Slava Naprienko -> Solvable Lattice Models Seminar
A state in a solvable lattice model representing a value of a metaplectic Iwahori Whittaker function
A state in a solvable lattice model representing a value of a metaplectic Iwahori Whittaker function. (source.)

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

Videos from the seminar are available on YouTube.

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

Date Speaker Title Abstract and Materials
6/23/2021 at 2:00 pm PT Alisa Knizel Stationary measure for the open KPZ equation:

The Kardar-Parisi-Zhang (KPZ) equation is the stochastic partial differential equation that models stochastic interface growth. I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. This construction is a joint work with Ivan Corwin. In the talk I will mostly focus on the algebraic tools that are used in the study of stationary measures.

Zoom link: https://stanford.zoom.us/j/91274693500?pwd=bmtHZnhTMG1HQ3pTOHYxUWJ2Z1ZjQT09
Zoom ID: 912 7469 3500
Password: 3628800 (= 10!).

6/16/2021 at 2:00 pm PT Arun Ram Level 0 modules of affine Lie algebras

The favourite R-matrices and transfer matrices (which give the 6 vertex model) arise from the evaluation representation of the standard representation of the quantum group of sl(n). This is a level 0 representation of quantum affine sl(n) and the lattice models are based on tensor products of this representation. With Finn McGlade and Yaping Yang we have written a survey about the classification of these modules (by dominant weights for extremal weight modules and by Drinfeld polynomials for finite dimensional modules), their characters (which are q-Whittaker functions) and their crystals. I will try to sketch how I think this category (of level 0 modules) is the controlling structure for vertex operators, Fock spaces (Kyoto path model) and the Algebraic Bethe ansatz.

video recording | notes

6/9/2021 at 2:00 pm PT Andy Hardt Hamiltonian operators and free fermionic lattice models

We will explore Hamiltonian operators from the “symmetric function” perspective favored by Lam (arXiv:0507341). In the same way that Schur polynomials can be built up from power sum symmetric functions, we can produce other symmetric functions by replacing the power sums with other functions. The symmetric functions we obtain, which include Macdonald and LLT polynomials, will have “Schur-like” identities such as Cauchy and Pieri rules.

A question we will ask is “When can a lattice model partition function also be obtained from a Hamiltonian operator?” In the six-vertex case, the condition turns out to be that the Boltzmann weights are free fermionic. If we consider lattice models with charge, we obtain a similar condition that suggests that the weights of Brubaker, Bump, Buciumas, and Gustafsson (arXiv:1806.07776) may be (essentially) the only set of weights that corresponds to a Hamiltonian operator.

video recording (YouTube) | notes

Relevant papers: A combinatorial generalization of the Boson-Fermion correspondence

6/2/2021 at 2:00 pm PT Andrew Gitlin A vertex model for LLT polynomials

We describe a novel Yang-Baxter integrable vertex model, from which we construct a certain class of partition functions that are equal to the LLT polynomials of Lascoux, Leclerc, and Thibon. Using the vertex model formalism, we are able to prove many properties of these polynomials, including symmetry and a Cauchy identity. This is based on joint work with Sylvie Corteel, David Keating, and Jeremy Meza.

video recording (YouTube) | notes

Relevant papers: A vertex model for LLT polynomials, Colored Fermionic Vertex Models and Symmetric Functions, A Shuffle Theorem for Paths Under Any Line, A Combinatorial Formula for Macdonald Polynomials, Equivalences of LLT polynomials via lattice paths; Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties; Ribbon Tableaux and the Heisenberg Algebra, Ribbon lattices and ribbon function identities

5/26/2021 at 2:00 pm PT Nikolaos Zygouras Whittaker functions through polymer models and geometric RSK

I will describe how Whittaker functions related to both GLn(R) and SO2n+1(R) arise from a solvable random polymer model. The main tool towards this is A.N.Kirillov’s geometric lifting of the Robinson-Schensted-Knuth correspondence and certain variants of this, which include the geometric lifting of the Burge correspondence. A by-product of these studies is a combinatorial derivation of the Bump-Stade identities. In the core of this approach lies a volume preserving property of the geometric RSK and geometric Burge, which merits deeper investigation. This will be an overview of some works in collaboration with Bisi, O’Connell, Sepp ̈al ̈ainen.

video recording (YouTube) | notes

Relevant papers: Geometric RSK correspondence, Whittaker functions andsymmetrized random polymers; The geometric burge correspondence and the partition function of polymer replicas; Transition between characters of classical groups, decomposition of Gelfand-Tsetlin pattern and last passage percolation; Point-to-line polymers and orthogonal Whittaker functions; Introduction to tropical combinatorics; ArchimedeanL-factors on $GL(n) \times GL(n)$ and generalized Barnes integrals

