# Ring Theory and Representation Theory Seminar in 2018

In this page, we announce information on Ring Theory and Representation Theory Seminar held from April 2018 to March 2019. During this period, this seminar is organized by Osamu Iyama, Ryo Takahashi, Laurent Demonet, Erik Darpö, and Sota Asai, and also by Toshiya Yurikusa from February 2019.

Following seminars: 2019.
Previous seminars: 2017, 2015–16, 2014, 2013–.

## 11th March, 2019

Date: 2019/03/11 (Monday), 13:00–14:30
Room: 309 math.
Speaker: Atsushi Takahashi (Osaka University)
Title: Maximally graded matrix factorizations for an invertible polynomial of chain type
Abstract: In 1977, Orlik-Randell proposed a conjecture on the existence of certain distinguished basis of vanishing cycles in the Milnor fiber associated to an invertible polynomial of chain type. Under the homological mirror symmetry, it is expected from their conjecture that the triangulated category of maximally-graded matrix factorizations for the Berglund--H\"{u}bsch transpose admits a full exceptional collection with a nice numerical property. The purpose of this talk is to prove this algebraic analogue of Orlik-Randell conjecture.

## 6th March, 2019

Date: 2019/03/06 (Wednesday), 13:00–14:30
Room: 409 math.
Speaker: Shunsuke Kano (Tokyo Institute of Technology)
Title: 曲面の擬Anosov写像類と圏論的エントロピー
Abstract: 曲面の写像類のうち、最も一般的な種類として擬Anosov写像類と呼ばれるものがある。擬Anosov写像類の不変量として位相的エントロピーが有名である。Dimitrov--Haiden--Katzarkov--Kontsevich は位相的エントロピーの圏化として、三角圏の自己関手の圏論的エントロピーを定義した。また、曲面の三角形分割 T から三角圏 D(T) を構成する方法が知られている。今回は擬Anosov写像類 f を D(T) 上の同値関手に持ち上げ、その圏論的エントロピーが元の f の位相的エントロピーと一致することを解説する。この同値関手の構成は、三角圏 D(T) の安定性条件への作用や、クラスター多様体の境界への作用と密接に関連するため、これらについても時間が許せば話したい。

## 1st March, 2019

Date: 2019/03/01 (Friday), 10:30–12:00
Room: 552 math.
Speaker: Sota Asai (Nagoya University)
Title: The wall-chamber structures of the real-valued Grothendieck groups
Abstract: We consider a finite-dimensional algebra $A$ over a field and the real-valued Grothendieck group of the category of finite-dimensional projective $A$-modules. The real-valued Grothendieck group can be identified with a Euclidean space, and Br\"{u}stle--Smith--Treffinger defined a wall-chamber structure of the real-valued Grothendieck group via the semistability conditions by King. In this talk, I will introduce my new combinatorial algorithm to obtain the wall-chamber structure in the case $A$ is a path algebra. I will also explain my result that the chambers of the wall-chamber structure bijectively correspond to the 2-term silting objects of the perfect derived category, by using the numerical torsion(-free) classes defined by Baumann--Kamnitzer--Tingley.

Date: 2019/03/01 (Friday), 13:30–15:00
Room: 552 math.
Speaker: Hiroki Matsui (Nagoya University)
Title: On the equivariant smash nilpotence theorem
Abstract: One of the important approaches to understand the structure of a given triangulated category is to classify its thick subcategories. Originally, such an approach is considered by Devinatz, Hopkins, and Smith in their studies of stable homotopy theory. After that Hopkins, Neeman, and Thomason gave a corresponding classification result of the thick tensor ideals of the perfect derived category of a noetherian scheme. On the other hand, Benson, Carlson, Iyengar et. al. classified the thick tensor ideals of the bounded derived category of finitely generated representations of a finite group. In this talk, we consider schemes X admitting an action of a finite group G and assume |G| acts as a unit on X. Under this assumption, we classify thick tensor ideals of the category of equivariant perfect complexes D_{perf}(X)^G. If we take G to be the trivial group, then this result recovers Thomason's result and if we consider X to be the spectrum of a commutative ring R, then we obtain the classification of thick tensor ideals of the perfect derived category of a skew group algebra R*G. The key point to prove the theorem is to establish the equivariant version of the smash nilpotence theorem.

