Takuma Aihara
The notion of silting mutation was introduced by Iyama and the author. In this paper we mainly study silting mutation for self-injective algebras and prove that any representation-finite symmetric algebra is tilting-connected. Moreover we give some sufficient conditions for a Bongartz-type Lemma to hold for silting objects.
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Examples of tilting-discrete symmetric algebras
本文通过构造若干例子研究倾斜-离散对称代数，给出一个反例否定τ-倾斜有限对称代数必为倾斜-离散的猜想，并讨论对称代数的倾斜-不连通性，提供新的倾斜-不连通对称代数实例。
Two-sided tilting complexes over symmetric algebras sending simple modules to shifts of modules
本文针对代数闭域上有限维对称代数的倾斜复形，其不同分次分量无公共不可分解投射直和项，从诱导Morita型稳定等价的双模出发，显式地（重新）构造了对应于该倾斜复形的双侧倾斜复形。
On $τ$-tilting finiteness of symmetric algebras of polynomial growth
本文研究代数闭域上多项式增长对称代数等有限维代数的$τ$-倾斜有限性，发现导出等价保持该性质，并证明了相关代数中表示有限性与$τ$-倾斜有限性的一致性。
We show that $τ$-tilting finite simply connected algebras are representation-finite. Then, some related algebras are considered, including iterated tilted algebras, tubular algebras and so on. We also prove that the $τ$-tilting finiteness of non-sincere algebras can be reduced to that of sincere algebras. This motivates us to give a complete list of $τ$-tilting finite sincere simply connected algebras.
We give a necessary and sufficient condition for that the support $τ$-tilting poset of a finite dimensional algebra is isomorphic to the poset of symmetric group with weak order. Moreover we show that there are infinitely many finite dimensional algebras whose support $τ$-tilting posets are isomorphic to the poset of symmetric group with weak order.
If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in $\mathrm{K^b}(\mathrm{proj}A)$ are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.
In this paper, we compute the number of two-term tilting complexes for an arbitrary symmetric algebra with radical cube zero over an algebraically closed field. Firstly, we give a complete list of symmetric algebras with radical cube zero having only finitely many isomorphism classes of two-term tilting complexes in terms of their associated graphs. Secondly, we enumerate the number of two-term tilting complexes for each case in the list.
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.
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, and $M$ be a partial tilting $A$-module. We prove that the Bongartz $τ$-tilting complement of $M$ coincides with its Bongartz complement, and then we give a new proof of that every almost complete tilting $A$-module has at most two complements. Let $A=kQ$ be a path algebra. We prove that the support $τ$-tilting quiver $\overrightarrow{Q}({\rm s}τ$-${\rm tilt} A)$ of $A$ is connected. As an application, we investigate the conjecture of Happel and Unger in [9] which claims that each connected component of the tilting…
We discuss finiteness/infiniteness of $τ$-tilting modules over tensor products of two symmetric algebras. As an application, we discuss that over block algebras of direct products of finite groups.
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 $\mathscr{E}$.
In this paper, we study the conjecture II.1.9 of Cluster structures for 2-Calabi-Yau categories and unipotent groups, which said that any maximal rigid object without loops or 2-cycles in its quiver is a cluster tilting object in a connected Hom-finite triangulated 2-CY category C. We obtain some conditions equivalent to the conjecture, and using them we proved the conjecture.
Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $δ(T)$ the number of non isomorphic indecomposable summands of $T$. In this paper, we prove that a partial tilting $A^{(m)}$-module $T$ is a tilting $A^{(m)}$-module if and only if $δ(T)=δ(A^{(m)})$, and that every partial tilting $A^{(m)}$-module has complements. As an application, we deduce that the tilting quiver $\mathscr{K}_{A^{(m)}}$ of $A^{(m)}$ is connected. Moreover, we investigate the number of complements to almost til…
Gluing support $τ$-tilting modules via symmetric ladders of height 2
该研究针对阿贝尔范畴recollement的粘合问题，通过高度为2的对称阶梯给出支撑τ-倾斜模的显式粘合构造，并将其应用于三角矩阵代数，完善Jasso约化且研究最大绿色序列。
Let $\mathbb{X}$ be a weighted projective line of tubular type and $\operatorname{coh}\mathbb{X}$ the category of coherent sheaves on $\mathbb{X}$. The main purpose of this note is to show that the subgraph of the tilting graph consisting of all basic tilting bundles in $\operatorname{coh}\mathbb{X}$ is connected. This yields an alternative proof for the connectedness of the tilting graph of $\operatorname{coh}\mathbb{X}$. Our approach leads to the investigation of the change of slopes of a tilting sheaf in $\operatorname{coh}\mathbb{X}$ under (co-)APR mutations, which may be of independent in…
We build a bijection between the set $\sttiltΛ$ of isomorphism classes of basic support $τ$-tilting modules over the Auslander algebra $Λ$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttiltΛ$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $Λ$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $Γ$ of Dynkin type $A…
Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence between the Galois coverings with group G of A and the Galois coverings with group G of B. The hypothesis on T is expressed using the Hasse diagram of basic tilting A-modules and is always verified if A is of finite representation type. Then, we use the above correspondence to prove that A is simply connected if and only if B is simply connected, under the sam…
$τ$-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。
Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $Λ=\operatorname{End}_{\mathcal{D}}R$ be the endomorphism algebra of $R$. We introduce the notion of mutation of maximal rigid objects in the two-term subcategory $R\ast R[1]$ via exchange triangles, which is shown to be compatible with mutation of support $τ$-tilting $Λ$-modules. In the case that $\mathcal{D}$ is the cluster category arising from a punctured marked surface, it is shown that the graph of mutations of support $τ$-tilting $Λ$-modules is isomorphic to the graph of fli…
We formulate a conjecture for the second generation characters of indecomposable tilting modules for ${\rm SL}_3$. This gives many new conjectural decomposition numbers for symmetric groups. Our conjecture can be interpreted as saying that these characters are governed by a discrete dynamical system ("billiards bouncing in alcoves"). The conjecture implies that decomposition numbers for symmetric groups display (at least) exponential growth.
We consider tilting mutations of a weakly symmetric algebra at a subset of simple modules, as recently introduced by T. Aihara. These mutations are defined as the endomorphism rings of certain tilting complexes of length 1. Starting from a weakly symmetric algebra A, presented by a quiver with relations, we give a detailed description of the quiver and relations of the algebra obtained by mutating at a single loopless vertex of the quiver of A. In this form the mutation procedure appears similar to, although significantly more complicated than, the mutation procedure of Derksen, Weyman and Zel…
Important objects of study in $τ$-tilting theory include the $τ$-tilting pairs over an algebra on the form $kQ/I$, with $kQ$ being a path algebra and $I$ an admissible ideal. In this paper, we study aspects of the combinatorics of mutation quivers of support $τ$-tilting pairs, simply called mutation quivers. In particular, we are interested in identifying connected components of the underlying graphs of such quivers. We give a class of algebras with two simple modules such that every algebra in the class has at most two connected components in its mutation quiver, generalizing a result by Demo…
Tilting Completion and Full-Rank Self-Orthogonal Modules
该研究针对有限维代数的两个倾斜完备化问题给出否定答案，构造反例反驳相关猜想，并证明两个自正交猜想间的等价关系。
In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $τ$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $\mathbb{A}_{1}, \mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type $\mathbb{A}_{1}$.
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 prove a theorem which gives a bijection between the support $τ$-tilting modules over a given finite-dimensional algebra $A$ and the support $τ$-tilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the center of $A$ and the radical of $A$. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are $τ$-tilting finite wild blocks with more than one simple module. We then go on to classify all support $τ$-tilting modules for all algeb…
Let C be a connected noetherian hereditary abelian Ext-finite category with Serre functor over an algebraically closed field k, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from math.RT/9911242, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C has a tilting object.
Comparing the module categories of an algebra and of the endomorphism algebra of a given support $τ$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $τ$-tilting theory. Afterwards we define $τ$-slices and prove that complete slices of tilted algebras and local slices of cluster tilted algebras are examples of complete $τ$-slices. Then we apply this concept to the study of simply connected tilted algebras. Finally, we study the one-point extensions and the split-by-nilpotent extensions of an algebra with $τ$-slices.
The Nakayama conjecture states that an algebra of infinite dominant dimension should be self-injective. Motivated by understanding this conjecture in the context of derived categories, we study dominant dimensions of algebras under derived equivalences induced by tilting modules, specifically, the infinity of dominant dimensions under tilting procedure. We first give a new method to produce derived equivalences from relatively exact sequences, and then establish relationships and lower bounds of dominant dimensions for derived equivalences induced by tilting modules. Particularly, we show that…
We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras.
