Representations and Lie Theory Seminar

Schedule of talks:

Abstracts

September 22, 2026. The six vertex model, a model originating in statistical mechanics, has weights related to the R-matrix of qunatum affine sl(n). When the model has an open boundary, additional weights satisfying the so-called reflection equation are needed, and this is related to the K-matrix of a quantum symmetric space. Fusion is a procedure to obtain more complex weights from simpler ones (or a method of obtaining R-matrices for Verma modules from the standard representation). While the general procedure is understood for a while, for many probablistic and combinatorial applications one requires explicit tractable formulas. These have been found for bulk weights, but not for the boundary ones.
In joint work in progress with Amol Aggarwal, Ivan Corwin, Milinde Hegde, and Michael Wheeler, we extend fusion to boundary weights in a special case known as the boson model. This is already enough for many interesting applications. I will explain the physical background and connections to quantum groups, provide an overview of fusion, and explain some of the possible applications within probability and combinatorics that our work may allow.

September 29, 2026. A fundamental structural property of Drinfeld-Jimbo quantum (super)groups is their realization as a Drinfeld double, implying quasitriangularity via the universal R-matrix. Evaluating this R-matrix on a specific representation gives rise to the RLL (or RTT) realization, a quantum analogue of the matrix realization of classical Lie (super)algebras. This talk discusses the relationship between the Drinfeld-Jimbo and RLL realizations for orthosymplectic quantum supergroups. This talk consists of two parts. First, we compute the finite and affine R-matrices. By embedding the nilpotent half into a shuffle superalgebra, we utilize combinatorial tools to construct dual PBW bases. This also enables the factorization of the reduced R-matrix into an ordered product of local q-exponents. Furthermore, applying Yang-Baxterization to these finite R-matrices yields the explicit affine R-matrix R(z). Second, these R-matrices are used to define the RLL realization, constructing a direct Hopf superalgebra isomorphism from the Drinfeld-Jimbo presentation. The primary difficulty is establishing injectivity; classical proofs in the non-super setup (e.g., Ding-Frenkel for finite A-type) relied heavily on faithful representation arguments, which become more subtle in the super setup. To circumvent these difficulties, we leverage the fact that both realizations admit (generalized) Drinfeld double structures. By exploiting the non-degeneracy of the skew-pairing between the Borel subalgebras, we prove injectivity and establish the isomorphism.