2020年度
第12回
日時:2021年3月19日(金) 15:00~16:30
場所:Zoom
講演者:直井克之氏(東京農工大学)
Title:Equivalence via generalized quantum affine Schur-Weyl duality
Abstract: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.
第11回
日時:2021年1月22(金) 15:00~16:00
場所:Zoom
講演者:Sin-Myung Lee氏(Seoul National University)
Title:Super duality for quantum affine algebras of type A
Abstract:We introduce a new approach to study finite-dimensional representations of the quantum group of the affine Lie superalgebra L(gl_{M|N}) (M≠N). More precisely, there exists an exact monoidal functor called truncation, which gives an equivalence of categories between certain subcategory of modules over the generalized quantum groups introduced by [Kuniba-Okado-Sergeev 15], and a subcategory of those over the quantum affine algebra of type A. The key observation is that spectral decompositions of R-matrices coincide, so that the construction of the generalized quantum affine Schur-Weyl duality functor by [Kang-Kashiwara-Kim 18] works equally well in either case. The result can be viewed as an affine analogue of super duality of type A. This talk is based on a joint work (arXiv:2010.06508) with Jae-Hoon Kwon.
第10回
日時:2020年12月18日(金) 15:00~16:30
場所:Zoom
講演者:村上浩大氏(京都大学 理学研究科)
タイトル:PBW parametrizations and generalized preprojective algebras
アブストラクト:Generalized preprojective algebras are introduced by [Gei\ss-Leclerc-Schr\”oer, 2017], motivated from the study of $q$-characters of Kirillov-Reshetikhin modules in [Hernandez-Leclerc, 2016]. These algebras are defined for symmetrizable GCMs $C$ and their symmetrizers $D$, and their module theoretical concepts often know information about Kac-Moody algebras or their quantum groups associated with $C$. In this talk, we categorify Weyl chambers via module categories of generalized preprojective algebras. Then, we compare some filtrations determined by these chamber structures with combinatorial data from crystal bases of quantum groups for finite symmetrizable cases.
第9回
日時:2020年12月4日(金) 15:00~16:30
場所:Zoom
講演者:井上玲氏(千葉大学)
タイトル:Cluster realization of Weyl groups and q-characters of quantum affine algebras
アブストラクト:We introduce the Weyl group realization as a subgroup of the cluster modular group for some periodic quiver, and discuss its applications, based on arXiv: 2003.04491. We consider an infinite quiver Q(g) and a family of periodic quivers Q_m(g) for a finite dimensional simple Lie algebra g and an integer m bigger than one. The quiver Q(g) is essentially same as what introduced in [Hernandez-Leclerc 16] in studying the q-characters for quantum non-twisted affine algebras. For the quiver Q_m(g) we construct the Weyl group W(g) in a similar way as [I-Ishibashi-Oya 19], and study its applications to the q-characters [Frenkel-Reshetikhin 99], and to the lattice g-Toda field theory [I-Hikami 00]. In particular, when q is a root of unity, we prove that the q-character is invariant under the Weyl group action. We also show that the A-variables for Q(g) correspond to the tau-functions for the lattice g-Toda field equation.
第8回
日時:2020年11月13日(金) 15:00~16:30
場所:Zoom
講演者:藤田直樹氏(東京大学)
タイトル:Schubert calculus from polyhedral parametrizations of Demazure crystals
アブストラクト:One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced (dual) Kogan faces. In this talk, we discuss its generalization to other Lie types through the theory of Kashiwara crystal bases. We first explicitly describe string parametrizations of opposite Demazure crystals in general Lie type, which give a natural generalization of reduced dual Kogan faces. We then relate reduced Kogan faces with Demazure crystals in type A through the theory of mitosis operators on reduced pipe dreams. This relation is naturally extended to the case of type C, which leads to the theory of Schubert calculus on symplectic Gelfand-Tsetlin polytopes.
第7回
日時:2020年10月23日(金) 15:00~16:30
場所:Zoom
講演者:河野隆史氏(東京工業大学)
タイトル:Inverse K-Chevalley formula for type A semi-infinite flag manifolds
アブストラクト:The semi-infinite flag manifold associated to a connected, simply-connected simple algebraic group, is a reduced ind-scheme that is the semi-infinite analog of the (ordinary) flag manifold. In our recent works, we studied the equivariant K-group of a semi-infinite flag manifold, and described a Chevalley formula explicitly; our Chevalley formula gives the expansion of the (tensor) product of a Schubert class and the class of a line bundle into a (possibly infinite) linear combination of Schubert classes twisted by characters of the maximal torus. The purpose of this talk is to describe the ``inverse Chevalley formula'', which gives the expansion of a Schubert class twisted by a character of the maximal torus into a finite linear combination of products of Schubert classes and line bundles. In this talk, we give an explicit description of the inverse Chevalley formula in the case that the algebraic group is of type A and the character is an element of the Weyl group orbit of the 1st fundamental weight. This talk is based on a joint work with Satoshi Naito, Daniel Orr, and Daisuke Sagaki.
