Sibylle Schroll, Hipolito Treffinger
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.
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Stable Brauer--Thrall II$'$ conjecture for finite-dimensional Jacobian algebras
本文证明了有限维雅可比代数满足稳定布拉uer-Thrall II'猜想，并验证了该猜想在该类代数中的成立。
The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and strongly derived unbounded algebras naturally. The first and second Brauer-Thrall type theorems for the bounded derived category of a finite-dimensional algebra over an algebraically closed field are obtained. The first Brauer-Thrall type theorem says that derived bounded algebras are just derived finite algebras. The second Brauer-Thrall type theorem says that an …
Remarks on $τ$-tilted versions of the second Brauer-Thrall Conjecture
本文提出第二 Brauer–Thrall 猜想的稳定版与 $τ$-约化版，说明前者蕴含后者，并在 Asai–Iyama 的 E-tame 代数类上证明反向蕴含。
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 first Brauer-Thrall conjecture for extriangulated length categories
本文在外三角长度范畴中引入关于 $Θ$ 的 Gabriel-Roiter 测度并推广 Gabriel 性质，在技术性条件下证明第一 Brauer-Thrall 猜想。
Representation embeddings and the second Brauer-Thrall conjecture
本文通过构造从经典 Kronecker 模范畴到任意表示无限代数模范畴的表示嵌入，并确定表示无限代数的最小类，推进了第二 Brauer-Thrall 猜想的研究。
Let R be a commutative noetherian local ring that is not Gorenstein. It is known that the category of totally reflexive modules over R is representation infinite, provided that it contains a non-free module. The main goal of this paper is to understand how complex the category of totally reflexive modules can be in this situation. Local rings (R,m) with m^3=0 are commonly regarded as the structurally simplest rings to admit diverse categorical and homological characteristics. For such rings we obtain conclusive results about the category of totally reflexive modules, modeled on the Brauer-Thra…
Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexiv…
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。
A Proof of the Second brick-Brauer-Thrall Conjecture for $E$-infinite algebras
本文证明对代数闭域上的有限维代数，$E$-无限性蕴含第二 brick-Brauer-Thrall 猜想，并由此推出稳定离散性与 $E$-有限性等价，同时证明 Demonet 猜想蕴含稳定版本猜想，解决了多个开放问题。
Two-sided tilting complexes for generalized Brauer tree algebras
针对广义Brauer树代数，显式构造出与Membrillo-Hernández树到星倾斜复形相对应的双侧倾斜复形。
Two-term tilting complexes of biserial fractional Brauer graph algebras
本文研究双序列分数Brauer图代数的两项倾斜复形和Kauer移动，证明其斜群代数是斜Brauer图代数，并给出倾斜离散性的充要条件及导出等价封闭性。
We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the simplicial complexes as lattice polytopes. Via a geometric interpretation of derived equivalences, we show that the $f$-vector of simplicial complexes of Brauer tree algebras only depends the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support $τ$-tilting modules, is a derived invariant. Moreover, we apply our result to the …
In this paper we describe all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application we describe all two-term tilting complexes over Brauer star algebra and compute their endomorphism rings.
Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.
In this paper all two-term tilting complexes over a Brauer tree algebra with multiplicity one are described using a classification of indecomposable two-term partial tilting complexes obtained earlier in a joint paper with M. Antipov. The endomorphism rings of such complexes are computed.
In this paper, motivated by a $τ$-tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $τ$-tilting finite if and only if no band module is a brick. We also recover a criterion for the $τ$-tilting finiteness of Brauer graph algebras in terms of the Brauer graph.
We show that any Brauer tree algebra has precisely $\binom{2n}{n}$ $2$-tilting complexes, where $n$ is the number of edges of the associated Brauer tree. More explicitly, for an external edge $e$ and an integer $j\neq0$, we show that the number of $2$-tilting complexes $T$ with $g_e(T)=j$ is $\binom{2n-|j|-1}{n-1}$, where $g_e(T)$ denotes the $e$-th of the $g$-vector of $T$. To prove this, we use a geometric model of Brauer graph algebras on the closed oriented marked surfaces and a classification of $2$-tilting complexes due to Adachi-Aihara-Chan.
Tilting mutations as generalized Kauer moves for (skew) Brauer graph algebras with multiplicity
本文将广义Kauer移动与倾斜突变诱导导出等价的结果推广到任意重数Brauer图代数及斜Brauer图代数，并证明重数为1的斜Brauer图代数是斜gentle代数的平凡扩张。
In this paper we introduce mutation of Brauer trees. We show that our mutation of Brauer trees explicitly describes the tilting mutation of Brauer tree algebras introduced by Okuyama and Rickard.
