Wilberd van der Kallen
Let G be a connected reductive linear algebraic group over a field k of characteristic p>0. Let p be large enough with respect to the root system. We show that if a finitely generated commutative k-algebra A with G-action has good filtration, then any noetherian A-module with compatible G-action has finite good filtration dimension.
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
Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to $p^n$-filtrations for $n > 1$, to modules containing the Steinberg module as a tensor factor, and to the Donkin conjecture on modules having $p$-filtrations.
The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for $GL_n(k)$. This paper is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new r…
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。
Canonical filtrations on Harish-Chandra modules
本文证明包括所有不可约模在内的一大类Harish-Chandra (g,K)-模都具有某种典范滤过结构。
Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski constructed a filtration of weight spaces of a G module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. The goal of this article is to generalize these results to positive characteristics.
We obtain alternative explicit Specht filtrations for the induced and the restricted Specht modules in the Hecke algebra of the symmetric group (defined over the ring $A=\mathbb Z[q^{1/2},q^{-1/2}]$ where $q$ is an indeterminate) using $C$-bases for these modules. Moreover, we provide a link between a certain $C$-basis for the induced Specht module and the notion of pairs of partitions.
We study the PBW filtration on the irreducible highest weight representations of simple complex finite-dimensional Lie algebras. This filtration is induced by the standard degree filtration on the universal enveloping algebra. For certain rectangular weights we provide a new description of the associated graded module in terms of generators and relations. We also construct a basis parametrized by the integer points of a normal polytope. The main tool we use is the Hasse diagram defined via the standard partial order on the positive roots. As an application we conclude that all representations …
In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the usual BGG category $\mathcal{O}$. Combined with Jantzen coefficients, we determine the radical filtrations for all basic generalized Verma modules. The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of generalized Verma modules in any (possibly s…
The space of homomorphisms from a projective object to a Verma module in category O inherits an induced filtration from the Jantzen filtration on the Verma module. On the other hand there is the Andersen filtration on the space of homomorphisms from a Verma module to a tilting module as described in [Soe07]. The tilting equivalence from [Soe98] induces an isomorphism of these kinds of Hom-spaces. We will show that this equivalence even identifies both filtrations.
Let $G$ be a connected reductive group over an algebraically closed field $\Bbbk$. Under mild restrictions on the characteristic of $\Bbbk$, we show that any $G$-module with a good filtration also has a good filtration as a module for the reductive part of the centralizer of a nilpotent element $x$ in its Lie algebra.
Let G be a connected and reductive algebraic group over an algebraically closed field of characteristic p > 0. An interesting class of representations of G consists of those G-modules having a good filtration -- i.e. a filtration whose layers are the standard highest weight modules obtained as the space of global sections of G-linearized line bundles on the flag variety of G. Let H be a connected and reductive subgroup of G. One says that (G,H) is a Donkin pair, or that H is a good filtration subgroup of G, if every G-module with good filtration also has a good filtration as an H-module. In th…
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.
This paper considers Weyl modules for a simple, simply connected algebraic group over an algebraically closed field $k$ of positive characteristic $p\not=2$. The main result proves, if $p\geq 2h-2$ (where $h$ is the Coxeter number) and if the Lusztig character formula holds for all (irreducible modules with) regular restricted highest weights, then any Weyl module $Δ(λ)$ has a $Δ^p$-filtration, namely, a filtration with sections of the form $Δ^p(μ_0+pμ_1):=L(μ_0)\otimesΔ(μ_1)^{[1]}$, where $μ_0$ is restricted and $μ_1$ is arbitrary dominant. In case the highest weight $λ$ of the Weyl module $Δ…
Studied are the composition series of the standard Whittaker (g,K)-modules. For a generic infinitesimal character, the structures of these modules are completely understood, but if the infinitesimal character is integral, then there are not so many cases in which the structures of them are known. In this paper, as an example of the integral case, we determine the socle filtrations of the standard Whittaker (g,K)-modules when G is the group Spin(r,1) and the infinitesimal character is regular integral.
Given a quasi-hereditary superalgebra $A$, the first author and R. Muth have defined generalized Schur bi-superalgebras $T^A(n)$ and proved that these algebras are again quasi-hereditary. In particular, $T^A(n)$ comes with a family of standard modules. Developing the work of Donkin and Mathieu on good filtrations, we prove that tensor product of standard modules over $T^A(n)$ has a standard filtration.
