了 FI - 模导出范畴的望远镜猜想。
On Donkin's Tilting Module Conjecture II: Counterexamples
本文构造了Jantzen问题和Donkin倾斜模猜想的无穷反例族，提供显式例子及从小秩到大秩的构造方法，覆盖除A_n和B_2型外的所有群。
A conjecture going back to the eighties claims that there are no non-trivial self-extensions of irreducible modules over symmetric groups if the characteristic of the ground field is not equal to $2$. We obtain some partial positive results on this conjecture.
We prove that a certain eventually homological isomorphism between module categories induces a triangle equivalence between their singularity categories, Gorenstein defect categories and the stable categories of Gorenstein projective modules. Further, we show that Auslander-Reiten conjecture and Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms. Applying the results to arrow removal and vertex removal, we describe the Gorenstein projective modules over some non-monomial algebras, and we verify the Auslander-Reiten conjecture for certain algebras.
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…
The exceptional infinite-dimensional linearly compact simple Lie superalgebra ${E}(5,10)$, which Kac believes, is the algebra of symmetries of the ${SU}_{5}$ Grand Unified Model. In this paper, we give a proof of Kac and Rudakov's conjecture about the classification of all the degenerate generalized Verma module over ${E}(5,10)$. Also, we work out all the nontrivial singular vectors degree by degree. It is a potential that the representation theory of ${E}(5,10)$ will shed new light on various features of the the ${SU}_{5}$ Grand unified model.
In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module is n!, and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for the module which would give a simple proof of the dimension. Using the theory of Orbit Harmonics developed by Garsia and Haiman, we present a "kicking basis" for Garsia-Haiman modules indexed by a partition with at most two columns.
On self-extensions of irreducible modules over symmetric groups, II
本文改进对称群在奇特征域上不可约表示无非平凡自扩张猜想的已有部分结果，提供更多证据。
First extension groups of Verma modules and $R$-polynomials
针对 Gabber-Joseph 猜想被 Booe 证伪的问题，本文研究 Verma 模第一扩张群维数与 Kazhdan-Lusztig 的 $R$-多项式系数之间的偏差程度。
We determine explicitly the maximal dominant weights for the integrable highest weight $\hat{sl}(n)$-modules $V((k-1)Λ_0 + Λ_s)$, $0 \leq s \leq n-1$, $ k \geq 2$. We give a conjecture for the number of maximal dominant weights of $V(kΛ_0)$ and prove it in some low rank cases. We give an explicit formula in terms of lattice paths for the multiplicities of a family of maximal dominant weights of $V(kΛ_0)$. We conjecture that these multiplicities are equal to the number of certain pattern avoiding permutations. We prove that the conjecture holds for $k=2$ and give computational evidence for the …
A counterexample to the periodicity conjecture for finite-dimensional algebras
针对有限维代数周期性猜想，构造了一个36维反例，其单模周期为四，但存在2维非周期模，证明代数本身非周期。
We study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between is composition of elementary homomorphisms. The purpose of this article is to show the conjecture is affirmative for many parabolic subalgebras under the assumption that the infinitesimal characters are regular.
Dave Benson conjectured in 2020 that if $G$ is a finite $2$-group and $V$ is an odd-dimensional indecomposable representation of $G$ over an algebraically closed field $\Bbbk$ of characteristic $2$, then the only odd-dimensional indecomposable summand of $V \otimes V^*$ is the trivial representation $\Bbbk$. This would imply that a tensor power of an odd-dimensional indecomposable representation of $G$ over $\Bbbk$ has a unique odd-dimensional summand. Benson has further conjectured that, given such a representation $V$, the function sending a positive integer $n$ to the dimension of the uniqu…
We study the decomposability of Specht modules labelled by bihooks, bipartitions with a hook in each component, for the Iwahori--Hecke algebra of type $B$. In all characteristics, we determine a large family of decomposable Specht modules, and conjecture that these provide a complete list of decomposable Specht modules indexed by bihooks. We prove the conjecture for small $n$.