第6回
日時:2020年7月17日(金) 15:00~16:30
場所:Zoom
講演者: 水野勇磨氏(東京工業大学)
タイトル: T-systems and Y-systems in cluster algebras
アブストラクト:We study T-systems (and Y-systems) arising from cluster algebras. We find that these systems are characterized by triples of matrices, which we call T-data, that satisfy a certain symplectic relation. We show that any T-datum corresponding to a periodic T-system has simultaneous positivity, which can be considered as a generalization of the characterization of finite type Cartan matrices. As an application, we discuss the relation between periodic Y-systems and Nahm's conjecture.
第5回
日時:2020年7月3日(金) 15:00~16:30
場所:Zoom
講演者: 百合草寿哉氏(東北大学 理学研究科)
タイトル:Tame algebras have dense g-vector fans
アブストラクト: The g-vector fan of a finite-dimensional algebra is a fan whose rays are the g-vectors of its 2-term presilting objects. We prove that the g-vector fan of a tame algebra is dense. Moreover, we apply this result to obtain a near classification of quivers for which the cluster g-vector fan is dense, using the additive categorification of cluster algebras by Jacobian algebras. This talk is based on a joint work with Pierre-Guy Plamondon.
第4回
日時:2020年6月19日(金) 15:00~16:00
場所:Zoom
講演者: 石橋典氏(京都大学数理解析研究所)
タイトル:Algebraic entropy of sign-stable mutation loops
アブストラクト: A mutation loop is a certain equivalence class of a sequence of mutations and permutations of indices. They form a group called the cluster modular group, which can be regarded as a combinatorial generalization of the mapping class groups of marked surfaces. We introduce a new property of mutation loops which we call the “sign stability” as a generalization of the pseudo-Anosov property of a mapping class. A sign-stable mutation loop has a numerical invariant which we call the “cluster stretch factor”, in analogy with the stretch factor of a pA mapping class. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two are estimated by the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.
日時:2020年6月19日(金) 16:15~17:15
場所:Zoom
講演者: 狩野隼輔氏(東京工業大学)
タイトル:Pseudo-Anosov mapping classes are sign-stable
アブストラクト: We introduced the sign stability of mutation loops as a generalization of the pseudo-Anosov property of mapping classes of on surfaces. In this talk, I will explain the equivalence between the pseudo-Anosov property and the sign stability for a mapping class. If time permits, I will explain the relationship between the signs of mutations and train track splittings. This talk is based on a joint work with Tsukasa Ishibashi.
第3回
日時:2020年6月5日(金) 15:00~16:00, 16:15~17:15
場所:Zoom
講演者:阿部紀行氏(東京大学 大学院数理科学研究科)
タイトル:On Soergel bimodules
アブストラクト:Soergel introduced the category which is now called the Soergel bimodules. It gives a categorification of the Hecke algebra with, roughly speaking, characteristic zero field. Williamson gave a singular version of his theory. It still works only with the same assumption as Soergel. I will explain how to modify this category to work with positive characteristic field. I also explain relations with other categories which categorify the Hecke algebra.
第2回
日時:2020年5月22日(金) 15:00~16:30
場所:Zoom
講演者:藤田遼氏(京都大学理学研究科)
タイトル:Singularities of R-matrices, graded quiver varieties and generalized quantum affine Schur-Weyl duality
アブストラクト:The R-matrices are realized as intertwining operators between tensor products of two finite-dimensional simple modules of the quantum affine algebras. They can be seen as matrix-valued rational functions in spectral parameters, whose singularities strongly reflect the structure of tensor product modules. In this talk, we present a simple unified formula expressing the denominators of the R-matrices between the fundamental modules of type ADE and explain its relation to the representation theory of the Dynkin quivers / the geometry of Nakajima's graded quiver varieties. As an application, we obtain a geometric interpretation of Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor when it arises from a family of fundamental modules. Such a geometric interpretation sometimes provides an efficient way to understand the monoidal structure of the module category of the quantum affine algebras.
第1回
日時:2020年5月8日(金) 15:00~16:30
場所:Zoom
講演者: 榎本 悠久氏(名古屋大学多元数理科学研究科)
タイトル:Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras
アブストラクト: For an element w of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott introduced a subcategory F(w) of a module category over a preprojective algebra. This category plays an important role in the representation theory of algebras (torsion-free classes), as well as in the Lie theory (a categorification of the cluster structure on the coordinate ring of the unipotent cell). In this talk, I will study the structure of F(w) as an exact category. I will explain the classification of simple objects in F(w) in terms of the root system, and give some applications. If time permits, I will discuss how to classify simple objects in torsion-free classes over general finite-dimensional algebras.