Conference talks will take place in room STM 464, located in the STEM Complex.

Conference organizers: Siddhartha Sahi, Hadi Salmasian.

Acknowledgement. The workshop is sponsored by funding from the National Science Foundation and a University of Ottawa CWCO grant.

Funding and Registration. We have some funds to sponsor the travel and accommodation
expenses of participants, with priority given to participants with no funding or those that meet the Equity, Diversity, and Inclusion criteria (EDI). If you would like to request funding, please register for the activity using this URL.

Schedule of Talks

Date Speaker Title and Abstract (hover your mouse pointer on titles)
Aug 31, 9:00 a.m. Daniel Bump (Stanford University) Solvable Lattice Models and Quantum Groups Abstract: We will consider solvable lattice models whose partition functions are (for example) Schur functions, Hall-Littlewood polynomials, nonsymmetric variants such as Demazure atoms, or more interesting polynomials related to algebraic geometry or p-adic representation theory. In the basic paradigm, every edge of a planar graph such as a lattice is assigned to a module in a quantum group. The Yang-Baxter equation leads to interesting properties of the partition function such as Demazure recurrences that will be discussed in Brubaker's talk. In this lecture, after introducing the topic, we will focus on the local aspects, including monochrome factorization related to the PBW theorem for quantum groups, and some interesting examples that stretch or break the basic paradigm.
Aug 31, 10:00 a.m. Ben Brubaker (University of Minnesota) Special Functions via Lattice Models and Quantum Groups Abstract: We survey connections between quantum group modules and certain classes of special functions, where the bridge is made using solvable lattice models. Lattice models are defined as generating functions on certain admissible states of the lattice, but admit multiple modes of analysis. Here we discuss one such mode - the Yang-Baxter equation. In recent examples (including those in collaboration with Buciumas, Bump, and Gustafsson) these lead to special functions defined by divided difference operators and are flexible enough to include a wide range of special functions. Together with Dan Bump's talk connecting quantum group modules to lattice models, this suggests many open questions about how the theory of quantum group representations and the theory of special functions inform each other.
Aug 31, 11:00 a.m. Hadi Salmasian (University of Ottawa) Quantum groups, quantum Weyl algebras, and a new FFT for Uq(gl(n)) Abstract: The First Fundamental Theorem (FFT) is one of the highlights of invariant theory of reductive groups and goes back to the works of Schur, Weyl, Brauer, etc. For the group GL(V), the FFT describes generators for polynomial invariants on direct sums of several copies of the standard module V and its dual V*. Then R. Howe pointed out that the FFT is closely related to a double centralizer statement inside a Weyl algebra (a.k.a. the algebra of polynomial-coefficient differential operators). In this talk we first construct a quantum Weyl algebra and then present a quantum analogue of the FFT. As a special case of this FFT we obtain a double centralizer statement inside our quantum Weyl algebra. We remark that the FFT that we obtain is different from the one proved by G. Lehrer, H. Zhang, and R.B. Zhang for Uq(gl(n)). Time permitting, I will explain the connection between this work and the Capelli Eigenvalue Problem for classical quantum groups. This talk is based on a joint project with Gail Letzter and Siddhartha Sahi.
Aug 31, 1:00 p.m. Milen Yakimov (Northeastern University) Quantum cluster algebra structures on quantum nilpotent algebras Abstract: Quantum nilpotent algebras form a large axiomatically defined class of algebras that contains many interesting subfamilies, such as quantum Schubert cell algebras and Kashiwara's q-boson algebras. Cluster algebras and their quantum counterparts play an important role in representation theory, combinatorics and topology. We will describe a result that constructs quantum cluster algebra structures on all quantum nilpotent algebras. This proves that various algebras in the theory of quantum groups have quantum cluster algebra structures, such as quantum Schubert cell algebras, quantum double Bruhat cell (the Berenstein-Zelevinsky), and others. This is a joint work with Ken Goodearl (UCSB).
Sept 1, 9:00 a.m. Jimmy He (MIT) Applications of integrability in half space models Abstract: The study of quantum groups and integrable systems have long been connected. Recently, quantum groups have appeared in probability, in the study of integrable probabilistic models. I will discuss an integrable model called the six vertex model with a single open boundary, which is connected to an affine quantum symmetric pair via the reflection equation. I will also explain a connection to the type BC Hecke algebra, and applications to probabilistic symmetries of the model.
Sept 1, 10:00 a.m. Huanchen Bao (NUS, Singapore) Symmetric subgroups schemes, Frobenius splittings, and quantum symmetric pairs Abstract: Let G be a connected reductive group over an algebraically closed field. Such groups are classified via root data and can be parameterised via Chevalley group schemes over integers. In this talk, we shall first recall the construction of Chevalley group schemes by Lusztig using quantum groups. Then we shall discuss the construction of symmetric subgroup schemes parameterising symmetric subgroups K of G using quantum symmetric pairs. The existence of such group schemes allows us to apply characteristic p methods to study the geometry of K-orbits on the flag variety of G, which we shall discuss as well. This is based on joint work with Jinfeng Song (NUS).