The center of SL2 tilting modules
Daniel Tubbenhauer, Paul Wedrich
本文计算了素特征下 SL2 的 tilting 模、量子群及射影 GgT-模的中心，涵盖 g=1,2 情形。
本文计算了素特征下 G=SL2 的 tilting 模范畴的中心、相应量子群在复单位根处的 tilting 模的中心，以及 g=1,2 时射影 GgT-模的中心。
In this note we compute the centers of the categories of tilting modules for G=SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g=1,2.
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Let $G$ be the quantum $GL_n$ over a field of characteristic $p\neq 0$. In this paper we define the ring of twisted tilting modules of $G$. We give generators and relations for the ring of twisted tilting modules of quantum $GL_2(k)$. We also show that this is a reduced ring
Balanced semisimple filtrations for tilting modules
设$U_l$是在$l$第单位根上的量子群. $U_l$的许多倾斜模都被证明具有我们称之为平衡半单滤子的性质，或者是一个Loewy级数，其半单层关于某个中间层对称。如果已知不可约特征标，则这种过滤的存在表明了计算这些特征标的一种非常简单的算法。首先证明了该算法的结果与Soergel正则倾斜模的特征公式一致。然后，我们表明，这些平衡的半单滤子确实存在这些倾斜模。
In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a (strictly object-adapted) cellular category. We use this result to specify a subset of cellular basis elements, which generates all morphisms in this category. In a different direction we generalize the earlier results to the case where G is replaced by the infinitesimal thickenings G_rT of a maximal torus T in G by the Frobenius subgroup schemes G_r. Here …
We show that tilting modules for quantum groups over local Noetherian domains exist and that the indecomposable tilting modules are parametrized by their highest weight. For this we introduce a model category ${\mathcal X}={\mathcal X}_{\mathscr A}(R)$ associated with a Noetherian ${\mathbb Z}[v,v^{-1}]$-domain ${\mathscr A}$ and a root system $R$. We show that if ${\mathscr A}$ is of quantum characteristic $0$, the model category contains all $U_{\mathscr A}$-modules that admit a Weyl filtration. If ${\mathscr A}$ is in addition local, we study torsion phenomena in the model category. This le…
Rigidity of tilting modules
设$U_q$表示有限维半单李代数的量子群.设$q$是奇数阶单位的复根，$U_q$是通过Lusztig的$q$ -分幂构造得到的量子群形式.我们证明了$U_q$的所有正则投射（倾斜）模都是刚性的，即，有相同的根和柱的过滤。此外，我们还得到了$U_q$的一类Weyl模的相同结果.另一方面，我们给出了非刚性不可分解倾斜模和非刚性Weyl模的例子。这些例子是针对类型$B_2$的，在这种情况下以及对于类型$A_2$，我们明确地计算了所有正则Weyl模的Loewy结构。我们还表明，这些结果结转到模块化的情况下，当最高权的问题是在所谓的Jantzen区域。同时，我们通过例子表明，一旦我们离开这个区域，非刚性倾斜模确实出现。
The higher APR tilting modules and higher BB tilting modules were introduced and studied in higher Auslander-Reiten theory. Our objective is to consider these tilting modules by the corresponding simple modules, and show that the tensor product of higher APR (BB) tilting modules is a higher APR (BB) tilting module.
Quivers for SL(2) tilting modules
本文利用图方法定义了一个依赖于素数p的箭图代数，并证明它是特征p下SL(2)倾斜模范畴的底层代数，同时给出了p-Jones-Wenzl投影子之间态射的表示。
$τ$-tilting modules over one-point extensions by a projective module
设$A$是代数$B$通过投射$B$ -模的单点扩张.我们证明了一个给定的支持$τ$ -倾斜$B$ -模的扩张是一个支持$τ$ -倾斜$A$ -模;反之，一个给定的支持$τ$ -倾斜$A$ -模的限制是一个支持$τ$ -倾斜$B$ -模.证明了支撑$τ$ -倾斜模的偏序集之间存在箭图的完全嵌入.
