Ryo Fujita

Title: Isomorphisms among quantum Grothendieck rings and their applications

Abstract: Quantum Grothendieck ring is a one-parameter deformation of the usual Grothendieck ring of the monoidal category of finite-dimensional modules over the quantum loop algebras, endowed with a canonical basis. In the case of type ADE, thanks to the geometry of quiver varieties, the canonical basis is known to compute the q-characters of simple modules (via the analog of Kazhdan-Lusztig algorithm) and enjoy some positivity properties. In this talk, we discuss a collection of isomorphisms between the quantum Grothendieck ring of type BCFG and that of ``unfolded” type ADE, respecting the canonical bases. They propagate the positivity properties to type BCFG and verify the analog of Kazhdan-Lusztig conjecture for several new cases. This talk is based on a joint work with David Hernandez, Se-jin Oh, and Hironori Oya.

Stefan Kolb

Title: Quantum symmetric pairs

Abstract: The theory of quantum symmetric pairs was initiated by Gail Letzter and provides coideal subalgebras of Drinfeld-Jimbo quantum groups. These coideal subalgebras are quantum group analogs of Lie subalgebras fixed under an involution. Over the past decade it became clear that many constructions for quantum groups allow generalisations for quantum symmetric pairs.

Drinfeld-Jimbo quantum groups give rise to a universal R-matrix which provides solutions of the quantum Yang-Baxter equation in representations. Quantum symmetric pairs provide additionally a universal K-matrix which gives rise to solutions of the quantum reflection equation. A main ingredient in the construction of a universal K-matrix is the quasi K-matrix which can be elegantly obtained via a star-product interpretation of quantum symmetric pair coideal subalgebras.

The three lectures on quantum symmetric pairs aim to give an introduction to the subject and to explain the above constructions. The titles for the individual lectures are as follows: 1) Quantum symmetric pairs: Motivation, definition and first properties. 2) The star product interpretation of quantum symmetric pairs. 3) Universal K-matrix and braided module categories.

Jae-Hoon Kwon

Title: Crystal bases for a general linear Lie superalgebra

Abstract: The crystal base theory for the quantum group associated with a symmetrizable Kac-Moody algebra has been one of the most important tools in the representation theory, reflecting its fundamental combinatorial structure. For a classical Lie superalgebra and its quantized enveloping algebra, there have been several works on the existence of a crystal base, where its representation theory is no longer parallel to that of a Kac-Moody algebra. In this lecture, we introduce the crystal base theory for general linear Lie superalgebra focusing on a polynomial representation, a Kac module, and the negative part of the quantum group. If time permits, then we introduce several related topics.

Hironori Oya

Title: Relations among the $q$-characters of simple modules over quantum loop algebras of several Dynkin types

Abstract: In our previous work, we gave a systematic construction of algebra isomorphisms among quantum Grothendieck rings of the monoidal categories of finite-dimensional modules over quantum loop algebras of several Dynkin types. In this talk, I explain the cluster algebraic nature of our isomorphisms. As an application, we realize our isomorphism as a birational change of variables. Since our isomorphisms respect the $(q, t)$-characters of simple modules, this realization reveals non-trivial relations among the $(q, t)$-characters of simple modules of several Dynkin types. This talk is based on a joint work with Ryo Fujita, David Hernandez, and Se-jin Oh.

Anne Schilling

Title: Crystal bases in statistical mechanics, representation theory and combinatorics

Abstract: Crystal bases originated from the quantum group analysis of statistical mechanical models. Professor Okado has contributed many important results in this area of mathematics. I will review some of the applications of crystal bases in connection with symmetric functions and representation theory and will discuss some recent results in invariant theory.

Mark Shimozono

Title: Parabolic Quiver Hall-Littlewood polynomials

Abstract: In 2017 Dan Orr and I introduced Hall-Littlewood (HL) polynomials for arbitrary quivers (which we later generalized to the parabolic case). We are primarily interested in cyclic quivers. Special cases include parabolic HL polynomials studied by the author with Weyman and with Zabrocki, which include all affine type A one-dimensional sums (Kirillov-Reshetikhin characters), and the Shoji-Finkelberg-Ionov polynomials. We will discuss how wreath HL polynomials (taking the coefficient of the smallest appearing power of q in Haiman's wreath Macdonald H polynomials) arise as parabolic HL polynomials for the cyclic quiver.

Akihito Yoneyama

Title: Tetrahedron equations associated with quantized six-vertex models

Abstract: The Zamolodchikov tetrahedron equation is a three-dimensional analog of the Yang-Baxter equation, where the latter equation is the central object of two-dimensional integrability. Unlike the Yang-Baxter equation, there is no systematic way to obtain non-trivial solutions to the tetrahedron equation, but there are also solutions that have a close relationship with quantum algebras. The solution by Bazhanov and Baxter and by Bazhanov and Sergeev are seminal examples of such solutions. The former is closely related to the generalized Chiral Potts model via reductions to two dimensions, which is associated with representations of $U_q(A_{n-1}^{(1)})$ at roots of unity. On the other hand, the latter is characterized as an intertwiner of the quantum coordinate ring $A_q(A_2)$, and also gives $R$-matrices associated with the symmetric tensor representations of $U_q(A_{n-1}^{(1)})$ via reductions similarly. Notably, both solutions share an origin as solutions to the tetrahedron equation of the form $RLLL=LLLR$. Here, $L$ is a quantized version of the six-vertex model, which is obtained by replacing its matrix elements with operator-valued ones. In this talk, we will discuss a generalization of this framework and obtain a new family of solutions to the tetrahedron equation, where their matrix elements are either factorized or expressed as $q$-hypergeometric series. This talk is based on a joint work with Atsuo Kuniba and Shuichiro Matsuike.