We investigate rational $G$-modules $M$ for a linear algebraic group $G$ over an algebraically closed field $k$ of characteristic $p > 0$ using filtrations by sub-coalgebras of the coordinate algebra $k[G]$ of $G$. Even in the special case of the additive group $\mathbb G_a$, interesting structures and examples are revealed. The "degree" filtration we consider for unipotent algebraic groups leads to a "filtration by exponential degree" applicable to rational $G$ modules for any linear algebraic group $G$ of exponential type; this filtration is defined in terms of 1-parameter subgroups and is r…
We construct a cellular basis of the walled Brauer algebra which has similar properties as the Murphy basis of the group algebra of the symmetric group. In particular, the restriction of a cell module to a certain subalgebra can be easily described via this basis. Furthermore, the mixed tensor space possesses a filtration by cell modules -- although not by cell modules of the walled Brauer algebra itself, but by cell modules of its image in the algebra of endomorphisms of mixed tensor space.
On the space of homomorphisms from a Verma module to an indecomposable tilting module of the BGG-category O we define a natural filtration following Andersen and establish a formula expressing the dimensions of the filtration steps in terms of coefficients of Kazhdan-Lusztig polynomials.
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。
We study the structure of the $(\mathfrak{g},K)$-modules of the principal series representations of $SL(3,\mathbb{R})$ and $Sp(2,\mathbb{R})$ induced from minimal parabolic subgroups, in the case when the infinitesimal character is nonsingular. The composition factors of these modules are known by Kazhdan-Lusztig-Vogan conjecture. In this paper, we give complete descriptions of the socle filtrations of these modules.
The classical Pieri formula gives a multiplicity free expansion of an irreducible module with a fundamental one for the complex general linear group. In this article we replace the tensor product by the fusion product and prove an analogue Pieri formula for higher level Demazure modules for the affine Lie algebra $\widehat{\mathfrak{sl}}_{n+1}$. To be more precise, we show that the fusion product of an arbitrary stable Demazure module with a fundamental module admits a multiplicity free excellent filtration and the successive quotients are described explicitly. As a consequence, we derive recu…
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。
Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $\ggg$ with $p$-character $χ$ of standard Levi-form associated with an index subset $I$ of simple roots. With aid of support variety theory we prove a theorem that a $U_χ(\ggg)$-module is projective if and only if it is a strong "tilting" module, i.e. admitting both $\cz_Q$- and $\cz^{w^I}_Q$-filtrations (to see Theorem \ref{THMFORINV}). Then by analogy of the arguments in \cite{…
We reprove the main result of our joint work [arXiv:0706.0604] with Srinivas, concerning finite Schur filtration dimension, but now with the base field replaced by a commutative noetherian ring k. This leads to finiteness results for the cohomology of a reductive group scheme G over k with coefficients in a finitely generated commutative k-algebra with G action.
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.
Jantzen filtration of Weyl modules for general linear supergroups
本文为奇特征域上一般线性超群$GL(m|n)$的典型权构造Weyl模的Jantzen滤过，导出特征标求和公式，并证明不可约模为Kac模当且仅当权为$p$-典型。
Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p>0$, $Δ(λ)$ denote the Weyl module of $G$ of highest weight $λ$ and $ι_{λ,μ}:Δ(λ+μ)\to Δ(λ)\otimesΔ(μ)$ be the canonical $G$-morphism. We study the split condition for $ι_{λ,μ}$ over $\mathbb{Z}_{(p)}$, and apply this as an approach to compare the Jantzen filtrations of the Weyl modules $Δ(λ)$ and $Δ(λ+μ)$. In the case when $G$ is of type $A$, we show that the split condition is closely related to the product of certain Young symmetrizers and, under some mild conditions, is further character…
In this paper, we will introduce a subcategory of totally reflexive modules that have a saturated filtration by other totally reflexive modules. We will prove these are precisely the totally reflexive modules with an upper-triangular presentation matrix. We conclude with an investigation of the ranks of $\operatorname{Ext}^1$ of two such modules over a specific ring.
In this paper we study a generalization of the Jacquet module of a parabolic induction and construct a filtration on it. The successive quotient of the filtration is written by using the twisting functor.
Let $kE$ denote the group algebra of an elementary abelian $p$-group of rank $r$ over an algebraically closed field of characteristic $p$. We investigate the functors $\mathcal{F}_i$ from $kE$-modules of constant Jordan type to vector bundles on $\mathbb{P}^{r-1}(k)$, constructed by Benson and Pevtsova. For a $kE$-module $M$ of constant Jordan type, we show that restricting the sheaf $\mathcal{F}_i(M)$ to a dimension $s-1$ linear subvariety of $\mathbb{P}^{r-1}(k)$ is equivalent to restricting $M$ along a corresponding rank $s$ shifted subgroup of $kE$ and then applying $\mathcal{F}_i$. In the…
In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in graded restricted module categories of basic classical Lie superalgebras over an algebraically closed field of odd characteristic, and prove a sum formula in the corresponding Grothendieck groups. We finally obtain a strong linkage principle in such categories.