Taking tilting modules from the poset of support tilting modules
C. Ingalls和H. Thomas定义的路代数的支撑倾斜模。从T.足立岛伊山和我。Reiten，D. Happel和L. Unger作为偏序在支撑倾斜模集上进行了推广。本文研究了基本倾斜模偏序集与基本支撑倾斜模之间的一种组合关系。我们将证明倾斜模的子偏序集是由支撑倾斜模的偏序集结构唯一决定的。
Distributive lattices of tilting modules and support $τ$-tilting modules over path algebras
本文分别研究了当$Q$是Dynkin图时基本倾斜$kQ$ -模的偏序集和当$Q$是连通无圈图时基本支撑$τ$ -倾斜$kQ$ -模的偏序集.证明了第一偏序集是分配格当且仅当$Q$是$\mathbb{A}_{1}, \mathbb{A}_{2}$型或具有非线性定向的$\mathbb{A}_{3}$型，第二偏序集是分配格当且仅当$Q$是$\mathbb{A}_{1}$型.
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。
Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of a locally finite building of type $\tilde{\text{A}}_{n-1}$ ($n\ge3$). From a type $\tilde{\text{A}}_{n-1}$ triangle presentation on a geometry of order $q$, we construct a fiber functor on the diagrammatic monoidal category $\text{Web}(\text{SL}^{-}_{n})$ over any field $\mathbb{k}$ with characteristic $p\ge n-1$ such that $q \equiv 1$ mod $p$. When $\mathbb{k}$ is algebraically closed and $n$ odd, this gives new fiber functors on the categ…
Using Kuperberg's $B_2/C_2$ webs, and following Elias and Libedinsky, we describe a "light leaves" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for $\mathfrak{so}_5\cong \mathfrak{sp}_4$ (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when $[2]_q\ne 0$, the Karoubi envelope of the $C_2$ web category is equivalent to the category of tilting modules for the divided powers quantum group $\mathcal{U}_q^{\mathbb{Z}}(\mathfrak{sp}_4)$.
The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, and infinitely generated relative tilting modules have recently been studied by many authors in the context of Gorenstein homological algebra. In this work, we build on the theory of infinitely generated Gorenstein tilting modules by developing "Gorenstein tilting approximations" and employing these approximations to study Gorenstein tilting classes and their associated relative cotorsion pairs. As applications of our results, we discuss the problem of existence of compleme…
On mutation of τ-tilting modules
τ-倾斜模的变异是从一个给定的支撑τ-倾斜模通过置换直被加数构造新的支撑τ-倾斜模的基本操作。本文的目的是对[AIR，问题2.31]中提出的关于τ-倾斜模的突变的问题给出一个肯定的回答。
In this paper, we study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey-Sauter, Nguyen-Reiten-Todorov-Zhu and Pressland-Sauter. Moreover, we give characterizations of almost $n$-Auslander-Gorenstein algebras and almost $n$-Auslander algebras by the existence of tilting modules. As an application, we describe a sufficient condition for almost $1$-Auslander algebras to be strongly quasi-hereditary by comparing such tilting modules and characteristic tilting modules.
Let $Λ$ be an artin algebra and $T$ a $τ$-tilting $Λ$-module. We prove that $T$ is a tilting module if and only if ${\rm Ext}_Λ^{i}(T,\Fac T)=0$ for all $i\geq 1$, where $\Fac T$ is the full subcategory consisting of modules generated by $T$. Consequently, a $τ$-tilting module $T$ of finite projective dimension is a tilting module if and only if ${\rm Ext}_Λ^{i}(T, T)=0$ for all $i\geq 1$. Moreover, we also give an example to show that a support $τ$-tilting but not $τ$-tilting module $M$ of finite projective dimension satisfying ${\rm Ext}_Λ^{i}(M, M)=0$ for all $i\geq1$ need not be a partial …
Tilting modules arising from two-term tilting complexes
我们证明了Artin代数上的每一个双项倾斜复形都有一个倾斜模在某个因子代数上作为同调群。另外，我们还确定了这样一个同调群的自同态代数，它是作为两项倾斜复形的自同态代数的一个因子代数给出的。因此，每一个由两项倾斜复形给出的Artin代数之间的导出等价都诱导出相应的因子代数之间的导出等价。
For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Geiß~ et al. (2016) introduced a generalized preprojective algebra $Π(C, D)$. We study tilting modules and support $τ$-tilting modules for the generalized preprojective algebra $Π(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $Π(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $τ$-tilting $Π(C,D)$-modules and the corresponding Weyl group $…
Let $Λ$ be an algebra with a indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support $τ$-tilting $Λ$-modules. In this paper, we will show that it can be restricted to $τ$-tilting modules.
