On the moduli space of G-local systems with some boundary data called the framings and pinnings on a marked surface, we introduce a new class of G-valued functions associated with the arcs between boundary intervals which we call the ``Wilson line" functions. These functions are regular G-valued functions, and their matrix coefficients in a finite-dimensional representation of G give rise to Laurent polynomials in each cluster Poisson chart. We show that, for a suitable choice of matrix coefficients, these Laurent polynomials have positive integral coefficients in the cluster Poisson chart associated with any decorated triangulation. This talk is based on a joint work with Hironori Oya.

A positive cluster complex is a simplicial complex whose vertex set is the cluster variables other than the initial variables and whose simplicial set consists of all subsets of clusters without the initial variables. In this talk, I will deal with the case where the number of faces of this simplicial complex is finite. It is known that this simplicial complex has natural counterparts in the representation theory of algebras or Lie theory in special cases. In the representation theory of algebras, it is a simplicial complex whose vertex set consists of indecomposable modules and whose simplicial set consists of partial tilting modules. On the other hand, in Lie theory, it is a full subcomplex of a generalized associahedron such that the vertex set consists of all positive roots. In this talk, I will introduce the properties of these simplicial complexes. I will also talk about the relation to Gabriel's theorem, which is a well-known theorem on the correspondence between positive roots and indecomposable modules.

I will talk about the construction of $q$-Painlev\'{e} systems using cluster theory. Specifically, I will explain the following: the construction of

the initial value spaces of $q$-Painlev\'{e} systems using cluster transformations, the construction of the root lattices associated with $q$-Painlev\'{e} systems

using cluster seeds and toric fans, the classification of the initial value spaces of $q$-Painlev\'{e} systems by mutation equivalence classes of cluster seeds,

and actions of the group of Cremona isometries on cluster varieties.

In this talk, we present a method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of the latter. By applying our method to Sergeev's electrical solution and a two-component solution associated with the discrete modified KP equation, we obtain new solutions to the 3D reflection equation. Our approach is closely related to a relation between the transition maps of Lusztig's parametrizations of the totally positive part of $SL_3$ and $SO_5$, which is obtained via folding the Dynkin diagram of $A_3$ into one of $B_2$.

In the theory of (g,K)-modules, the induction functor $I is a fundamental tool to construct representations. In the late 2010s, the speaker constructed the functor I over a general commutative base ring. However, it appears nontrivial to compute the resulting modules explicitly.

In this talk, we discuss certain models of the real parabolic induction of SU(1,1) over the complex numbers. In particular, we see that new phenomena occur when we work over the ring of integers and its localization by 2.

In representation theory of the quantum groups $U_q$, a based module is a $U_q$-module with a canonical basis. One of the most important properties of canonical bases is cellularity. In this talk, we generalize the notion of based modules to representation theory of the i-quantum group of type AI, and prove that the associated canonical bases are cellular. The i-quantum group of type AI is a certain coideal subalgebra of the quantum group of type A whose classical limit is the universal enveloping algebra of the special orthogonal Lie algebra $so_n$. As a byproduct of our main result, we obtain a new combinatorial formula describing the branching rules from the special linear Lie algebra $sl_n$ to $so_n$.

Koornwinder多項式はAskey-Wilson多項式の多変数版として導入された6つのパラメータを持つq直交多項式系であり, 今日ではMacdonald-Cherednik理論により$(C_n^\vee,C_n)$型アフィンルート系に付随したMacdonald多項式として理解されている. 講演ではKoornwinder多項式の積に関する構造定数, 即ちLittlewood-Richardson係数の明示公式を紹介する. 非捩れ型のアフィン ルート系に付随したMacdonald多項式についてはYipがアルコーブウォークと呼ばれる組合せ論的対象を用いたLittlewood-Richardson係数公式を導出しているが, 我々の公式はちょうどその$(C_n^\vee,C_n)$型類似になっている.

