# Ring Theory and Representation Theory Seminar in 2018

This seminar is organized by Osamu Iyama, Ryo Takahashi, Laurent Demonet, Erik Darpö, and Sota Asai now.
In this page, we announce information on seminars held from April 2018.
For previous seminars, see the following pages: 2017, 2015–16, 2014, 2013–.

## 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.

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