## Conference on Algebraic Representation Theory 2022 (CART 2022)

Date: November 18 (Fri) -- 20 (Sun), 2022 University of Tsukuba & Zoom (Hybrid) Institutes of Natural Sciences Building D, Room D509

• All participants (including speakers and organizers) are required to register by the following form.
Registration Form   The registration has been closed.

• We will send the Zoom link to the email address you write in the form above.
(Nov. 14th) Today, I sent the Zoom link to to the email address you write in the form above. If you registered, but haven't received the e-mail yet, please let me (sagaki *** math.tsukuba.ac.jp) know. (*** = @)

• Due to the relatively small size of the conference room in University of Tsukuba, we need to limit the number of in-person participants. Please do not come to the university without registration.
(Nov. 12th) The room will be filled up with registered participants (and hence those without registration can't enter the room).

• (Nov. 14th) Please check Extra Information

### Hotel information for those from abroad

We recommend, for example,

[Hotel Grand Shinonome]
https://www.hg-shinonome.co.jp/english/

[Daiwa Roynet Hotel Tsukuba]
https://www.daiwaroynet.jp/en/tsukuba/

## Program

UTC+9 UTC+8 Nov. 18
(Fri)
Nov. 19
(Sat)
Nov. 20
(Sun)
9:30 -- 10:308:30 -- 9:30 M.ChenZhang
10:30 -- 10:509:30 -- 9:50 Opening
10:50 -- 11:509:50 -- 10:50 KimShimizuLin
14:00 -- 15:0013:00 -- 14:00 TsaiArakawaOya
15:15 -- 16:1514:15 -- 15:15 AbeSongLee
16:30 -- 17:3015:30 -- 16:30 SpeyerC.-W.ChenScrimshaw

## Title and Abstract

Numbers in red indicate UTC+9.
Numbers in blue indicate UTC+8.

### November 18 (Fri)

#### 10:50 -- 11:50(9:50 -- 10:50)

Speaker: Myungho Kim (Kyung Hee University)

Title: Quiver Hecke algebras and localization of monoidal categories

Abstract: For each element $w$ in the Weyl group, there is a subcategory $C_w$ of the category of finite-dimensional modules over the quiver Hecke algebra, which categorifies the coordinate ring $C[N(w)]$ of the unipotent subgroup $N(w)$. We develop a localization process of a $k$-linear abelian monoidal category via a family of simple objects and apply it to the category $C_w$ and the family of simple modules corresponding to frozen variables of $C[N(w)]$. The Grothendieck ring of the localization $\tilde C_w$ of $C_w$ is isomorphic to the coordinate ring $C[N^w]$ of the unipotent cell. It turns out that the category $\tilde C_w$ is a rigid monoidal category;i.e., every object admits a left dual and a right dual. This is joint work with Masaki Kashiwara, Se-jin Oh, and Euiyong Park.

#### 14:00 -- 15:00(13:00 -- 14:00)

Title: Wave-front sets and graded Lie algebras

Abstract: For characters of p-adic reductive groups, there is the notion of wave-front set, which is a set of nilpotent orbits that describes the asymptotic or micro-local behavior of the character near the identity. There is a long-standing conjecture that any wave-front set is contained in a single geometric orbit, as worked out by many authors for several types of depth-0 representations. In this talk, we explain how the above conjecture cannot hold in general because an analogous assertion does not hold for graded Lie algebras.

#### 15:15 -- 16:15(14:15 -- 15:15)

Speaker: Noriyuki Abe (The University of Tokyo)

Title: A Hecke action on G_{1}T-modules

Abstract: A Hecke category means a (good) categorification of the Hecke algebra of a Coxeter system. We given an action of the Hecke category on the principal block of G_{1}T-modules, where G is a connected reductive group over an algebraically closed filed of positive characteristic, G_1 the kernel of Frobenius map and T a maximal torus. An ingredient is a new realization of the Hecke category.

#### 16:30 -- 17:30(15:30 -- 16:30)

Speaker: Liron Speyer (Okinawa Institute of Science and Technology)

Title: Schurian-infinite blocks of type $A$ Hecke algebras

Abstract: For any algebra A over an algebraically closed field F, we say that an A-module M is Schurian if End_A(M) is isomorphic to F. We say that A is Schurian-finite if there are only finitely many isomorphism classes of Schurian A-modules, and Schurian-infinite otherwise. I will present recent joint work with Susumu Ariki in which we determined that many blocks of type A Hecke algebras are Schurian-infinite, and a sequel with Sinéad Lyle that built on our techniques to complete the classification of Schurian-finiteness of blocks of type A Hecke algebras – when e ≠ 2, a block of the Hecke algebra is Schurian-finite if and only if it has finite representation type, if and only it has weight 0 or 1. I will give an overview of the techniques that were used in this project.

### November 19 (Sat)

#### 9:30 -- 10:30(8:30 -- 9:30)

Speaker: Miaofen Chen (East China Normal University)

#### 10:50 -- 11:50(9:50 -- 10:50)

Speaker: Kenichi Shimizu (Shibaura Institute of Technology)

Title: Nakayama functors for Frobenius tensor categories

Abstract: This talk is based on my joint work with Taiki Shibata. The Nakayama functor plays an important role in the representation theory of finite-dimensional algebras. Fuchs, Schaumann and Schweigert pointed out that the Nakayama functor has a certain universal property and defined the Nakayama functor for finite abelian categories by that property. As they also pointed out, such an abstract treatment of the Nakayama functor is useful in the study of finite tensor categories. In this talk, I will report recent progress for extending their approach to possibly infinite tensor categories. For a tensor category C, the Nakayama functor can be defined by the universal property as in the finite case. Generalizing an existence criterion for integrals on Hopf algebras, we show that the Nakayama functor of C is a non-zero functor if and only if C is Frobenius, that is, C has enough projective objects. This observation gives rise to some criteria for a tensor category to be Frobenius.