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 new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of Gorenstein properties of endomorphism rings.
We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p-Kazhdan-Lusztig polynomials, for p>h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if p>2h-3. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.
Let G be a semisimple, simply connected linear algebraic group over an algebraically closed field k of characteristic p > 0. In a recent paper [4], Doty introduces the notion of r-minuscule weight and exhibits a tensor product factorization of a corresponding tilting module under the assumption p >= 2h-2, where h is the coxeter number. We remove this restriction and consider some variations involving the more general notion of (p,r)-minuscule weights.
$τ$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras
该研究利用concealed代数理论，刻画了有限偏序集的$τ$-倾斜有限关联代数与单连通偏序集的$\boldsymbol{g}$-驯顺关联代数，提出相关猜想并验证反例稀少，还确定了双曲型concealed代数$τ$-倾斜约化的表示型。
Inspired by the tropical duality in cluster algebras, we introduce c-vectors for finite-dimensional algebras via $τ$-tilting theory. Let $A$ be a finite-dimensional algebra over a field $k$. Each c-vector of $A$ can be realized as the (negative) dimension vector of certain indecomposable $A$-module and hence we establish the sign-coherence property of this kind of $c$-vectors. We then study the positive c-vectors for certain classes of finite-dimensional algebras. More precisely, we establish the equalities between the set of positive c-vectors and the set of dimension vectors of exceptional m…
On support $τ$-tilting graphs of gentle algebras
本文研究了代数闭域上有限维gentle代数的支撑$τ$-倾斜图的组合性质，证明了其连通性及面内可达性质，印证了cluster代数交换图的相关猜想。
On $τ$-tilting graphs for quasi-silted algebras
本文证明任意拟倾斜代数的$\ au$-倾斜图是连通的且具有可达面性质，通过$\ au$-约化与墙室结构给出商代数保持连通性的充分条件。
Tilting-completion for gentle algebras
本文利用曲面模型和曲面切割归纳法，证明了gentle代数上的几乎倾斜模均为部分倾斜模且可完备化，至多有$2n$个补模，从而证实了Happel的猜想，并构造了非部分倾斜的预倾斜模反例。
Let $A$ be a finite dimensional hereditary algebra over a field $k$ and $A^{(1)}$ the duplicated algebra of $A$. We first show that the global dimension of endomorphism ring of tilting modules of $A^{(1)}$ is at most 3. Then we investigate embedding tilting quiver $\mathscr{K}(A)$ of $A$ into tilting quiver $\mathscr{K}(A^{(1)})$ of $A^{(1)}$. As applications, we give new proofs for some results of D.Happel and L.Unger, and prove that every connected component in $\mathscr{K}({A})$ has finite non-saturated points if $A$ is tame type, which gives a partially positive answer to the conjecture of…
Dominant Auslander-Gorenstein algebras and mixed cluster tilting
本文引入支配 Auslander-Gorenstein 代数与混合（预）cluster tilting 模，建立两者间的 Auslander 型双射对应，并证明平凡扩张代数与迭代 SGC-扩张均允许混合 tilting 模。
In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $τ^2=\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve …
In this paper, we show that for an algebra $Λ$ with radical square zero and an indecomposable $Λ$-module $M$ such that $Λ$ is Gorenstein of finite type or $τM$ is $τ$-rigid, $M$ is $τ$-rigid if and only if the first two projective terms of a minimal projective resolution of $M$ have no on-zero direct summands in common. We also determined all $τ$-tilting modules for Nakayama algebras with radical square zero. Moreover, by giving a construction theorem we show that a basic connected radical square zero algebra admitting a unique $τ$-tilting module is local.