The blob algebra is a finite-dimensional quotient of the Hecke algebra of type $B$ which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic $0$ in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl multiplicities of the indecomposabl…
We introduce the notions of Gorenstein projective $τ$-rigid modules, Gorenstein projective support $τ$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support $τ$-tilting modules. More precisely, for an algebra $Λ$, We prove that there is a bijection between the set of Gorenstein projective support $τ$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$τ$-tilting finite algebras and show that $Λ$ is CM-$τ$-tilting finite if and only if…
Support $τ$-tilting modules correspond to some classes of categorical objects bijectively, such as two-term tilting complexes for any finite dimensional symmetric algebra. This fact motivates us to classify support $τ$-tilting modules over blocks of finite groups. Therefore we classify support $τ$-tilting modules over particular blocks of finite groups by using the modular representation theoretical approaches including Clifford theory, Green's indecomposability theorem and so on.
Relative contravariantly finite subcategories and relative tilting modules
设$A$是代数闭域$k$上的有限维代数.设$T$是倾斜$A$ -模，$B={\rm End}_A\ T$是$T$的自同态代数.本文研究了倾斜$A$ -模与倾斜$B$ -模之间的对应关系，证明了$T^{\perp}$中的基本$T$ -倾斜$A$ -模与$^{\perp}(D_BT)$中的基本倾斜$B$ -模之间存在一一对应关系.此外，我们还证明了$T^{\perp}$的$T$ -逆变有限$T$ -分解子范畴与包含在$T^{\perp}$中的基本$T$ -倾斜$A$ -模之间存在一一对应.作为应用，我们证明了当$A$是有限维遗传代数上的$1$ -Gorenstein代数或$m$ -复制代数时，$T^{\perp}$中的基本倾斜$A$ -模与$^{\perp}(D_BT)$中的基本倾斜$B$ -模之间存在一一对应.
Comparing classical tilting modules and $τ$-tilting modules
本文通过张量消失与 Tor¹ 消失条件建立 τ-倾斜模和 1-倾斜模的相互刻画，并利用 delooping levels 在自同态代数具有有限全局 delooping level 时证实自正交 τ-倾斜猜想。
We study (support) $τ$-tilting modules over the trivial extensions of finite dimensional algebras. More precisely, we construct two classes of (support)$τ$-tilting modules in terms of the adjoint functors which extend and generalize the results on (support) $τ$-tilting modules over triangular matrix rings given by Gao-Huang.
Classifying τ-tilting modules over preprojective algebras of Dynkin type
研究Dynkin型预投射代数上的支撑$τ$ -倾斜模。我们通过给出一个双射来分类基本支撑$τ$ -倾斜模，其中的元素在相应的外尔群中.并且证明了它们与挠类集、无挠类集以及表示论中的其他重要对象是双射的。我们还研究了支撑$τ$ -倾斜模的$g$ -矩阵，它表示不可分解直和项的极小投射表示.给出了$g$ -矩阵的显式描述，并证明了由$g$ -矩阵给出的锥与相关根系的腔室重合.
We give a construction of Gorenstein projective $τ$-tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective $τ$-tilting modules. Moreover, we show that a finite dimensional algebra $Λ$ over an algebraically closed field is $CM$-$τ$-tilting finite if $T_n(Λ)$ is $CM$-$τ$-tilting finite which gives a partial answer to a question on $CM$-$τ$-tilting finite algebras posed by Xie and Zhang.
Following the previous work, we determine exactly the highest weights for which a tensor product of two induced modules is a tilting module, for the algebraic group $SL_2$ over an algebraically closed field of positive characteristic.
We give an example of a finite-dimensional algebra with a 2-cluster tilting module and a simple module which has infinite complexity. This answers a question of Erdmann and Holm.
Tilting modules for cyclotomic Schur algebras
本文研究了分圆q-Schur代数的倾斜模和Ariki-Koike代数的Young模以及它们之间的联系。用来理解倾斜模的主要工具是矛盾对偶，以及某些置换模的Specht滤子和对偶Specht滤子。令人惊讶的是，外尔过滤--一般来说比斯佩希特过滤更强大--只起次要作用。
In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize the algebras with finite relative dominant dimension. As an application, we introduce the almost n-precluster tilting module and establish a correspondence between almost n-precluster tilting modules and almost n-minimal Auslander-Gorenstein algebras. Moreover, we give a description of the Gorenstein projective modules over almost n-minimal Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster tilting modules.
