# Solvable Lattice Models Seminar

Solvable lattice models are statistical mechanical systems that can be studied exactly by a method of Baxter, based on the Yang-Baxter equation. This can be understood in terms of quantum groups. Recently particular examples showing symmetry with respect to the Lie super group $U_q(\widehat{\mathfrak{gl}}(r|n))$ arose in two very different contexts: the representation theory of $p$-adic groups, where such models were used by Brubaker, Buciumas, Bump and Gustafsson to study Iwahori Whittaker functions on metaplectic groups; and in integrable probability, in work recent of Aggarwal, Borodin and Wheeler. We will introduce the topic starting with Baxter's work on six vertex model, then look at more recent work explaining some of the ideas.

The seminar serves both learning and research purposes. We explore applications of solvable lattice models in different areas of mathematics. First we give learning talks with history and first results in the area. After that we give several research talks on the recent results. Thus, the seminar is friendly for both students and researchers. Feel free to join every new topic even if you got lost last time because we start over with introductory talks.

Possible topics

• Representations of $p$-adic groups
• Spherical Whittaker functions
• Metaplectic spherical Whittaker functions
• Iwahori Whittaker functions
• Metaplectic Iwahori Whittaker functions
• Whittaker Functions for Sympelctic groups
• Probability theory
• Stochastic Six-vertex Models
• Colored Stochastic Vertex Models
• Colored Fermionic Vertex Models

Write me (naprienko@stanford.edu) if you want to be added to Google Group of the seminar and get Zoom link for upcoming talks.

Date Speaker Title Abstract and Materials
3/3/2021 at 2:00 pm PT Alexey Bufetov TBA

TBA

Write me (naprienko@stanford.edu) if you want to be added to the email list of the seminar and get Zoom links for upcoming talks.

2/24/2021 at 2:00 pm PT Chenyang Zhong Stochastic symplectic ice: a stochastic vertex model with U-turn boundary

In this talk, we present a novel solvable lattice model which we term "stochastic symplectic ice" with stochastic weights and U-turn right boundary. The model can be interpreted probabilistically as a new interacting particle system in which particles jump alternately between right and left. We also introduce two colored versions of the model -- one of which involves "signed color" -- and the related stochastic dynamics. We then show how the functional equations and recursive relations for the partition functions of those models can be obtained using the Yang-Baxter equation. Finally, we show that the recursive relations satisfied by the partition function of one of the colored models are closely related to Demazure-Lusztig operators of type C.

Related papers: Symplectic Ice, Stochastic symplectic ice

2/17/2021 at 2:00 pm PT Henrik Gustafsson A vertex model for Iwahori Whittaker functions

I will present a detailed description of a vertex model whose partition function computes all values of a basis of Iwahori Whittaker functions for unramified principal series of $GL_r(F)$ where $F$ is a non-archimedean field. In particular, there is a remarkable bijection between the data determining these values and the boundary data of the solvable lattice model. The vertex model can be described by colored paths in a grid and we will define its associated Boltzmann weights using a fusion process. We will show how its Yang-Baxter equation can be interpreted by Demazure-like operators on the representation theory side, and how the same model, but with different boundary conditions, also describes parahoric Whittaker functions.

Based on joint work with Ben Brubaker, Valentin Buciumas and Daniel Bump.

2/10/2021 at 2:00 pm PT Leonid Petrov Vertex models and stochastic particle systems

I will explain connections between stochastic particle systems (like q-TASEP or random polymers) and exactly solvable vertex models. More precisely, there is a whole family of results identifying random variables on the particle system side with certain quantities in a vertex model. There are several ways of establishing these results, including the Robinson-Schensted-Knuth correspondence. I will focus on bijectivisation of the Yang-Baxter equation for vertex models corresponding to q-Whittaker and the new spin q-Whittaker polynomials.

2/3/2021 at 2:00 pm PT Ben Brubaker Solvability and Special Functions

We survey the ways in which solvability informs the study of special functions in algebra and representation theory. In particular, we explain how various natural Hecke algebra actions - from p-adic representation theory and from Schubert calculus - are manifested as Yang-Baxter equations of associated solvable lattice models. Then we describe how formal diagram identities of solvable models lead to new identities for the associated special functions.

Related papers: Formal group laws, Algebraic Bethe Ansatz, Hecke algebras

1/27/2021 at 2:00 pm PT Amol Aggarwal The Stochastic Six-Vertex Model

This expository talk will concern various aspects of the stochastic six-vertex model. In particular, we will describe what sorts of asymptotic questions about this model integrable probabilists are interested in. Moreover, we will try to (at least partially) outline how its Yang-Baxter integrability can be useful in answering some of these questions.

1/20/2021 at 2:00 pm PT Daniel Bump Solvable lattice models and representations of $p$-adic groups

Solvable lattice models are statistical mechanical systems that can be studied exactly by a method of Baxter, based on the Yang-Baxter equation. This can be understood in terms of quantum groups. Recently particular examples showing symmetry with respect to the Lie super group $U_q(\widehat{\mathfrak{gl}}(r|n))$ arose in two very different contexts: the representation theory of $p$-adic groups, where such models were used by Brubaker, Buciumas, Bump and Gustafsson to study Iwahori Whittaker functions on metaplectic groups; and in integrable probability, in work recent of Aggarwal, Borodin and Wheeler. We will introduce the topic starting with Baxter's work on six vertex model, then look at more recent work explaining some of the ideas.