For any truncated path algebra $Λ$, we give a structural description of the modules in the categories ${\cal P}^{<\infty}(Λ\text{-mod})$ and ${\cal P}^{<\infty}(Λ\text{-Mod})$, consisting of the finitely generated (resp. arbitrary) $Λ$-modules of finite projective dimension. We deduce that these categories are contravariantly finite in $Λ\text{-mod}$ and $Λ\text{-Mod}$, respectively, and determine the corresponding minimal ${\cal P}^{<\infty}$-approximation of an arbitrary $Λ$-module from a projective presentation. In particular, we explicitly construct - based on the Gabriel quiver $Q$ and th…
Maximal ideals in the finitary symmetric group algebra in characteristic two
本文解决了特征2域上有限对称群代数极大理想分类的悬而未决问题，通过关联其素理想与高阶Verlinde范畴，借助研究一般线性群倾斜模的厚张量理想完成了该工作。
Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field $k$ and $\mathscr{C}_{F^m}$ be the repetitive cluster category of $H$ with $m\geq 1$. We investigate the properties of cluster tilting objects in $\mathscr{C}_{F^m}$ and the structure of repetitive cluster-tilted algebras. Moreover, we generalized Theorem 4.2 in \cite{bmrrt} (Buan A, Marsh R, Reiten I. Cluster-tilted algebra. Trans. Amer. Math. Soc., 359(1)(2007), 323-332.) to the situation of $\mathscr{C}_{F^m}$, and prove that the tilting graph $\mathscr{K}_{\mathscr{C}_{F^m}}$ of $\mathscr{C}_{F^m}$ is con…
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。
To every minimal model of a complete local isolated cDV singularity Donovan--Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions, obtaining a faithful group action on the bounded derived category. Further, we determine precisely those standard equivalences which are induced by two-term tilting complexes and show that any standard equivalence between contraction algebras (up to algeb…
Tilting realizations of derived-equivalent matrix centralizer algebras
该研究解决了矩阵中心化子的固定源实现问题，明确与给定矩阵中心化子代数导出等价的代数结构，分类了基础倾斜模并确定其性质，还刻画了相关偏序集与突变图的结构。
We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan-Iyama-Reiten-Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tu…
We say that an algebra $Λ$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $Λ$-modules. We show that when $R$ is $d$-dimensional local Gorenstein the $d$-CY algebras are exactly the symmetric $R$-orders of global dimension $d$. We give a complete description of all tilting modules of projective dimension at most one for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We show that t…
Fans and polytopes in tilting theory II: $g$-fans of rank 2
本文基于Gluing、Rotation和Subdivision定理，对秩2的完整$g$-fans进行了分类，并刻画了对应代数，同时证明了2-term silting complexes的Hasse quiver连通分支数可取任意正整数。
Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and each indecomposable tilting vector bundle $T$ on $C$ a certain complex $S(C,T)$ of $G \times \mathbb{G}_m$-equivariant coherent sheaves on $\mathcal{N}$. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-$t$-structure. We then show that if $p$ is larger than the Coxeter number, then the hypercoho…
Relative tilting, interlacing categories and rational Dyck posets
该研究构造了有理Dyck楼梯关联代数与A型高阶Auslander代数幂等拐角的显式倾斜等价，证明了互素正整数的Chapoton-Ladkani-Rognerud猜想，拓展了区间倾斜机制并实现了楼梯导出范畴的几何实现。
Mutating Species with Potentials and Cluster Tilting Objects
本文研究了带势的物种与突变之间的关系，证明了在完美域上，带势的物种突变与三角2-Calabi-Yau范畴中的簇倾斜对象有密切联系，并探讨了Nakayama自同构与自内射性相关的性质。
Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences
本文研究局部Φ-Beilinson-Green代数，利用n-exangulated范畴中的对称逼近序列和倾斜复形构造导出等价，推广了Chen-Xi结果并得到新的导出等价代数类。
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…
Functor-induced isomorphisms and $G$-matrices
本文研究函子诱导同构的 $G$-矩阵编码，证明倾斜模与 $2$-项 silting 复形诱导的 Grothendieck 群同构可由其 $G$-矩阵实现，并给出 Coxeter 变换、Nakayama 函子及 Auslander-Reiten 平移的 $G$-矩阵解释。
When is the silting-discreteness inherited?
本文研究了silting-离散性的继承条件，证明幂等截断与同调满射可传递该性质，并分类了silting-离散的单连通张量代数及silting-非离散的自内射Nakayama代数。
$τ_d$-tilting theory for linear Nakayama algebras
本文在具有 $d$-丛倾斜模的截断线性 Nakayama 代数上，通过组合方法分类了 $τ_d$-刚性对和 $d$-挠类，并建立了它们与 $(d+1)$-项 silting 复形的联系，同时研究了变异行为。
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