Program and Abstracts

Titles and Abstracts: (pdf version 8, Oct. 19

10/21 (Monday)

13:00--14:00 M. Okado (Osaka City University)

Title: Quantum super duality

Abstract: In a paper with Kuniba and Sergeev in 2015, we introduced a certain generalization of quantum affine algebras and good modules through the study of solutions of the tetrahedron equation. For type A and a homogeneous case it corresponds to the usual quantum affine algebra of type A and its Kirillov-Reshetikhin module. Our recent studies show that they also have a q-analogue of the super duality by Cheng-Lam-Wang. Later part is a joint work with Jae-Hoon Kwon.

14:20--15:20 K. Naoi (Tokyo University of Agriculture and Technology)

Title: Existence of Kirillov-Reshetikhin crystals for the near adjoint nodes in exceptional types

Abstract: Kirillov-Reshetikhin (KR) modules are a distinguished family of finite-dimensional simple modules over a quantum affine algebra $U_q(\mathfrak{g})$, and it has been conjectured for a long time that KR modules have crystal (pseudo)bases. When a given KR module has a multiplicity free classical decomposition (in particular, all KR modules in nonexceptional types), this conjecture has been proved. On the other hand, in exceptional types there are many non-multiplicity free KR modules, and the conjecture is still open for most of them.

We say a node in the Dynkin diagram of $\mathfrak{g}$ is "near adjoint node" if the distance from "0" (of Kac labelling) is 2. In this talk, I will introduce the recent result in which we give a proof to the conjecture for the KR modules associated with near adjoint nodes of type $E_n^{(1)}$ ($n=6,7,8$), $F_4^{(1)}$ and $E_6^{(2)}$, which are all non-multiplicity free. The proof is accomplished by applying the criterion for the existence of a crystal pseudobase introduced by Kang, Kashiwara, Misra, Miwa, Nakashima and Nakayashiki. The criterion reduces the existence of a crystal pseudobase to some assertions on the values of a bilinear form, but It is quite hard to calculate the values directly. Instead, the assertions are shown by using the theory of the global basis of extremal weight modules. This talk is based on a joint work with Travis Scrimshaw.

15:40--16:40 A. Mathas (The University of Sydney)

Title: The center of the cyclotomic Hecke algebras of type A

Abstract: The cyclotomic Hecke algebras of type A are an interesting family of algebras that generalise the Iwahori-Hecke algebras of types A and B. It is a long-standing conjecture that the center of these algebras is the set of symmetric polynomials in their Jucys-Murphy elements. I will give a survey of these algebras culminating in the proof of this conjecture. This is joint work with Catharina Stroppel.

10/22 (Tuesday)

9:00--10:00 J. Brundan (University of Oregon)

Title: Heisenberg and Kac-Moody categorification

Abstract: I will talk about recent joint work with Alistair Savage and Ben Webster in which we establish a direct relationship between Heisenberg categorical actions and Kac-Moody categorical actions. This gives a rather direct way to prove the existence of an action of the Kac-Moody 2-category on several fundamental Abelian categories appearing in representation theory, including representations of symmetric groups, general linear groups, Ariki-Koike algebras, rational Cherednik algebras and cyclotomic q-Schur algebras.

10:20--11:20 H. Oya (Shibaura Institute of Technology)

Title: Cluster algebras and calculation of $q$-characters of simple modules over quantum loop algebras of non-symmetric type

Abstract: In the first part of this talk, I present a ring isomorphism between ``$t$-deformed'' Grothendieck rings of finite-dimensional module categories of quantum affine algebras of type $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. This isomorphism implies the validity of Kazhdan-Lusztig algorithm in the calculation of $q$-characters of simple modules of type $B_n^{(1)}$. Secondly, I explain a cluster theoretic origin of our isomorphism and benefit arising from this perspective. This talk is based on a joint work with David Hernandez.

13:00--14:00 T. Ikeda (Okayama University of Science)

Title: Quantum K-theory of the Grassmannians, duality and its applications

Abstract: The quantum K-theory of Grassmannians has bee studied for decades. In 2011, Buch and Mihalcea proved a Pieri formulas for it. In principle, the formula is strong enough to determine the K-theoretic 3-point Gromov-Witten invariants (KGW invariants for short) of genus zero for Grassmannians. The aim of the talk is to give an algorithm to compute the KGW invariants. I will first show how the the ring of K-theory of a Grassmannian is identified with a certain quotient of the ring of symmetric functions, so that the Schubert class is represented by the corresponding Grothendieck polynomial. This means we can calculate the KGW invariants starting from the classical structure constants for ordinary K-theory. We will give some ``reduction rule’’, typically given by a procedure of removing hooks from a Young diagram, which lead to the KGW invariants. One of the tools in our argument is the classical duality for Grassmannians. As an application, we prove a Giambelli formula expressing the general Schubert class as a determinant. The talk is based on a joint work with D. Hiep and T. Matsumura.