#### 14:00 -- 15:00(13:00 -- 14:00)

Speaker: Tomoyuki Arakawa (RIMS, Kyoto University)

Title: 4D/2D duality and representation theory

Abstract: This talk is about the 4D/2D duality discovered by Beem et al. rather recently. It associates a VOA to any 4-dimensional superconformal field theory, which is conjecturally a complete invariant of the 4-dimensional theory. The VOAs appearing in this manner may be regarded as chiralization of various symplectic singularities and are expected to have some interesting representation theory,

#### 15:15 -- 16:15(14:15 -- 15:15)

Speaker: Arim Song (Seoul National University)

Title: Supersymmetric W-algebra

Abstract: In 2-dimensional conformal field theory, we have an algebraic object called W-algebra. As an N=1 supersymmetric(SUSY) analogue of it, we have a SUSY W-algebra. In this talk, we mainly deal with SUSY W-algebra and its algebraic structures. In particular, we observe how SUSY and nonSUSY structures are related. After that, we introduce the way to find its free generators. This talk is based on the joint work with Ragoucy and Suh.

#### 16:30 -- 17:30(15:30 -- 16:30)

Speaker: Chih-Whi Chen (National Central University)

Title: Whittaker modules and categories for quasi-reductive Lie superalgebras.

Abstract: We introduce a construction of simple and standard Whittaker modules for quasi-reductive Lie superalgebras in terms of their parabolic decompositions. In this talk, we present a solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category O. We explain certain categories of Whittaker modules realized as cokernel categories that fit into the framework of properly stratified categories. This is joint work with Shun-Jen Cheng and Volodymyr Mazorchuk.

### November 20 (Sun)

#### 9:30 -- 10:30(8:30 -- 9:30)

Speaker: Xiaoting Zhang (Capital Normal University)

Title: Geometric classification of total stability conditions and Reineke's conjecture

Abstract: We construct a geometric model for the root category associated to any Dynkin diagram Q and classify the space of total stability conditions on the bounded derived category of Q. As an application, we prove Reineke's conjecture, that for any Dynkin quiver, there is a stability function on its module category such that any indecomposable is stable. This is based on joint works with Wen Chang and Yu Qiu.

#### 10:50 -- 11:50(9:50 -- 10:50)

Speaker: Huang Lin (Beijing Institute of Technology)

Title: On the center conjecture of some cyclotomic KLR algebras

Abstract: In this talk, we present a cocenter approach to the center conjecture of cyclotomic KLR algebras of general Cartan types. As an application, we obtain the center conjecture for cyclotomic KLR algebras at some special element of the positive root lattice by constructing monomial bases for their cocenters.

#### 14:00 -- 15:00(13:00 -- 14:00)

Speaker: Hironori Oya (Tokyo Institute of Technology)

Title: Wilson lines on the moduli space of decorated twisted $G$-local systems on a marked surface

Abstract: Let $G$ be a simply-connected complex simple algebraic group $G$, and $\Sigma$ a compact oriented surface with marked points on its boundary. Fock and Goncharov introduced a moduli space $\mathcal{A}_{G, \Sigma}$ of decorated twisted $G$-local systems on $\Sigma$ as an algebro-geometric avatar of higher Teichm\"uller spaces. In this talk, we introduce a class of $G$-valued morphisms defined on the generic part'' $\mathcal{A}_{G, \Sigma}^{\times}$ of $\mathcal{A}_{G, \Sigma}$, which we call Wilson lines. We also explain its application to the A=U problem for the cluster algebras arising from the cluster $K_2$-structures on $\mathcal{A}_{G, \Sigma}$.

#### 15:15 -- 16:15(14:15 -- 15:15)

Speaker: Sin-Myung Lee (Seoul National University)

Title: Affinization of q-oscillator representations of U_q(gl_n)

Abstract: We introduce a category of q-oscillator representations of the quantum affine algebra of untwisted affine type A. This category can be viewed as a quantum affine analogue of the semisimple tensor category generated by unitarizable highest weight representations of gl_{u+v} appearing in the (gl_{u+v}, gl_\ell)-duality on a bosonic Fock space. We construct a family of irreducible representations in this category, which naturally corresponds to finite-dimensional irreducible representations of the same quantum affine algebra, and explain this connection using q-oscillator representations of quantum affine superalgebras or Kuniba-Okado-Sergeev's generalized quantum group of type A. This is based on a joint work with Jae-Hoon Kwon (arXiv:2203.12862). We also present some progress for type C and type D, which is an ongoing project with J.-H. Kwon and Masato Okado.

#### 16:30 -- 17:30(15:30 -- 16:30)

Speaker: Travis Scrimshaw (Hokkaido University)

Title: Representation theory of the quantum Clifford algebra

Abstract: The quantum Clifford algebra was introduced by Hayashi in 1990 as a q-analog of the classical Clifford algebra, where he used it to construct certain representations of certain quantum groups. In this talk, we introduce a slight generalization based on another parameter k that we call the twist parameter and study it over various fields. In particular, we describe some bases (including giving an algorithm to compute multiplication), its center, and its simple modules when there are sufficiently many roots of unity in the ground field. In particular, we show it is a split semisimple algebra when there are sufficiently many roots of unity. Finally, we push our generalization slightly further to show that the quantum Clifford algebra is an actual deformation of the usual Clifford algebra. This is based on joint work with Willie Aboumrad.

### Organizing Committee:

Susumu Ariki (Osaka University)