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…
We show that the Hodge filtration of a tempered Hodge module is generated by the lowest piece of its Hodge filtration. As a consequence, we prove the main conjecture of [SV] in the special case of tempered representations of real reductive Lie groups.
In continuation of work begun in \cite{FR}, we study in this article those Auslander--Reiten components of the algebras $\Dist(G_r)$ that contain simple modules or baby Verma modules, where $\Dist(G_r)$ is the algebra of distributions of the $r$-th Frobenius kernel of the smooth reductive group $G$, defined over an algebraically closed field of positive characteristic.
Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N < 6 or p > 2^N. Let G act rationally on a finitely generated commutative k-algebra A. Assume that A as a G-module has a good filtration or a Schur filtration. Let M be a noetherian A-module with compatible G action. Then M has finite good/Schur filtration dimension, so that there are at most finitely many nonzero H^i(G,M). Moreover these H^i(G,M) are noetherian modules over the ring of invariants A^G. Our main tool is a res…
We give a new proof that the restriction of a cell module of the Hecke algebra of the symmetric group on $n$ letters, to the Hecke algebra of the symmetric group on $n-1$ letters, has a filtration by cell modules.
Let $U_l$ be a quantum group at an $l$th root of unity. Many tilting modules for $U_l$ have been shown to have what we call a balanced semisimple filtration, or a Loewy series whose semisimple layers are symmetric about some middle layer. The existence of such filtrations suggests a remarkably straightforward algorithm for calculating these characters if the irreducible characters are already known. We first show that the results of this algorithm agree with Soergel's character formula for the regular tilting modules. We then show that these balanced semisimple filtrations really do exist for …
Tilting modules and tilting torsion pairs - filtrations induced by tilting modules
本文研究倾斜模如何通过诱导滤子揭示模块范畴的整体结构，探讨其在代数与表示论中的应用与贡献。
Some methods of computing first extensions between modules of graded Hecke algebras
本文利用Langlands分类、根滤过与Jantzen滤过，研究分次Hecke代数中单模的第一扩张群，并计算类型C3分次Hecke代数块内不可约模的Ext群。
We prove that in positive characteristic a module with good filtration for a group of type E6 restricts to a module with good filtration for a subgroup of type F4. (Recall that a filtration of a module for a semisimple algebraic group is called good if its layers are dual Weyl modules.) Our result confirms a conjecture of Brundan for one more case. The method relies on the canonical Frobenius splittings of Mathieu. Next we settle the remaining cases, in characteristic not 2, with a computer-aided variation on the old method of Donkin.
An example of the Jantzen filtration of a D-module
本文计算SL₂(ℂ)旗簇上D-模的Jantzen滤过，结合整体截面的𝔰𝔩₂(ℂ)-模结构给出代数图景，验证了Beilinson–Bernstein定理。
We study the PBW filtration on the highest weight representations $V(\la)$ of $\msl_{n+1}$. This filtration is induced by the standard degree filtration on $U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr V(\la)$ in terms of generators and relations. We also construct a basis of $gr V(\la)$. As an application we derive a graded combinatorial character formula for $V(\la)$, and we obtain a new class of bases of the modules $V(\la)$ conjectured by Vinberg in 2005.
We define a filtration indexed by the integers on the tensor product of an integrable highest weight module and a loop module for a quantum affine algebra. We prove that the filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the module is irreducible and in the second case we prove that the intersection of all the modules is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration. These are certain highest weight integrable modu…
New graded modules for the current algebra of $\mathfrak{sl}_n$ are introduced. Relating these modules to the fusion product of simple $\mathfrak{sl}_n$-modules and local Weyl modules of truncated current algebras shows their expected impact on several outstanding conjectures. We further generalize results on PBW filtrations of simple $\mathfrak{sl}_n$-modules and use them to provide decomposition formulas for these new modules in important cases.
The ADR algebra $R_A$ of an Artin algebra $A$ is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the $Δ$-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of $R_A$-modules. As an application, we give a counterexample to a claim by Auslander-Platzeck-Todorov, concerning projective resolutions over the ADR algebra.
We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for $\widehat{\mathfrak{sl}}_2$ in terms of partitions with bounded parts, which subsequently lead to certain partition identities.
