Title and abstracts
Speaker: Uly Alvarez (The University of Alabama)
Title: An explicit bijection between the cells of a quiver Grassmannian and the perfect matching of a snake graph
Abstract: It is an open problem to find cell decompositions of quiver Grassmannians associated to each cluster variable. We initiate a new approach to this problem by giving an explicit description for each cell. In this talk, we illustrate our approach for the Kronecker quiver.
Speaker: Ryo Fujita (Kyoto University)
Title: Isomorphisms among quantum Grothendieck rings and their cluster theoretical interpretation
Abstract: Quantum Grothendieck ring in this talk is a deformation of the Grothendieck ring of the monoidal category of finite-dimensional modules over the quantum loop algebra, endowed with a canonical basis consisting of the so-called simple (q,t)-characters. We discuss a collection of isomorphisms among the quantum Grothendieck rings of different types respecting the canonical bases, via which the (q,t)-characters of type BCFG inherit several good properties from those of the unfolded type ADE. We also discuss their cluster theoretical interpretation, which particularly yields non-trivial birational relations among the (q,t)-characters of different types. This is a joint work with David Hernandez, Se-jin Oh, and Hironori Oya.
Speaker: Hyun Kyu Kim (KIAS)
Title: A cluster-variety quantization of 3-dimensional gravity
Abstract: For an oriented punctured surface $S$, I will present a quantization of the moduli space of Lorentzian metrics on the 3-manifold $S\times \mathbb{R}$ with constant sectional curvature $\Lambda$ satisfying some natural conditions. This is based on the Meusburger-Scarinci coordinates on this moduli space, which behave like cluster X-coordinates with values in the ring of generalized complex numbers $\mathbb{R}_\Lambda = \mathbb{R}[\ell]/(\ell^2=-\
Speaker: Yoosik Kim (Pusan National University)
Title: Infinitely many monotone Lagrangian tori in flag manifolds
Abstract: In symplectic topology, constructing monotone Lagrangian tori that are not Hamiltonian isotopic to each other is interesting. In this talk, I discuss how to use cluster structures for constructing infinitely many monotone Lagrangian tori and distinguishing them in complete flag manifolds of arbitrary type (except in some low-dimensional cases).
Speaker: Kyu-Hwan Lee (University of Connecticut)
Title: Geometric description of C-vectors and real Lösungen
Abstract: We introduce real Lösungen as an analogue of real roots. For each mutation sequence of an arbitrary skew-symmetrizable matrix, we define a family of reflections along with associated
vectors, called L-vectors, which are real Lösungen and a set of curves on a Riemann surface. The matrix consisting of the L-vectors is called the L-matrix. We conjecture that (1) the L-matrix depends (up to signs of row vectors) only on the seed, and that (2) the curves can be drawn without self-intersections. We prove conjecture (1) for the quivers mutation equivalent to type A_n quivers. This is a joint work with Tucker Ervin, Blake Jackson, Kyungyong Lee and Matthew Mills.
Speaker: Kyungyong Lee (University of Alabama)
Title: Unexpected structures of rank 3 cyclic quivers
Abstract: This is based on joint projects with Jae-Hoon Kwon, Kyu-Hwan Lee, Tucker Ervin, Blake Jackson, Son Nguyen, and Jihyun Lee. We have discovered numerous unexpected properties and structures of rank 3 cyclic quivers. We will report these features in terms of C-matrices, G-matrices, and L-matrices.
Speaker: Sin-Myung Lee (Seoul National University)
Title: Oscillator representations of quantum affine orthosymplectic superalgebras
Abstract: We introduce a category of q-oscillator representations over the quantum affine superalgebras of type D and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible q-oscillator representations of affine type X and the finite-dimensional irreducible representations of affine type Y for (X,Y) = (C,D), (D,C) under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finite-dimensional representations of classical Lie algebras arising from the Howe’s reductive dual pairs (g,G), where g = sp_2n, so_2n and G = O_l, Sp_2l. This talk is based on a joint work with Jae-Hoon Kwon and Masato Okado (arXiv:2304.06215).
Speaker: Li Li (Oakland University)
Title: Cluster algebras and Nakajima’s graded quiver varieties
Abstract: Nakajima's graded quiver varieties are complex algebraic varieties associated with quivers. They are introduced by Nakajima in the study of representations of universal enveloping algebras of Kac-Moody Lie algebras, and can be used to study cluster algebras. In the talk, I will describe how the geometry of Nakajima's graded quiver varieties can be used to study the supports of the triangular basis of skew-symmetric rank 2 quantum cluster algebras, thus prove a conjecture proposed by Lee-Li-Rupel-Zelevinsky.
Speaker: Yuma Mizuno (Chiba University)
Title: Classification of periodic Y-systems of rank 2
Abstract: Y-systems are difference equations associated with automorphisms on cluster algebras. They are characterized by pairs of square matrices satisfying the symplectic property. In this talk, I will talk about the classification of periodic Y-systems of rank 2.
Speaker: Se-jin Oh (Sungkyunkwan University)
Title: Discussion on quantum tori
Abstract: Quantum cluster algebra is an $\Z[q^{\pm 1}]$ subalgebra of a quantum torus. In this talk, we discuss the quantum tori and their relationships with the representation theory of quiver Hecke algebras. This is a joint work with Kashiwara, Kim and Park.
Speaker: Fan Qin (Shanghai Jiao Tong University)
Title: From dual canonical bases to triangular bases of quantum cluster algebras
Abstract: For any symmetrizable Kac-Moody algebra and any Weyl group element, the corresponding quantum unipotent subgroup possesses the dual canonical basis. We show that the dual canonical basis is the (common) triangular basis of the quantum cluster algebra. Consequently, we deduce that the basis contains all quantum cluster monomials, extending previous results by the author and Kang-Kashiwara-Kim-Oh. If time permits, we will briefly explain how to extend the result to other Lie theoretic varieties.
Notes Lecture1 Lecture2 Lecture3