Let $V(\lambda)$ be the extremal weight module of extremal weight $\lambda$ over the quantum universal enveloping algebra associated to a symmetrizable Kac-Moody algebra $g$, which was introduced by Kashiwara as a natural generalization of integrable highest weight modules. This module has crystal basis $B(\lambda)$. For the case that $g$ is of finite or affine type, it is known that the crystal basis $B(\lambda)$ is isomorphic to the crystal of (semi-infinite) Lakshmibai-Seshadri path of shape $\lambda$. In this talk, I will explain the relationship between the crystal of Lakshmibai-Seshadri paths of shape $\lambda$ and the crystal basis $B(\lambda)$ of $V(\lambda)$ in the case that $g$ is hyperbolic Kac-Moody algebras of rank 2.

The quantum K-theory introduced by Givental and Lee is the generalization of the K-theory defined by using Gromov-Witten theory. We study the quantum K-theory of the Grassmannian.

In this talk, we give a presentation of the quantum K-theory of the Grassmannian as a quotient of the ring of symmetric polynomials modulo an ideal. The generators of the ideal is described by a generalization of the rim hook algorithm due to Bertram, Ciocan-Fontanine and Fulton that works for the quantum cohomology ring of the Grassmannian. This result means that Grothendieck polynomials represent Schubert classes in the quantum K-theory of the Grassmannian.

As an application, we obtained a Giambelli formula for the Schubert class that expresses an arbitrary Schubert class as a determinant whose entries are written in terms of certain special Schubert classes. This is a joint work with D. Hiep, T. Ikeda, T. Matsumura and S. Sugimoto.

We propose a partial strategy to calculate an elliptic analogue of the canonical bases of equivariant K-theory of conical symplectic resolutions equipped with a duality data developed in our previous work and accomplish this strategy for the cotangent bundle of the Grassmannian variety Gr(2,4) and its symplectic dual. As an application, we find another formula for the elliptic stable bases of them which make the symmetry of elliptic stable bases under symplectic duality apparent.

Marked chain-order polytopes are convex polytopes constructed from a marked poset, which give a discrete family relating a marked order polytope with a marked chain polytope. In this talk, we consider the Gelfand-Tsetlin poset of type A, and realize the associated marked chain -order polytopes as Newton-Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as Newton-Okounkov bodies in a uniform way. We also construct a specific basis of an irreducible highest weight representation which is naturally parametrized by the set of lattice points in a marked chain-order polytope.

この講演では, $2d$ $N=(4,4)$と$(2,2)$ vortex partition functionsの関数等式を紹介する. これらの分配関数は, $A_1$型のhandsaw quiver varieties 上の積分により定義される. この関数等式は望月拓郎氏の壁越え公式により得られる.

非定常Ruijsenaars関数は, あるクラスの特殊関数達の (かなり) 上位に位置するものと考えられる. それの属する退化スキームの中には, Macdonald多項式, Schur多項式, affine Lie環の既約指標, 楕円型 (定常) Ruijsenaars差分作用素の固有関数, 定常及び非定常の affine Toda系の固有関数 (Whittaker関数) など, さまざまな関数が含まれている (はずである). 非定常Ruijsenaars関数に関する予想は色々あるが, 時間の関係で, Edwin Langmann氏, 野海正俊氏との共同研究に基づいて次の3項目について説明する.

(1) 非定常方程式とは？ 特に, 非定常affine Toda方程式とPainlev\'{e}方程式との関係

(2) Macdonald多項式の満たす方程式 (再考)

(3) 非定常Ruijsenaars関数の満たす非定常方程式についての予想

来年度の世話人を発表します。

This talk is based on joint work in progress with Osamu Iyama (the University of Tokyo). Let $A$ be a finite-dimensional algebra over an algebraically closed field $K$. The Grothendieck group $K_0(proj A)$ of the category of finitely generated projective $A$-modules is an abelian free group whose canonical basis is given by the indecomposable projective modules. Derksen-Fei introduced the notion of direct sums of elements in $K_0(proj A)$ by using the presentation space defined for each element $\theta \in K_0(proj A)$. On the other hand, King and Baumann-Kamnitzer-Tingley associated the semistable subcategory and the numerical torsion pairs in the category $\mod A$ of finitely generated modules to each $\theta$. We investigate their relationship in our work. I will report some of our results in this talk.