The aim of this paper is to study a poset isomorphism between two support $τ$-tilting posets. We take several algebraic information from combinatorial properties of support $τ$-tilting posets. As an application, we treat a certain class of basic algebras which contains preprojective algebras of type $A$, Nakayama algebras, and generalized Brauer tree algebras. We provide a necessary condition for that an algebra $Λ$ share the same support $τ$-tilting poset with a given algebra $Γ$ in this class. Furthermore, we see that this necessary condition is also a sufficient condition if $Γ$ is either a…
Motivated by a new conjecture on the behavior of bricks, we start a systematic study of minimal $τ$-tilting infinite algebras. In particular, we treat minimal $τ$-tilting infinite algebras as a modern counterpart of minimal representation infinite algebras and show some of the fundamental similarities and differences between these families. We then relate our studies to the classical tilting theory and observe that this modern approach can provide fresh impetus to the study of some old problems. We further show that in order to verify the conjecture it is sufficient to treat those minimal $τ$-…
The notion of mutation plays crucial roles in representation theory of algebras. Two kinds of mutation are well-known: tilting/silting mutation and quiver-mutation. In this paper, we focus on tilting mutation for symmetric algebras. Introducing mutation of SB quivers, we explicitly give a combinatorial description of tilting mutation of symmetric special biserial algebras. As an application, we generalize Rickard's star theorem. We also introduce flip of Brauer graphs and apply our results to Brauer graph algebras.
Mutation of Brauer configuration algebras
本文将 Brauer graph 代数中倾斜突变与翻转相容的结果推广至 Green 和 Schroll 引入的 Brauer configuration 代数，在特定条件下引入了 Brauer configuration 的翻转并证明其与倾斜突变相容。
Constructible Subcategories and Unbounded Representation Type (with an appendix by Andres Fernandez Herrero)
本研究针对域上有限生成代数的可构造子范畴，证明有界型蕴含有限型（第一Brauer-Thrall猜想变体），借助Ziegler谱与有限维模概形的联系展开论证，讨论第二Brauer-Thrall猜想变体的成立条件，并给出矩阵约化到正合结构约化的转化。
Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel [Ha] proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard [Ri] to develop a general Morita theory of derived categories. In the other hand, functor categories were introduced in representation theory by M. Auslander and used in his proof of the first Brauer- Thrall conjecture and later on, used systematically in his joint w…
Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $Λ^+$ be an indexing set of all simple modules in $A\lmod$. We show that if there is an integer $r\in\N$ such that for any $λ\inΛ^+$, there is an embedding $Δ(λ)\hookrightarrow T^{\oplus r}$ as well as an epimorphism $T^{\oplus r}\twoheadrightarrow\overline{\nabla}(λ)$ as $A$-modules, then $T$ is a faithful $A$-module and $A$ has the double centraliser property with respect to $T$. As applications, we prove that if $A$ is quasi-hereditary with a simple preserving duality and $T$ a given faithful …
On $τ$-tilting modules over trivial extensions of gentle tree algebras
本研究刻画了gentle树代数平凡扩张的结构（对应无例外顶点的Brauer树代数等），并得出其支撑$τ$-倾斜模数量仅由单模数量决定的结论。
We give a survey on the theory of representation-finite and certain minimal representation-infinite algebras.The main goals are the existence of multiplicative bases and of coverings with good properties. Both are attained via ray-categories. As applications we include a proof of a sharper version of the second Brauer-Thrall conjecture and of the fact that there are no gaps in the lengths of the indecomposable modules over an algebra.
Brauer tree algebras are important and fundamental blocks in the modular representation theory of groups. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between the ordinary Brauer tree algebra and a particular algebra called the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tr…
In this paper we construct non-negative gradings on a basic Brauer tree algebra $A_Γ$ corresponding to an arbitrary Brauer tree $Γ$ of type $(m,e)$. We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra $A_S$, whose tree is a star with the exceptional vertex in the middle, to $A_Γ$. The grading on $A_S$ comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green's walk around $Γ$ (cf. [\ref{Zak}]). By computing endomorphism rings of these tilting complexes …
Infinite $τ$-tilting theory
通过Ziegler谱的cosilting子集分类小abelian范畴中的torsion对，推广到Artin代数上得到无限τ-倾斜理论，并应用于研究generic bricks和generic τ⁻-rigid模。
We show how to use Jantzen's sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\textbf{U}_q$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in\mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\textbf{A}$ and $\textbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.
On algebras of finite general representation type
本文引入有限一般表示型概念，构造野表示型但有限一般表示型的代数族，并证明离散一般表示型的局部代数必为有限表示型，最后提出Brauer-Thrall风格猜想。
We introduce the notion of reflections for selfinjective algebras from the point of view of torsion theories induced by two-term tilting complexes. As an application, we determine the transformations of Brauer trees associated with reflections. In particular, we provide a way to transform every Brauer tree into a Brauer line.
In this paper we establish a connection between ribbon graphs and Brauer graphs. As a result, we show that a compact oriented surface with marked points gives rise to a unique Brauer graph algebra up to derived equivalence. In the case of a disc with marked points we show that a dual construction in terms of dual graphs exists. The rotation of a diagonal in an m-angulation gives rise to a Whitehead move in the dual graph, and we explicitly construct a tilting complex on the related Brauer graph algebras reflecting this geometrical move.
