Tatsuyuki Hikita (RIMS)
TBA
Mao Hoshino (The University of Tokyo)
Title: Semisimple module categories with fusion rules of the compact full flag manifold type
Abstract: In the operator algebraic context, semisimple module categories over the representation category of quantum groups are natural subjects to study since there is a duality theorem between them and quantum group actions on operator algebras. Especially, if such a module category has a specific fusion rule, it would be regarded as a noncommutative analog of a compact full flag manifold. It would also provide a noncommutative analog of a semisimple coadjoint orbit if we consider the algebraic setting.
In this talk, I will explain a construction of such categories, based on a certain generalization of the BGG category O. I will also explain a classification result in the type A case, with Poisson geometric background.
Penghui Li (Tsinghua University)
Title: Hecke categories and TQFT
Abstract: The 2d TQFT associated to a finite group can be explicitly understood in term of representation theory of the group. We shall investigate the analoguous result for Hecke category, and give a conjectural formula for value of TQFT on surfaces. This talk is based on joint work with Quoc P. Ho.
Zhe Sun (University of Science and Technology of China)
Title:Webs and their intersections
Abstract: G-Webs as certain trivalent graphs on the surfaces, appear naturally in the G-skein algebra and its classical limit the regular function ring of the G character variety. I will explain how we introduce their intersection number to parameterize a basis of the algebra for SL3 (joint work with Daniel Douglas, Linhui Shen, Daping Weng) and Sp4 (joint work with Tsukasa Ishibashi, Wataru Yuasa).
Jinwei Yang (Shanghai Jiao Tong University)
Title: Representation and tensor category of affine sl_2 Lie algebra at positive rational levels
Abstract: In a series of work, D. Kazhdan and G. Lusztig constructed rigid braided tensor category structures over the categories of finite length modules for the affine Lie algebras at non-positive-rational levels and showed that they are braided tensor equivalent to the categories of finite dimensional weight modules for the quantum groups.
We use the tensor category theory of vertex operator algebras developed by Huang-Lepowsky-Zhang to study the tensor category structure over representation categories of affine sl_2 Lie algebras at positive rational levels. We consider the category of ordinary modules and the category of weight modules for the simple affine vertex operator algebra, and the category of ordinary modules for the universal affine vertex algebra. We show the first two categories have a rigid braided tensor category structure and are equivalent to certain module categories for the quantum group over sl_2 at the root of unity. The third is not rigid, but the subcategory of rigid objects is exactly the category of projective objects, and consequently it is derived equivalent to the category of finite dimensional weight modules for the quantum group over sl_2. These work are based on the joint work with T. Creutzig, Y.-Z. Huang and R. McRae.
Eunjeong Lee (Chungbuk National University)
Title: On Białynicki-Birula decompositions of regular semisimple Hessenberg varieties
Abstract: Hessenberg varieties are subvarieties of the flag variety that provide a rich connection between geometry, the representation theory of finite groups, and combinatorics. In particular, the symmetric group acts on the cohomology of a regular semisimple Hessenberg variety, and studying this representation is closely related to the Stanley--Stembridge conjecture on chromatic symmetric functions. Recently, Hikita proved the non-graded version of this conjecture, while the graded version remains open.
In this talk, we study a basis of the equivariant cohomology of a regular semisimple Hessenberg variety obtained via the Białynicki-Birula decomposition. The maximal torus acts on each cell, and by analyzing this torus action, we present a combinatorial description of the support of each basis element. Moreover, we discuss how this analysis provides a geometric construction of the permutation module decomposition of the equivariant cohomology of permutohedral varieties and the second cohomology of a regular semisimple Hessenberg variety.
This talk is based on joint work with Soojin Cho and Jaehyun Hong.
Uhi Rinn Suh (Seoul University of Science)
Title: SUSY extension of W-algebras
Abstract: Supersymmetry in the context of vertex algebras refers to a special endomorphism that facilitates their interpretation via superfields—structured as pairs of even and odd elements. A vertex algebra equipped with a supersymmetry is called a SUSY vertex algebra, and a SUSY vertex algebra containing a given vertex algebra is called a SUSY extension of that vertex algebra. Recently, it is proved that any W-algebra admits a SUSY extension, known as a SUSY W-algebra. Notably, this SUSY W-algebra can be constructed by adjoining a specific number of free fields to the original W-algebra. In this talk, I will present the embedding of a W-algebra into the corresponding SUSY W-algebra through their free field realizations.
Taiki Shibata (Okayama University of Science)
Title: Irreducible representations of algebraic superroups and their parameter sets
Abstract: The representation theory of split reductive algebraic groups can, in principle, be described in terms of the associated root data and Weyl groups, and research on the classification of irreducible representations and character theory has continued actively up to the present day. On the other hand, as Deligne pointed out, algebraic supergroups play an essential role in the theory of symmetric tensor categories, but the study of their internal structure and representation theory is still in its early stages and far from being as well understood as in the non-super case. For example, the fundamental question of describing irreducible representations in terms of the associated root system has been solved only for certain specific supergroups. One major difficulty in developing a unified theory of representations of supergroups lies in the peculiar behavior of roots and Borel subgroups in the super setting, and in the lack of a sufficiently well-developed framework to handle them uniformly.