14:20--15:20 Y. Oshima (Osaka University)

Title: Unitary representations of real reductive groups and the method of coadjoint orbits

Abstract: The method of coadjoint orbits, introduced by Kirillov around 1960, relates irreducible unitary representations of a Lie group and its coadjoint orbits. For real reductive Lie groups, a correspondence between large subsets of these two is known to exist. I would like to talk about how characters, inductions and restrictions of unitary representations of a real reductive group are related to the corresponding geometric picture of coadjoint orbits.

15:40--16:40 S. Naito (Tokyo Institute of Technology)

Title: Description of the Chevalley formula for the torus-equivariant K-group of partial flag manifolds of (co-) minuscule type in terms of the parabolic quantum Bruhat graph

Abstract: TBA

10/23 (Wednesday)

9:00--10:00 M. Shimozono (Virginia Tech)

Title: Wreath Macdonald polynomials

Abstract: In 2003 Haiman defined wreath Macdonald polynomials in terms of plethystic opera- tions on tensor products of symmetric functions. His construction arose from Nakajima varieties for the cyclic quiver. We will discuss some recent developments, including wreath analogues of Mac- donald operators in tensor symmetric function operator form coming form the quantum toroidal algebra, and also of the famous nabla operator. This is based on ongoing joint work with Dan Orr and with Mark Haiman.

10:20--11:20 R. Fujita (Kyoto University)

Title: Graded quiver varieties and normalized $R$-matrices for fundamental modules

Abstract: The normalized $R$-matrix is an intertwining operator between tensor products of two finite-dimensional simple modules of the quantum affine algebra. It can be seen as a matrix-valued rational function in the spectral parameter, whose denominator determines when the tensor product module becomes reducible. In this talk, we give a uniform formula expressing the denominators of the normalized $R$-matrices for fundamental modules of type $ADE$ by using the geometry of graded quiver varieties. As a by-product, we give a geometric interpretation of Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor when it arises from a family of fundamental modules.

13:00--14:00 W. Wang (University of Virginia)

Title: Categorification of i-quantum groups via Hall algebras

Abstract: A quantum symmetric pair consists of a quantum group and its coideal subalgebra (called an i-quantum group). A quantum group can be viewed as an example of i-quantum groups associated to symmetric pairs of diagonal type, and various fundamental constructions for quantum groups (such as R-matrices and canonical bases) have been generalized to i-quantum groups. In this talk, we present a Hall algebra realization of (universal) i-quantum groups in the framework of ``modified Ringel-Hall algebras” developed by Lu-Peng. A central reduction of universal i-quantum groups recovers G. Letzter’s i-quantum groups with parameters. Our approach leads to PBW bases and braid group actions for i-quantum groups. For symmetric pairs of diagonal type, our work reduces to a reformulation of Bridgeland's Hall algebra realization of (Drinfeld double of) a quantum group, which is in turn based on earlier constructions of Ringel and Lusztig for half a quantum group. This is joint work with Ming LU (Sichuan, China).

14:20--15:20 T. Kuwabara (University of Tsukuba)

Title: Deformed TCDO over Grassmannian and simple affine VOA

Abstract: Sheaves of CDO (chiral differential operators) never exist globally over a homogeneous space $G/P$ where $G$ is a semisimple algebraic group $P$ is a parabolic subgroup but not Borel by the result of Gorbounov, Malikov and Schechtman. However, sheaves of deformed TCDO (twisted CDO), introduced by Chebotarov may (or may not) exist over $G/P$. In this talk, I will show explicit construction of sheaves of deformed TCDOs over Grassmannians. These deformed TCDOs are compatible with the generalized Wakimoto realization given by Frenkel, and they give localization of simple affine VOAs of type A at certain levels.