We introduce Brauer complex of symmetric SB-algebra, and reformulate in terms of Brauer complex the so far known invariants of stable and derived equivalence of symmetric SB-algebras. In particular, the genus of Brauer complex turns out to be invariant under derived equivalence. We study transformations of Brauer complexes which preserve class of derived equivalence. Additionally, we establish a new invariant of derived equivalence of symmetric SB-algebras. As a consequence, symmetric SB-algebras with Brauer complex of genus 0 are classified. Keywords: Brauer tree algebras, special biserial al…
We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of simple modules under a stable equivalence, which is given by the restriction of a standard derived equivalence induced by a two-term tilting complex. We achieve this by exploiting and connecting the mutation theories from the combinatorics of Brauer tree, configurations of stable translations quivers of type A, and triangulations of a punctured convex regular…
To study the set of torsion classes of a finite dimensional basic algebra, we use a decomposition, called sign-decomposition, parametrized by elements of $\{\pm1\}^n$ where $n$ is the number of simple modules. If $A$ is an algebra with radical square zero, then for each $ε\in \{\pm1\}^n$ there is a hereditary algebra $A_ε^!$ with radical square zero and a bijection between the set of torsion classes of $A$ associated to $ε$ and the set of faithful torsion classes of $A_ε^!$. Furthermore, this bijection preserves the property of being functorially finite. As an application in $τ$-tilting theory…
Let k be an algebraically closed field of characteristic p>0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GL_n-modules to the category of modules for the walled Brauer algebra. As a corollary we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook lengt…
$τ$-Tilting finiteness of group algebras over generalized symmetric groups
本研究通过证明弱对称τ-倾斜有限代数的Cartan矩阵正定性，得到自内射代数τ-倾斜无限的矩阵条件，并将其应用于广义对称群的群代数，给出τ-倾斜无限的判定条件及p'-子群情形下的决定因素。
Decomposition numbers of cyclotomic Brauer algebras over the complex field, I
本文在特定条件下建立分圆Brauer代数右胞腔模与抛物线Verma模的联系，利用B_n、C_n和D_n型抛物线Kazhdan-Lusztig多项式实现了其分解数的显式计算。
Let $G$ be a connected reductive algebraic group over a field of positive characteristic $p$ and denote by $\mathcal T$ the category of tilting modules for $G$. The higher Jones algebras are the endomorphism algebras of objects in the fusion quotient category of $\mathcal T$. We determine the simple modules and their dimensions for these semisimple algebras as well as their quantized analogues. This provides a general approach for determining various classes of simple modules for many well-studied algebras such as group algebras for symmetric groups, Brauer algebras, Temperley--Lieb algebras, …
Brick infinite algebras admit infinitely many non-$τ$-rigid bricks
本文证明，若代数$A$的所有砖块除有限个外都是$\ au$-刚性的，则$A$是砖有限的；等价地，任何砖无限代数都有无限多个非$\ au$-刚性砖，从而验证了砖-Brauer-Thrall第二猜想的一个较弱版本。
On the bricks (Schur representations) of finite dimensional algebras
本文综述有限维代数中砖块（Schur表示）在τ-倾斜理论、挠理论、几何表示论和不变量理论中的应用与性质，并围绕第二个砖块Brauer-Thrall猜想证明若干新结果。
Derived Picard groups of symmetric representation-finite algebras of type $D$
本文研究了类型 $D$ 的对称表示有限代数的导出皮卡群，通过球形扭动生成，并扩展了布拉uer树和Kauer移动的概念。
In this paper, a notion of cyclotomic (or level $k$) walled Brauer algebras $\mathscr B_{k, r, t}$ is introduced for arbitrary positive integer $k$. It is proven that $\mathscr B_{k, r, t}$ is free over a commutative ring with rank $k^{r+t}(r+t)!$ if and only if it is admissible. Using super Schur-Weyl duality between general linear Lie superalgebras $\mathfrak{gl}_{m|n}$ and $\mathscr B_{2, r, t}$, we give a classification of highest weight vectors of $\mathfrak{gl}_{m|n}$-modules $M_{pq}^{rt}$, the tensor products of Kac-modules with mixed tensor products of the natural module and its dual. …
For a finite-dimensional gentle algebra, it is already known that the functorially finite torsion classes of its category of finite-dimensional modules can be classified using a combinatorial interpretation, called maximal non-crossing sets of strings, of the corresponding support $τ$-tilting module (or equivalently, two-term silting complexes). In the topological interpretation of gentle algebras via marked surfaces, such a set can be interpreted as a dissection (or partial triangulation), or equivalently, a lamination that does not contain a closed curve. We will refine this combinatorics, w…
Fans and polytopes in tilting theory I: Foundations
本文为倾斜理论中的 g-单纯复形、g-扇和 g-多胞形建立基础理论，证明 g-凸代数的 g-多胞形为自反多胞形，并分类了光滑 Fano 情形及预投射代数和 Brauer 图代数的 g-凸性。
A generic classification of locally free representations of affine GLS algebras
本文研究仿射 GLS 代数的局部自由表示，通过广义一点扩张构造稳定表示单参数族，并折叠路代数的 Kac 典范分解，建立一般分类并证明其 $E$-驯顺。
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