Ring Theory and Representation Theory Seminar in 2019

This seminar is coorganized by Osamu Iyama, Ryo Takahashi and Erik Darpö.
Previous seminars: 2018, 2017, 2015-16, 2014, 2013--.


20 December (Friday) 16:30-18:00
Room: 409 math.
Speaker: Kenta Ueyama (Hirosaki University)
Title: Combinatorial study of stable categories of graded CM modules over skew quadrics

Abstract: In this talk, we study the stable category of graded maximal Cohen-Macaulay modules over S/(f) where S is a (plus/minus 1)-skew polynomial algebra generated in degree 1, and f is the Fermat quadric in S. Our method is to use a certain graph associated with S. We discuss four graphical operations called mutation, relative mutation, Knörrer reduction, and two points reduction, and show that the above stable category can be completely computed by using these graphical operations. We also explore the consequences of this result.
This talk is based on joint work with Izuru Mori, and on joint work with Akihiro Higashitani.
6 December (Friday) 14:45-16:15
Room: 409 math.
Speaker: Ryo Kanda (Osaka University)
Title: Feigin-Odesskii's elliptic algebras

Abstract: This talk is based on ongoing joint work with Alex Chirvasitu and S. Paul Smith. Feigin and Odesskii introduced a family of noncommutative graded algebras, which are parametrized by an elliptic curve and some other data, and claimed a number of remarkable results in their series of papers. The family contains all higher dimensional Sklyanin algebras, which have been widely studied and recognized as important examples of Artin-Schelter regular algebras. In this talk, I will explain some properties of Feigin-Odesskii's algebras, including the nature of their point schemes and algebraic properties obtained by using the quantum Yang-Baxter equation.
6 December (Friday) 16:30-18:00
Room: 409 math.
Speaker: Ryo Takahashi (Nagoya University)
Title: Cones spanned by maximal Cohen-Macaulay modules of rank one

Abstract: Let A be a Cohen-Macaulay normal local ring. Let H(A) be the quotient of the Grothendieck group of finitely generated A-modules by modules with codimension at least two. Let D(A) be the convex cone in H(A)_R, the tensor product of H(A) with the real number field R, spanned by the maximal Cohen-Macaulay modules of rank one. We give various equivalent conditions for A to have only finitely many maximal Cohen-Macaulay modules of rank one in terms of topological properties of D(A). We then consider maximal Cohen-Macaulay modules of rank one as elements of the divisor class group Cl(A).
7 November (Thursday) 16:30-18:00
Room: 455 math.
Speaker: Martin Herschend (Uppsala University)
Title: Wide subcategories of d-Cluster tilting subcategories for higher Auslander algebras of type A

Abstract: In joint work with Peter Jorgensen and Laertis Vaso we introduced the notion of a wide subcategory of a d-abelian category, as one being closed under direct sums, d-kernels, d-cokernels and d-extensions. We also provided tools for classifying such wide subcategories for the most well-known kind of d-abelian category: a d-cluster tilting subcategory of the category of finitely generated modules over some finite dimensional algebra.

In my talk I will deal with higher Auslander algebras of type A. This is a family of algebras introduced by Osamu Iyama, each admitting a unique d-cluster tilting subcategory. Moreover, objects, morphisms and d-extensions of these d-cluster tilting subcategories have been described combinatorially by Steffen Oppermann and Hugh Thomas. In my talk I will present a classification of wide subcategories for these algebras. The classification is obtained in joint work with Peter Jorgensen.
26 July (Friday) 16:30-18:00
Room: 409 math.
Speaker: Cheol-Hyun Cho (Seoul National University)
Title: AR quivers of ADE curve singularities and Lagrangian Floer theory

Abstract: Given ADE singularity of two variables. we investigate equivariant Lagrangian Floer theory of the Milnor fiber of dual singularity. Recall that indecomposable Cohen-Macaulay modules of ADE singularity form an AR quiver, which was computed by Yoshino in mid 80's. We explain a geometric way to obtain these matrix factorizations in the AR quiver via counting suitable polygons.
5 July (Friday) 16:30-18:00
Room: 409 math.
Speaker: Haruhisa Enomoto (Nagoya University)
Title: The Jordan-Holder property, Grothendieck monoids and Bruhat inversions

Abstract: In this talk, I will talk about the Jordan-Holder property (JHP) for exact categories, which is a natural generalization of the uniqueness property of decompositions of modules into simples. First, I introduce a new invariant of exact categories, the Grothendieck monoids, show that (JHP) is equivalent to the free-ness of this monoid, and give a convenient numerical criterion for this.
Second, I will apply them to representation theory of artin algebras. Under a mild assumption, (JHP) is equivalent to that the number of projectives is equal to that of simples. For torsion-free classes of type A quiver, simple objects are described in terms of the combinatorics of the symmetric group: Bruhat inversions of c-sortable elements. Thus we can check the validity of (JHP) in a purely combinatorial way in this case.
31 May (Friday) 15:30-17:00
Room: 552 math.
Speaker: Yasuaki Gyoda (Nagoya University)
Title: F-matrices in cluster algebras

Abstract: Cluster algebras are a class of commutative algebras introduced by Fomin and Zelevinsky, which are generated by some distinguished elements called the cluster variables. The cluster variables are given by applying the mutations repeatedly starting from the initial cluster variables. Thanks to the separation formulas, the cluster variables and the coefficients are described by the C-matrices, the G-matrices, and the F-polynomials. These matrices and polynomials are studied to know the properties of the cluster variables or mutations. In particular, I consider the “degree matrices” of F-polynomials, (F-matrices). In this talk, I will introduce the F-matrices and these properties.
17 May (Friday) 16:30-
Room: 409 math.
Speaker: Yasuyoshi Yonezawa (Nagoya University)
Title: 変形Webster代数のある構造について

Abstract: 変形Webster代数に入るある構造について講演する。
11 March (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.
6 March (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。
1 March (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.
1 March (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.
1 March (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 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].