15:40--16:40 C. Stroppel (University of Bonn)

Title: Fusion rings and DAHA actions

Abstract: Tensor categories arising from tilting modules for quantum groups at roots of unities are a well-studied object in representation theory, but are also interesting from a topological point of view (they give for instance rise to 3-manifold invariants). In the talk I would like to revisite some of the constructions and then in particular study the Grothendieck/fusion rings of semisimple quotients. The main goal is to connect them via ideas from integrable systems with Cherednik’s double affine Hecke algebras (DAHAs).

10/24 (Thursday)

9:00--10:00 S. Kato (Kyoto University)

Title: Equivariant quantum $K$-groups of flag manifolds

Abstract: We first exhibit some structural results and our definitions of equivariant $K$-groups of semi-infinite partial flag manifolds. Then we explain its connection with equivariant quantum $K$-groups (or rather quantum $D$-modules) of partial flag manifolds, and equivariant $K$-groups of affine Grassmanians. These yield ways to understand the multiplication rules of equivariant quantum $K$-groups with respect to the Schubert bases.

10:20--11:20 T. Hikita (Kyoto University)

Title: Elliptic and $K$-theoretic canonical bases for hypertoric varieties

Abstract: Lusztig defined a notion of canonical basis in equivariant $K$-theory of Springer resolutions or Slodowy varieties and conjectured that they control the modular representation theory of Lie algebras, which was proved by Bezrukavnikov and Mirkovi\'c. In this talk, I will explain a reformulation and conjectural generalizations of the notion of this $K$-theoretic canonical basis for good conical symplectic resolutions equipped with their symplectic duals. Main tool here is the $K$-theoretic analogue of the notion of stable basis introduced by Maulik and Okounkov. Also, if time permit, I will show an explicit example of the elliptic analogue of these constructions for hypertoric varieties.

13:00--14:00 E. Park (University of Seoul)

Title: Cluster algebra structures on module categories over quantum affine algebras

Abstract: We study monoidal categorifications of certain monoidal subcategories $C_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type $A_\infty$ via the generalized quantum Schur-Weyl duality. When the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $C_\g^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras. This is joint work with M. Kashiwara, M. Kim and S.-j. Oh (arXiv:1904.01264)

14:20--15:20 S. Ariki (Osaka University)

Title: Modular Representation Theory of Hecke algebras

Abstract: When I was young, I asked standard questions on Hecke algebras and their block algebras such as classification of irreducible modules, representation type, decomposition numbers etc. Then, I found that those algebras were appropriate objects for testing usefulness of various developments in representation theory. Namely, each question which I could answer required broad ideas from other fields. We expect that this feature is true in current and future research on Hecke algebras. In this talk, I explain an ongoing project with Ryoichi Kase, Kengo Miyamoto and Qi Wang, whose aim is to determine tau-tilting finiteness/infiniteness of block algebras of Hecke algebras.

10/25 (Friday)

9:00--10:00 T. Shoji (Tongji University)

Title: Diagram automorphism and canonical basis of quantum affine algebras

Abstract: Let $\mathfrak g$ be the Kac-Moody Lie algebra of symmetric type and $\ul \mathfrak g$ the fixed point subalgebra of $\mathfrak g$ by the diagram automorphism $\sigma : \mathfrak g \to \mathfrak g$. Let $U^-$ be the negative part of the quantum enveloping algebra associated to $\mathfrak g$, and $\ul U^-$ the corresponding algebra for $\ul \mathfrak g$. Let $B$ be the canonical basis of $U^-$, and $\ul B$ the canonical basis of $\ul U^-$. Lusztig proved, by using his geometric construction of canonical basis, that there exists a canonical bijection between $B^{\sigma}$ and $\ul B$, where $B^{\sigma}$ is the set of $\sigma$-fixed elements in $B$. In this talk, we give an elementary proof of this result in the case of quantum affine algebras, by using Beck-Nakajima's PBW-basis. Here by the elementry proof we mean that we don't appeal to Lusztig's geometric theory of canonical basis nor Kashiwara's theory of crystal basis. We also discuss the correspondence in PBW-bases.

10:20--11:20 M. Geck (University of Stuttgart)

Title: Computing Green functions of finite groups of Lie type

Abstract: Green functions for finite groups of Lie type were introduced by Deligne and Lusztig in the 1970's, using cohomological methods. The computation of these functions is a crucial step in the more general programme of determining the whole character tables of those groups. We report on some recent progress, which essentially relies on a combination of Lusztig's theory of character sheaves and computer algebra methods.