This is a survey of some recent results on Sapovalov elements and the Jantzen filtration for contragredient Lie superalgebras. The topics covered include the existence and uniqueness of the Sapovalov elements, bounds on the degrees of their coefficients and the behavior of Sapovalov elements when the Borel subalgebra is changed. There is always a unique term whose coefficient has larger degree than any other term. This allows us to define some new highest weight modules. If X is a set of orthogonal isotropic roots and $λ\in h^*$ is such that $λ+ρ$ is orthogonal to all roots in X, we construct …
We define {\bf primitive derivations} for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the {\bf primitive filtrations} of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration has already been studied, which generalizes the Hodge filtration introduced by K. Saito.
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.
In a previous paper, the authors studied the radical filtration of a Weyl module $Δ_ζ(λ)$ for quantum enveloping algebras $U_ζ(\overset\circ{\mathfrak g})$ associated to a finite dimensional complex semisimple Lie algebra $\overset\circ{\mathfrak g}$. There $ζ^2=\sqrt[e]{1}$ and $λ$ was, initially, required to be $e$-regular. Some additional restrictions on $e$ were required---e.g., $e>h$, the Coxeter number, and $e$ odd. Translation to a facet gave an explicit semisimple series for all quantum Weyl modules with singular, as well as regular, weights. That is, the sections of the filtration are…
We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis. In the favourable case a basis of the module is parameterized by the lattice points of a normal polytope. The filtrations induce flat degenerations of the corresponding flag variety to its abelianized version and to a toric variety, the specia…
We introduce and study new categories T(g,k)of integrable sl(\infty)-modules which depend on the choice of a certain reductive subalgebra k in g=sl(\infty). The simple objects of these categories are tensor modules as in the previously studied category, however, the choice of k provides more flexibility of nonsimple modules. We then choose k to have two infinite-dimensional diagonal blocks, and show that a certain injective object K(m|n) in T(g,k) realizes a categorical sl(\infty)-action on the integral category O(m|n) of the Lie superalgebra gl(m|n). We show that the socle of K(m|n) is genera…
Filtration of tensor product of local Weyl modules for $\mathfrak{sl}_{n+1}[t]$
本文研究sl_{n+1}[t]特定最高权局部Weyl模的张量积，借助局部Weyl模分次特征标确定张量积的分次特征标，证明其存在商为截尾Weyl模或Demazure模融合乘积的滤过，还确立了sl_3[t]一类融合乘积模与赋值参数无关。
Let $k$ be an algebraically closed field of positive characteristic, $G$ a reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $B$ be a Borel subgroup of $G$, and $U$ its unipotent radical. We prove that if $S=\Sym V$ has a good filtration, then $S^U$ is $F$-pure.
We characterise the symplectic Weyl group elements such that the FFLV basis is compatible with the PBW filtration on symplectic Demazure modules, extending type A results by the second author. Surprisingly, the number of such elements depends not on the type A or C of the Lie algebra but on the rank only.
The purpose of this paper is to prove necessary and sufficient criteria for a $GL(m|n)$-supermodule to have a good or Weryl filtration. We also introduce the notion of a Steinberg supermodule analogous to the classical notion of Steinberg module. We prove that the Steinberg supermodule inherits some properties of the Steinberg module. Some new series of finite-dimensional tilting supermodules are found.
Let $k$ be an algebraically closed field of positive characteristic, $G$ a reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $P$ be a parabolic subgroup of $G$, and $U_P$ its unipotent radical. We prove that if $S$=\textyen $Sym V$ has a good filtration, then the ring of invariants $S^{U_P}$ is strongly $F$-regular.
Let $\H_n$ be a (degenerate or non-degenerate) Hecke algebra of type $G(\ell,1,n)$, defined over a commutative ring $R$ with one, and let $S(\bmu)$ be a Specht module for $\H_n$. This paper shows that the induced Specht module $S(\bmu)\otimes_{\H_n}\H_{n+1}$ has an explicit Specht filtration.
This paper investigates the tilting modules of the cyclotomic q-Schur algebras, the Young modules of the Ariki-Koike algebras, and the interconnections between them. The main tools used to understand the tilting modules are contragredient duality, and the Specht filtrations and dual Specht filtrations of certain permutation modules. Surprisingly, Weyl filtrations -- which are in general more powerful than Specht filtrations -- play only a secondary role.
Categories of split filtrations and graded quiver varieties
本文研究分裂滤波器范畴与分层有向无环图品种，通过几何方法实现n-折叠张量积，建立稳定范畴与三角矩阵代数的等价关系。
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。