Date: 2019/03/01 (Friday), 15:30–17:00
Room: 552 math.
Speaker: Laurent Demonet (Nagoya University)
Title: Torsion class over Brauer graph algebras and gentle algebras
Abstract: [jt. work in progress with A. Chan] Applying techniques introduced in a joint Applying techniques introduced in a joint work with Iyama, Reading, Reiten and Thomas [DIRRT], we classify torsion classes over Brauer graph algebras and finite dimensional gentle algebras. More specifically, we give a combinatorial-geometric realization of the complete lattice of torsion classes. After giving several examples for concrete algebras, we explain the strategy of the proof, recalling some important results of [DIRRT].

## 6th February, 2019

*** The room has been changed ***

Date: 2019/02/06 (Wednesday), 13:00–14:30
Room: 453 math.
Speaker: Yu Liu (Southwest Jiaotong University)
Title: Abelian categories arising from cluster tilting subcategories
Abstract: For a triangulated category (or an exact category) T , if C is a cluster-tilting subcategory of T , then the quotient category T/[C] is an abelian category. Under certain conditions, the converse also holds. This is an very important result of cluster-tilting theory, due to Koenig-Zhu, Demonet-Liu and Beligiannis. Now let B be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory C of B, which is a generalization of cluster tilting subcategory. We show that C is cluster tilting if and only if B/[C] is abelian. We also consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and full, in another word, the ideal quotient becomes abelian. Moreover, a new equivalent characterization of cluster tilting subcategories is given by applying homological methods according to this functor. As an application, we show that in a connected 2-Calabi- Yau triangulated category B, a functorially finite, extension closed subcategory D of B is cluster tilting if and only if B/[D] is an abelian category.

## 16th January, 2019

*** Yu Zhou's talk, which was scheduled for 13:00–14:30, has been cancelled ***

Date: 2019/01/16 (Wednesday), 14:45–16:15
Room: 409 math.
Speaker: Yu Qiu (Tsinghua University)
Title: Cluster exchange groupoids and quadratic differentials
Abstract: We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case of the decorated marked surface S, the universal cover of this groupoid can be constructed using decorated triangulations of S. Such a covering graph is a skeleton for a space of suitably framed quadratic differentials on S, which in turn models the space Stab(S) of Bridgeland stability conditions for the 3-Calabi-Yau category associated to S. Finally, we show that Stab(S) is simply connected.

## 11th January, 2019

*** The schedule has been changed ***

Date: 2019/01/11 (Friday), 16:30–18:00
Room: 409 math.
Speaker: Haibo Jin (Nagoya University)
Title: Cohen-Macaulay differential graded modules and Negative Calabi-Yau configurations
Abstract: In this talk, we introduce the class of Cohen-Macaulay (=CM) dg (=differential graded) modules over Gorenstein dg algebras and study their basic properties. We show that the category of CM dg modules forms a Frobenius extriangulated category, in the sense of Nakaoka and Palu, and it admits almost split extensions. We also study representation-finite d-self-injective dg algebras A in detail. In particular, we classify the Auslander-Reiten (=AR) quivers of CM A for those A in terms of (−d − 1)-Calabi-Yau (=CY) configurations, which are Riedtmann’s configuration for the case d = 0. In type A, for any given (−d−1)-CY configuration C, using a bijection between (−d−1)-CY configurations and certain purely combinatorial objects which we call maximal d-Brauer relations given by Coelho Simoes, we construct a Brauer tree dg algebra A such that the AR quiver of CMA is given by C.

## 12th December, 2018

Date: 2018/12/12 (Wednesday), 13:00–14:30
Room: 409 math.
Speaker: Toshitaka Aoki (Nagoya University)
Title: Classifying torsion classes and $\tau$-tilting modules for algebras with radical square zero
Abstract: In representation theory of finite dimensional algebras, the importance of torsion classes is increasing due to the recent development of $\tau$-tilting theory and the study of t-structures on the derived category. $\tau$-tilting theory introduced by Adachi-Iyama-Reiten provides an effective tool to study a particular class of torsion classes called functorially finite. In contrast, it is very hard to classify non functorially finite torsion classes in general. In this talk, we introduce an approach, which we call the sign-decomposition, of the set of torsion classes to the classification problem of all torsion classes. We apply the sign-decomposition to algebras with radical square zero and give an explicit description of each subset appearing in this decomposition. We also consider to restrict our results to support $\tau$-tilting modules.

