| Date: | November 18 (Fri) -- 20 (Sun), 2022 |
|---|---|
| Venue: | University of Tsukuba & Zoom (Hybrid) |
| Room: | Institutes of Natural Sciences Building D, Room D509 |
[Hotel Grand Shinonome]
https://www.hg-shinonome.co.jp/english/
[Daiwa Roynet Hotel Tsukuba]
https://www.daiwaroynet.jp/en/tsukuba/
| UTC+9 | UTC+8 | Nov. 18 (Fri) |
Nov. 19 (Sat) |
Nov. 20 (Sun) |
|---|---|---|---|---|
| 9:30 -- 10:30 | 8:30 -- 9:30 | M.Chen | Zhang | |
| 10:30 -- 10:50 | 9:30 -- 9:50 | Opening | ||
| 10:50 -- 11:50 | 9:50 -- 10:50 | Kim | Shimizu | Lin |
| 14:00 -- 15:00 | 13:00 -- 14:00 | Tsai | Arakawa | Oya |
| 15:15 -- 16:15 | 14:15 -- 15:15 | Abe | Song | Lee |
| 16:30 -- 17:30 | 15:30 -- 16:30 | Speyer | C.-W.Chen | Scrimshaw |
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.
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.
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.
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.
Title: Newton stratification and weakly admissible locus in p-adic Hodge theory
Abstract: The p-adic period domain (also called the admissible locus) is the image of the p-adic period mapping inside the rigid analytic p-adic flag varieties. The weakly admissible locus is an approximation of the admissible locus in the sense that these two spaces have the same classical points. On the flag variety, we have the Newton stratification which has p-adic period domain as its unique open stratum. In this talk, we consider the condition that the weakly admissible locus is maximal (i.e. the weakly admissible locus is a union of Newton strata). This unifies the extreme cases when the weakly admissible locus equals to the admissible locus or the whole flag variety. We will give several equivalent criterions for the condition that the weakly admissible locus is maximal. Moreover, we give a criterion when a single Newton stratum is contained in the weakly admissible locus. This is a joint work with Jilong Tong.
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.
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,
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.
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.
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.
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.
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 $\mathcal{A}_{G, \Sigma}$.
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.
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.