傾理論とは与えられた三角圏と代数上の導来圏を結び付ける手法であり, 多元環の表現論や代数幾何学にあらわれる三角圏を調べる際に重要な役割を果たす. 傾理論が上手く機能しない圏として周期三角圏がある. 周期三角圏とはシフト関手のある累乗が恒等関手になる三角圏であり, Cohen-Macaulay表現論や自己移入多元環の表現論で自然に姿を現す. 本講演では傾理論の周期三角圏における類似である周期傾理論について紹介する. まず導来圏の周期類似である周期導来圏について説明し, 周期三角圏と周期導来圏が三角同値になる十分条件を与える周期傾定理とその応用を紹介する.

Support $\tau$-tilting modules are introduced by Adachi, Iyama and Reiten as a generalization of classical tilting modules. One of the importance of these modules is that they are bijectively corresponding to many other objects, such as two-term silting complexes and left finite semibricks. Let $V$ be an $n$-dimensional vector space over an algebraically closed field $F$ of characteristic $p$. Then, the Schur algebra $S(n,r)$ is defined as the endomorphism ring $End_{F G_r} ( V^{\otimes r} )$ over the group algebra $F G_r$ of the symmetric group $G_r$. In this talk, we discuss when the Schur algebra $S(n,r)$ has only finitely many pairwise non-isomorphic basic support $\tau$-tilting modules.

Rogers-Ramanujan type identities are identities represented in the form of (infinite sum) = (infinite product) in terms of the Pochhammer symbols. Since the work by Lepowsky-Wilson, there has been an expectation that Rogers-Ramanujan type identities can be obtained by assigning a non-negative integer to each vertex of the affine Dynkin diagram. Related to this expectation, in this talk we will explain how to represent the basis of the vacuum space for the principal Heisenberg subalgebra of $A^{(2)}_{odd}$ type level 2 module in terms of Z-operators, focusing on the case of $A^{(2)}_7$ type level 2.

2 × 2の量子行列のべき乗の, 第二種Chebyshev多項式による明示公式を与える. またその応用として量子行列のベキ乗の成分の関係式の別証も与える.

群の指標(正定値・中心的な連続関数)の研究はユニタリ表現論で重要で，特に群がコンパクト群の帰納極限で与えられるとき，VershikとKerovはその指標を確率論的に研究する枠組みを与えた．それは漸近的表現論と呼ばれ，近年integrable probabilityと呼ばれ盛んに研究されている表現論・確率論・統計力学の境界領域の研究の一つの源流である．本講演ではコンパクト群を(Woronowiczの意味での)コンパクト量子群に取り替えたとき，漸近的表現論がどう拡張されるか議論する．具体的には，コンパクト量子群の帰納極限やその指標の定式化，各ABCD型のコンパクト量子群の帰納極限の指標が実際どう確率論的に取り扱われるか議論する．特にBCD型のコンパクト量子群の帰納極限の指標の研究と，CuencaとGorinによるBC型 q-Gelfand-Tsetlinグラフ上の解析との関係を説明する

Motivated by the generalized AGT conjecture, in this talk I will construct surjective homomorphisms from Guay's affine Yangians to the universal enveloping algebras of rectangular W-algebras of type A. This result is a super affine analogue of a result of Ragoucy and Sorba, which gave surjective homomorphisms from finite Yangians of type A to rectangular finite W-algebras of type A.

With a finite dimensional simple Lie algebra, a nilpotent element, and a complex number, we may associate a vertex algebra, called a W-algebra. It is well-known that the W-algebras for principal nilpotent elements enjoy a duality (Feigin-Frenkel duality), which is related to the geometric Langlands program. Recently, physicists found its generalization for other types of nilpotent elements from study on 4d supersymmetric gauge theory. I will explain the case between the subregular W-algebra and the principal W-superalgebra, conjectured originally by Feigin and Semikhatov. This talk is based on joint works with T. Creutzig, N. Genra and R. Sato.

