Solvable lattice models, representation theory of quantum groups and algebraic combinatorics

Solvable lattice models originated in statistical mechanics to explore global behavior (e.g., phase-transitions in matter) from local interactions (energy measured in nearest neighbor particle interactions). When those models admit certain symmetries, encoded in so-called Yang–Baxter equations, one can often solve for closed form solutions for the total energy of the system — hence we refer to these models as “solvable.” This workshop takes an algebraic approach to solvable lattice models (through modules of quantum groups) in search of families of models whose total energy are now special functions from algebraic combinatorics. The last several years have witnessed an explosion of research in this area, and the workshop will bring together researchers at the forefront of these developments.

Drinfeld and Jimbo were initially inspired to define quantum groups to explain and provide new examples of Yang–Baxter equations. It has long been understood that modules for quantum groups would, by construction, give rise to solvable lattice models. But it is only in the last decade or so that this has been systematically directed at prominent classes of algebro-geometric special functions. In brief, if one can show that the partition function (the weighted sum over all admissible states) of a family of models is a given family of special functions, then one can exploit symmetries of the model to prove desirable properties of the family.

Solvable lattice models have thus outgrown their origin in statistical mechanics, finding far reaching applications to integrable probability, different species of representation theory, algebraic combinatorics, and algebraic geometry. This conference will focus primarily on the underlying combinatorics of these related topics. Every subfield brings their own tools with the potential to impact other fields, and a fruitful cross-pollination of ideas is expected. The conference will provide an inclusive environment for a diverse set of backgrounds, with a variety of talks at different levels so that people with different levels of experience are welcomed into the field.