Reduction of $τ$-tilting modules and torsion pairs
支撑$τ$ -倾斜模类是最近由Adachi，Iyama和Reiten提出的.这些模从突变的角度完善了倾斜模类。给定一个有限维代数$A$，研究了所有以基本$τ$ -刚性$A$-模为直和项的基本支撑$τ$ -倾斜$A$ -模.证明了存在一个代数$C$，使得这些模与所有基本支集$τ$ -倾斜$C$ -模之间存在一个保序双射，我们称这个过程为$τ$ -倾斜约化.在这个过程中的一个重要步骤是形成$τ$ -垂直范畴，它是普通垂直范畴的类似物。最后，我们证明了$τ$ -倾斜归约与淤积归约和2-Calabi-Yau归约在适当的三角划分范畴内是相容的。
Universal derived equivalences of posets of tilting modules
我们表明，对于两个箭图没有定向循环相关的BGP反射，其倾斜模块的偏序集相关的一个简单的组合结构，我们称之为触发器。证明了无定向圈箭图的导等价路代数的倾斜模的偏序集是泛导等价的。
Let $B$ be the one-point extension algebra of $A$ by an $A$-module $X$. We proved that every support $τ$-tilting $A$-module can be extended to be a support $τ$-tilting $B$-module by two different ways. As a consequence, it is shown that there is an inequality $$|\stilt B|\geqslant 2|\stilt A|.$$
We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa-Suzuki functor [Arakawa-Suzuki, J. of Alg. 209 (1998)] gives a fully faithful embedding of a block of the deformed BGG category of $\mathfrak{gl}_{m}$ into the module category of a suitable completion of degenerate affine Hecke algebra of $GL_{n}$.
Tilting Modules over Tame Hereditary Algebras
给出了驯服遗传代数R上无限维倾斜模的一个完全分类。首先考虑T=R_U \oplus R_U /R型倾斜模，其中U是管的并，R_U表示R在U上的Schofield和Crawley-Boevey意义下的泛局部化.这里R_U/R是U中管子对应的Prüfer模的直和。在Kronecker代数上，大倾斜模在除了一种情况之外的所有情况下都是这种形式，例外是Lukas倾斜模L，其倾斜类Gen L由所有没有不可分解的预投射和项的模组成。在任意驯服遗传代数上，T可以有有限维的和项，但T的无穷维部分仍然由泛局部化、Prüfer模和Lukas倾斜模（的局部化）构成。我们还恢复了Buan和Krause对无限维余倾斜R-模的分类。
Let $B$ be an one-point extension of a finite dimensional $k$-algebra $A$ by a simple $A$-module at a source point $i$. In this paper, we classify the $τ$-tilting modules over $B$. Moreover, it is shown that there are equations $$|\tilt B|=|\tilt A|+|\tilt A/\langle e_i\rangle|\quad \text{and}\quad |\stilt B|=2|\stilt A|+|\stilt A/\langle e_i\rangle|.$$ As a consequence, we can calculate the numbers of $τ$-tilting modules and support $τ$-tilting modules over linearly Dynkin type algebras whose square radical are zero.
Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. We establish an isomorphism of $G$-modules between a direct sum of modules $\text{St} \otimes \text{St}$ and a direct sum of tensor products of simple modules of restricted highest weight with tilting modules that are projective over the Frobenius kernel of $G$. This isomorphism holds precisely when Donkin's Tilting Module Conjecture does, and thus can be seen as providing a $G$-module theoretic characterization of this conjecture.
Geometric models for endomorphism algebras of tilting modules over gentle algebras
研究温和代数上倾斜模及其自同态代数，利用标记曲面与平铺框架，证明平铺忠实剖分对应倾斜模，定义辅助代数并证其与自同态代数同构，构造新平铺，引入保持翻转的倾斜模。
A construction of support $τ$-tilting modules over $τ$-tilting finite algebras
本文研究τ-倾斜有限代数上支撑τ-倾斜模的构造，借助中间扩张函子沿recollement粘合半砖，探讨τ-倾斜有限性在模范畴粘合下的性质，并给出构造实例。
In this article, we prove that induced modules of support $τ$-tilting modules over blocks of finite groups satisfying inertial-invariant condition are also support $τ$-tilting modules.
On the character of certain tilting modules
设G是特征p>0的代数闭域上的半单群.在p很大的条件下，我们给出了许多不可分解倾斜有理G-模的特征标的一个（部分代数的）简单的闭公式.
We describe the semisimplification of the monoidal category of tilting modules for the algebraic group GL_n in characteristic p > 0. In particular, we compute the dimensions of the indecomposable tilting modules modulo p.