Abstract: Representation theory of vertex superalgebras has a characteristic feature in its monoidal structure with respect to the fusion product.

In the affine vertex algebra case, the monoidal structure is known to be equivalent to that of the corresponding quantum enveloping algebra of finite type.

In this talk I will explain a correspondence between the monoidal structures for subregular W-algebras and principal W-superalgebras by using simple current extensions, coset constructions, and relative semi-infinite cohomology.

This is a joint work with T. Creutzig, N. Genra, and S. Nakatsuka.

Let $I$ be a finite set. Let $Z \Pi$ be the free abelian group with a basis $\{a_i | i \in I \}$. Let $\chi: Z \Pi \times Z\Pi \to C^\times$ be a group bi-homomorphism. Let $q_{i j} :=\chi(a_i, a_j)$. To $\chi$, we can associate the generalized quantum group $U(\chi)$. To $U(\chi)$, we can associate the generalized root system $R(\chi)$ in the Kharchenko sense.

Assume that $R(\chi)$ is a finite set. Let $q \in C^\times$ be not a root of unity. In the case of $q_{i j}=q^{(a_i,a_j)}$ with a Killing form $(\, , \,): Z\Pi \times Z\Pi \to Z$ of a finite-dimensional simple complex Lie algebra $g$, $U(\chi)$ is virtually the same as $U_q(g)$. In the case of $q_{i j}=q^{(a_i, a_j)} (-1)^{p(a_i) p(a_j)}$ with a Killing form $(\, , \,)$ and a parity $p: \Pi \to \{0,1\}$ of a finite-dimensional basic classical simple complex Lie superalgebra $b$, $U(\chi)$ is virtually the same as $U_q(b)$. We also have two or three other types of $U(\chi)$ with $\chi$ involving $q$ which is not a root of unity. $U_q(b)$ has many finite-dimensional irreducible modules which cannot be taken by $q \to 1$.

The speaker has obtained a Weyl-Kac-type character formula for the finite-dimensional typical irreducible modules of $U(\chi)$, see [H. Yamane, Typical Irreducible Characters of Generalized Quantum Groups, J. Algebra Appl. 20 (2021), no. 1, 2140014, 58 pp., arXiv:1909.08881]. We also mention Hamilton circuits of Cayley graphs of Weyl groupoids of generalized quantum groups, see [H. Yamane, arXiv.2103.16126].

The Type A Schur algebra admits a presentation as the quotient of Lusztig's dotted quantum group via extra relations by Doty-Giaquinto. Its categorification, given by Mcckaay-Stosic-Vaz, uses a diagrammatic incarnation of these relations as a quotient of the Khovanov-Lauda-Rouquier 2-category. We give an analogue of both results in the case of cyclotomic Schur algebra, i.e. the centralizer algebra of the Ariki-Koike algebra. Specialized to Type B, this is compared to another geometric categorification given by Bao-Kujawa-Li-Wang, which uses isotropic double flag varieties and convolution products. Our result links the two approaches in the classical case, and suggests new relations for the Schur algebra in the geometric setting. This is based on two projects joint respectively with J. Kujawa and Y. Li.

In this talk, we show in full generality that the generalized quantum affine Schur-Weyl duality functor, introduced by Kang-Kashiwara-Kim, gives an equivalence between the category of finite-dimensional modules over a quiver-Hecke algebra and a certain full subcategory of finite-dimensional modules over a quantum affine algebra which is a generalization of a Hernandez-Leclerc's category.

In untwisted ADE types, this was previously proved by Ryo Fujita using the geometric representation theory on quiver varieties, which is not applicable in general types. In our approach, we study a certain endomorphism ring of a bimodule appearing in the construction of the generalized quantum affine Schur-Weyl duality functor. This is purely algebraic, and hence can be extended uniformly to general types.