In this talk, I will start from the definition of algebraic supergroups and look at several concrete examples together with their root systems. I will then introduce a method of constructing irreducible representations using induced representations (the Borel–Weil approach), and explain, through examples, what is currently known about irreducible representations and what difficulties remain. After that, I will explain how induced representations behave when the Borel subgroup is replaced by what is called an *odd reflection*, an operation introduced by Serganova and others as a kind of complement to Weyl groups. By observing this behavior, we can see that the parameter sets of irreducible representations can be determined, at least partially.
Hideya Watanabe (Rikkyo University)
Title: Symplectic tableaux, Berele's row insertion, and quantum symmetric pairs
Abstract: Symplectic tableaux, introduced by King, are combinatorial objects that can count the weight multiplicities of all finite-dimensional irreducible modules of symplectic groups. Berele modified the Schensted row-insertion in order to explain the irreducible decompositions of the tensor product of irreducible modules and the vector representation in terms of symplectic tableaux. In this talk, I will report a concrete connection between these combinatorics and representation theory of quantum symmetric pairs.
Masao Oi (National Taiwan University)
Title: Positive-depth Deligne-Lusztig induction for p-adic reductive groups
Abstract: In this talk, I would like to discuss a comparison of two kinds of representations of p-adic reductive groups arising from different origins. One is Yu's algebraic construction, further developed by Kim-Yu and Kaletha. The other is a geometric construction recently established by Chan-Ivanov, which generalizes the classical Deligne-Lusztig construction for finite groups of Lie type.
The strategy is to compare the trace characters of those two representations. A central idea for this is to introduce an analogue of the classical Green function (in the context of Deligne-Lusztig theory) for both the algebraic and geometric representations, which enables us to get a character formula completely parallel to the classical Deligne-Lusztig character formula.
This talk is based on my joint work with Charlotte Chan (University of Michigan).
Soo-Hong Lee (Seoul National University)
Title: Infinite-level Fock space, crystal bases, and tensor products of extremal weight modules of type A+∞
Abstract: We introduce an infinite-level Fock space, defined as a limit of higher-level Fock spaces. On this space, we construct commuting actions of the quantum group of type A+∞ and the parabolic q-boson algebra of type (gl∞,gl∞∖0), extending the classical Howe duality on higher-level Fock spaces. The study of this action revisits a subtle phenomenon in which the crystal base of a proper submodule coincides with that of the entire module — a behavior previously observed in level-zero representations of affine type by Kashiwara, Beck, and Nakajima. In type A+∞, we show that this combinatorial phenomenon is closely tied to the socle filtration of ambient representations. Building on this connection, we construct saturated crystal bases on the infinite-level Fock space, and provide an explicit description of the socle filtration of tensor products of extremal weight modules of type A+∞. This is joint work with Jae-Hoon Kwon.
Cailan Li (Academia Sinica)
Title: A categorification of Grothendieck differential operators on the affine line
Abstract: Given a Hopf pairing between two bialgebras K and G, one can define the Hesienberg double h(G), which recovers the classical Heisenberg algebra in the case K=G=Sym, the ring of symmetric functions in infinitely many variables. A classical result of Geissinger identifies Sym with the grothendieck ring K_0 on the tower of group algebras of the symmetric group \oplus_n C[S_n]. Replacing this tower of symmetric group algebras by a tower T of (finite-dimensional) algebras satisfying certain axioms, Savage and Yacobi constructed a categorification of h(K_0(T)) acting on its Fock space representation. In this talk we will extend this framework to towers of infinite-dimensional algebras and as an application, categorify the action of Grothendieck differential operators acting on A^1_{Z[v,v^{-1}]}.
This talk is based on joint work with Chun-Ju Lai.
Ngau Lam (National Cheng Kung University)
Title: The Gaudin model for unitarizable modules over the general linear Lie superalgebra
Abstract: Gaudin algebra $\cA^\z$ is a commutative subalgebra in the $\ell$-fold tensor product of the universal enveloping (super)algebra of a Lie (super)algebra $\fg$, depending on the $\ell$ distinct parameters in the complex field. Each Gaudin algebra naturally acts on any $\ell$-fold tensor product of $\fg$-modules. Let $\fg$ be the general linear Lie superalgebra and let $\un L$ be any $\ell$-fold tensor product of (infinite-dimensional) unitarizable highest weight $\fg$-modules. For each singular weight $\mu$ of $\un L$, we show that the singular space $\un L^\sing_\mu$ of weight $\mu$ is a cyclic $\cA^\z$-module, and that the image of the Gaudin algebra $\cA^\z$ in ${\rm End}_\C(\un L^\sing_\mu)$ is a Frobenius algebra. We also show that $\cA^\z$ is diagonalizable with a simple spectrum on $\un L^\sing_\mu$ for a generic $\z$. This is joint work with Bintao Cao and Wan-Keng Cheong.