On Certain Tilting Modules for SL2
我们给出了一个完整的图片时，一个诱导模和Weyl模块的张量积是一个倾斜模块的代数群$SL_2$上的代数闭域的特征$p$。虽然结果是递归的性质，我们给出了一个明确的声明中的$p$ -进的扩展的最高权的每个模块。
We construct families of representations for quantum groups over $\mathbb{Z}[v,v^{-1}]$-algebras that interpolate between Weyl modules and tilting modules. These families might be candidates for objects with characters satisfying the {\em generations of characters} philosophy of Lusztig and Lusztig-Williamson.
On tilting modules over cluster-tilted algebras
在本文中，我们证明了倾斜模上的集群倾斜代数$A$提升到倾斜对象在相关的集群范畴$\mathcal{C}_H$。作为第一个应用，我们描述了倾斜$A$ -模的诱导交换关系产生于$\mathcal{C}_H$中倾斜对象的交换关系。作为第二个应用，我们展示了倾斜$A$ -模簇倾斜自同态代数。
Rigidity of tilting modules in category O
本文给出了与约化李代数相关的BGG范畴O中倾斜模的刚性性质和Loewy长度。这些结果是众所周知的几个专家，但似乎很难找到现有的文献。
Virtual algebraic Lie theory: Tilting modules and Ringel duals for blob algebras
我们构建了一个表示的斑点代数在一个环，允许基地的变化，每一个有趣的（即非--半单）专业化，在准遗传专业化，传递到一个完整的倾斜模块。
In this article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in \cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(n\geq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.
Non-singular weighted surface algebras satisfy the necessary condition found in [6] for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiver is bipartite, has a module satisfying the necessary ext vanishing condition. We show that it is 3-cluster tilting precisely for the non-singular triangular or spherical algebras but not for any other weighted surface algebra with bipartite Gabriel quiver.
Strongly $n$-AIR-tilting modules
本文定义作为 support $τ$-tilting 模高维推广的强 $n$-AIR-tilting 模，梳理其与多种 silting、tilting 对象及广义两项 silting 复形的关系，并为否定秩问题提供思路。
Parity sheaves and tilting modules
当特征大于一个明确的界时，我们证明了仿射格拉斯曼算子上的倾斜模和宇称层是通过几何佐竹对应联系在一起的。
We study tilting and projective-injective modules in a parabolic BGG category $\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters of tilting modules in terms of those of simple modules in that category. We also obtain a classification of projective-injective modules in the full BGG category $\mathcal O$ for all simple classical Lie superalgebras. We then classify and give an explicit combinatorial description of parabolic subalgebras of the periplectic Lie superalgebras and …
The classification of τ-tilting modules over Nakayama algebras
本文研究了Nakayama代数上的τ-倾斜模。我们建立了τ-倾斜模、带穿刺的多边形三角剖分和某些整数序列之间的双射。此外，我们还利用Drozd-Kirichenko的拒绝引理给出了构造支撑τ倾斜模的Hasse箭图的算法。
Tilting modules arising from ring epimorphisms
给定一个环R，我们研究了对于某个内射环满射R \to S，具有S \oplus S/R形式的倾斜模.特别是，我们感兴趣的倾斜模块所产生的斯科菲尔德的普遍本地化。对于某些环，用这种方法可以得到所有倾斜模的分类。
Let $(\mbox{mod} Λ',\mbox{mod} Λ,\mbox{mod} Λ'')$ be a recollement of module categories for artin algebras $Λ'$, $Λ$ and $Λ''$. We provide a sufficient condition such that a glued torsion pair in $\mbox{mod} Λ$ is tilting when the given two torsion pairs are tilting in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ respectively. Using this result, we give a construction of gluing of tilting modules in $\mbox{mod} Λ$ with respect to tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ respectively.
Endomorphism algebras of semi-tilting modules
设$A$是代数闭域$k$上的有限维代数.研究了半倾斜$A$ -模的自同态代数的结构性质，证明了由半倾斜$A$ -模的突变产生的自同态代数可以实现为BB-倾斜模的自同态代数.
For any good tilting module $T$ over a ring $A$, there exists an $n$-symmetric subcategory $\mathscr{E}$ of a module category such that the derived category of the endomorphism ring of $T$ is a recollement of the derived categories of $\mathscr{E}$ and $A$ in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of $\mathscr{E}$.
We provide a classification of generalized tilting modules and full exceptional sequences for the dual extension algebra of the path algebra of a uniformly oriented linear quiver modulo the ideal generated by paths of length two with its opposite algebra. For the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration with respect to the relation arising from higher extensions.
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