In this paper we investigate Donkin's $(p,r)$-Filtration Conjecture, and present two proofs of the "if" direction of the statement when $p\geq 2h-2$. One proof involves the investigation of when the tensor product between the Steinberg module and a simple module has a good filtration. One of our main results shows that this holds under suitable requirements on the highest weight of the simple module. The second proof involves recasting Donkin's Conjecture in terms of the identifications of projective indecomposable $G_{r}$-modules with certain tilting $G$-modules, and establishing necessary co…
Hom-orthogonal modules and brick-Brauer-Thrall conjectures
本文利用 Hom-正交性及代数几何工具刻画同维无限 bricks 族的存在条件，证明新代数族上的 bBT 猜想，并推广已有预投射分支结果。
The Countable Telescope Conjecture arose in the framework of stable homotopy theory, as a tool conceived to study the chromatic filtration. It turned out, however, to trigger extremely fertile research within the framework of Module Categories. The project aims at presenting an almost self-contained review of the recent work of Saroch on the Countable Telescope Conjecture for Module Categories. After recalling some preliminaries, we report various devices of independent interest that will lead to a proof of the aforementioned result. This will be the outcome of inductive refinements of familie…
Using the theory of vertex operator algebras and intertwining operators, we obtain presentations for the principal subspaces of all the standard $\widehat{\goth{sl}(3)}$-modules. Certain of these presentations had been conjectured and used in work of Calinescu to construct exact sequences leading to the graded dimensions of certain principal subspaces. We prove the conjecture in its full generality for all standard $\widehat{\goth{sl}(3)}$-modules. We then provide a conjecture for the case of $\widehat{\goth{sl}(n+1)}$.
Self-Orthogonal $τ$-Tilting Modules and Tilting Modules III: Finitistic Dimension
本文证明在有限小有限维数和自扩张消失条件下，$\ au$-倾斜模成为经典倾斜模，并揭示自正交 $\ au$-倾斜猜想与有限维数猜想的关系，且对根立方为零的 Artin 代数成立。
Homological theory of orthogonal modules
本研究从稳定模范畴视角研究自入射Artin代数上的有限生成正交生成元，建立M-相对稳定范畴的粘合、刻画紧对象，给出Tachikawa第二猜想的等价刻画，还引入Gorenstein-Morita代数并证明Nakayama猜想对其成立。
Consider the middle perversity intersection cohomology groups of various compactifications of a Hermitian locally symmetric space. Rapoport and independently Goresky and MacPherson have conjectured that these groups coincide for the reductive Borel-Serre compactification and the Baily-Borel-Satake compactification. This paper describes the theory of L-modules and how it is used to solve the conjecture. More generally we consider a Satake compactification for which all real boundary components are equal-rank. Details will be given elsewhere (math.RT/0112251). As another application of L-modules…
Let $\mathscr{H}_n$ denote the Iwahori-Hecke algebra corresponding to the symmetric group $\mathfrak{S}_n$. We set up a Green correspondence for bimodules of these Hecke algebras, and a Brauer correspondence between their blocks. We examine Specht modules for $\mathscr{H}_n$ and compute the vertices of certain Specht modules, before using this to give a classification of the vertices of blocks of $\mathscr{H}_n$ in any characteristic. Finally, we apply this classification to resolve the Dipper-Du conjecture about the structure of vertices of indecomposable $\mathscr{H}_n$-modules.
The Cherednik-Orr conjecture expresses the $t\to\infty$ limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases.
We prove a conjecture by Dipper, James and Murphy that a bipartition is restricted if and only if it is Kleshchev. Hence the restricted bipartitions naturally label the crystal graphs of level two irreducible integrable $\mathcal{U}_v({\hat{\mathfrak{sl}}_e})$-modules and the simple modules of Hecke algebras of type $B_n$.
We study the restricted Verma modules of an affine Kac-Moody algebra at the critical level with special emphasis on their Jordan-H"older multiplicities. The Feigin-Frenkel conjecture gives a formula for these multiplicities that involves the periodic Kazhdan-Lusztig polynomials. We prove this conjecture for all subgeneric blocks and for the case of anti-dominant simple subquotients.
A conjecture of C. Bonnafé, M. Geck, L. Iancu, and T. Lam parameterizes Kazhdan-Lusztig left cells for unequal parameter Hecke algebras in type $B_n$ by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by G. Lusztig and the recent work of C. Bonnafé, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。