## 5th December, 2018

Date: 2018/12/05 (Wednesday), 13:00–14:30
Room: 409 math.
Speaker: Steffen Oppermann (Norges teknisk-naturvitenskapelige universitet)
Title: 0-cocompactness
Abstract: This talk is based on ongoing joint work with Chrysostomos Psaroudakis (Thessaloniki) and Torkil Stai (Trondheim). Compactly generated triangulated categories are a very nice class of triangulated categories in that they satisfy Brown representability. I will start my talk by recalling what this means, and what the main idea is for the proof. The starting idea of the project with Psaroudakis and Stai is to study the dual of compact generation. However, as we will see during my talk, few categories which occur naturally have any non-zero cocompact objects. Therefore we introduce a weaker version, which we call 0-cocompact, and investigate which properties of compact objects generalize to this weak dual.

## 28th November, 2018

Date: 2018/11/28 (Wednesday), 13:00–14:30
Room: 409 math.
Speaker: William Wong (Nagoya University)
Title: Okuyama’s tilting complexes and perverse equivalences
Abstract: In 1999 Okuyama generalised a construction of Rouquier and write down a complex of a symmetric algebra A, using only the stably equivalent image of simple modules of another symmetric algebra B. Given some conditions on these ‘simple image’ we have a tilting complex of A. With some further condition, it manages to create an algebra such that its simple modules are ‘some from A and some from B.’ In this talk, we introduce this construction and discuss when this construction gives a perverse equivalence, or a composition of perverse equivalences. In particular we will see how this applies to the group algebra SL(2,q) in defining characteristics.

## 4th June, 2018

Date: 2018/06/04 (Monday), 16:30–18:00
Room: 509 math.
Speaker: Hugh Thomas (Université du Québec à Montréal)
Title: Nilpotent endomorphisms of quiver representations and the Robinson-Schensted-Knuth correspondence
Abstract: The Robinson-Schensted-Knuth correspondence is a many-faceted jewel of algebraic combinatorics. In one variation, it provides a bijection between permutations of n and pairs of standard Young tableaux with the same shape, which is a partition of n. I will discuss an interpretation of (a more general form of) RSK in terms of the representation theory of type A quivers, which then also allows us to extend the correspondence to other Dynkin types. Let Q be a Dynkin quiver, and i a minuscule vertex. Let C_i consist of the additive category of sums of indecomposable representations all having the vertex i in their support. For X in C_i, we consider the Jordan forms of the linear maps induced from a generic nilpotent endomorphism of X, and show that this defines a bijection from isomorphism classes of objects in C_i to order-preserving maps from the corresponding minuscule poset into the non-negative integers. We study the action of reflections functors on the representations, and show that they correspond to certain piecewise-linear transformations on the order-preserving maps. This allows us to give a uniform proof of a periodicity result for a certain piece-wise linear transformation by relating it to the Auslander-Reiten translation. This is joint work with Al Garver and Rebecca Patrias.

## 24th April, 2018

Date: 2018/04/24 (Tuesday), 10:30–12:00
Room: 552 math.
Speaker: Tsutomu Nakamura (Okayama University)
Title: Cosupport and minimal pure-injective resolutions
Abstract: The notion of cosupport introduced by Benson, Iyengar and Krause can be regarded as a tool to understand completeness of commutative noetherian rings. It also has significant information of minimal pure-injective resolutions. However, computing of cosupports of rings is very difficult in general. In this talk, we prove that every affine ring has full-cosupport. Namely, the cosupport of an affine ring is equal to the spectrum of the ring. Using this fact, we give a partial answer to a conjecture by Gruson.

## 18th April, 2018

Date: 2018/04/18 (Wednesday), 13:00–14:30
Room: 452 math.
Speaker: Jongmyeong Kim (Nagoya University)
Title: A freeness criterion for spherical twists
Abstract: Spherical twists along spherical objects are autoequivalences of a triangulated category defined by Seidel and Thomas as a categorical analogue of Dehn twists along simple closed curves. Spherical twists share many properties with Dehn twists. On the other hand, there is a classical result by Humphries which states that if a collection of simple closed curves admits a "complete partition" and does not bound a disk then the group generated by the Dehn twists along them is isomorphic to the free product of free abelian groups. In this talk, we give a categorical analogue of Humphries' result.

Back to Homepage