5/19/2021 at 2:00 pm PT Daniel Bump Vertex Operators and Solvable Lattice Models

Vertex operators originally arose in mathematical physics (string theory and soliton theory). They were applied to construct representations of affine Lie algebras by I. Frenkel and Kac, and an important algebraic fact emerged, the "boson-fermion correspondence". This concerns a Hamiltonian operator on the "fermionic Fock space". In some cases such a Hamiltonian can be related to the row transfer matrix for a solvable lattice model. The archetype for such a relation is Baxter's work relating the XYZ Hamiltonian with the 8-vertex model. A recent (2017) example is arXiv:1806.07776 where Brubaker, Buciumas, Bump and Gustafsson proved that the row transfer matrices of certain models motivated by the theory of metaplectic Whittaker functions could be realized by vertex operators on a version of the fermionic Fock space invented by Kashiwara, Miwa and Stern, that was previously applied by Lascoux, Leclerc and Thibon in the theory of ribbon symmetric functions.

video recording (YouTube) | notes

Relevant papers: Vertex operators, solvable lattice models and metaplectic Whittaker functions

5/12/2021 at 2:00 pm PT Katherine (Katy) Weber Lattice models as Poincaré pairings

Geometric constructions of quantum groups and their associated R-matrices arose in the early 90's and have been generalized further in recent works of Maulik, Okounkov, and others, creating a bridge between geometry and integrable lattice models. One nice aspect of this bridge is that the "hard" basis of one theory corresponds to the "easy" basis of the other.

We will analyze some of the lattice models we have already encountered in this seminar from this perspective. We first describe how both the torus fixed point basis and the basis of Schubert classes in the equivariant cohomology of the flag variety are manifest in the "Frozen Pipes" lattice model. This analysis is a straightforward generalization of results in a paper of Gorbunov, Korff, and Stroppel (see also the notes of Zinn-Justin) for the Grassmannian.

Then we describe how the fixed point basis and the basis of motivic Chern classes in the equivariant K-theory of the cotangent bundle of the flag variety appear (in a more novel way) in the Tokuyama model and colored Iwahori Whittaker model. Recent work of Aluffi, Mihalcea, Schürmann, and Su identifies these geometric bases with the Casselman and standard bases, respectively, of the Iwahori fixed vectors in the principal series representation, so this perspective allows us to make contact with formulas from p-adic representation theory, such as the Gindikin-Karpelevich formula and Bump-Nakasuji-Naruse conjecture. These ideas will be detailed in my forthcoming doctoral thesis.

video recording (YouTube) | notes

Relevant papers: Maulik and Okounkov, a more accessible summary of Maulik and Okounkov, Rimanyi (h-deformed Schubert calculus), Gorbunov, Korff, and Stroppel, Zinn-Justin's notes, Tokuyama model, Iwahori Whittaker model, Frozen Pipes, Motivic Chern classes and Iwahori invariants

5/5/2021 at 2:00 pm PT Siddhartha Sahi Quasi-polynomial representations of double affine Hecke algebras and a generalization of Macdonald polynomials

Macdonald polynomials are a remarkable family of functions. They are a common generalization of many different families of special functions arising in the representation theory of reductive groups, including spherical functions and Whittaker functions.

In turn, Macdonald polynomials can be understood in terms of a certain representation of Cherednik's double affine Hecke algebra (DAHA), acting on polynomial functions on a torus.

Whittaker functions admit a natural generalization to the setting of metaplectic covers of reductive p-adic groups, which play a key role in the theory of Weyl group multiple Dirichlet series.

It turns out that Macdonald polynomials also admit a corresponding generalization, which can be understood in terms of a representation of the DAHA on the space of quasi-polynomial functions on a torus.

This is joint work with Jasper Stokman and Vidya Venkateswaran.

video recording (YouTube) | notes

Relevant papers: Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

4/28/2021 at 2:00 pm PT Travis Scrimshaw Refined dual Grothendieck polynomials from 3 perspectives

Dual Grothendieck polynomials are the symmetric functions that are dual (under the Hall inner product, where the famous Schur polynomials give an orthonormal basis) to the symmetric Grothendieck polynomials, which are used to describe the K-theory ring of the Grassmannian. We can introduce extra parameters to obtain the refined dual Grothendieck polynomials. In this talk, we will translate the combinatorics of refined dual Grothendieck polynomials into the language of integrable lattice models. We then use lattice model techniques to derive a number of identities. We conclude by relating refined dual Grothendieck polynomials to TASEP through the last-passage percolation random matrices.

video recording (YouTube) | notes

Relevant papers: Refined dual Grothendieck polynomials, integrability, and the Schur measure

4/21/2021 at 2:00 pm PT Donghyun Kim Schubert polynomials and the inhomogeneous TASEP on a ring

The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which adjacent numbers i and j swap places at rate x_i - y_j if the larger number is clockwise of the smaller. Conjecturally, steady state probabilities can be written as a positive sum of (double) Schubert polynomials. We will start by giving some background on this model, including Cantini's result showing that the inhomogeneous TASEP is a solvable lattice model. We will then use his result to show that a large number of states -- those corresponding to the "evil-avoiding" permutations (permutations avoiding patterns 2413, 4132, 4213, 3214) -- have steady state probabilities which are proportional to a product of Schubert polynomials.

Based on joint work with Lauren Williams.

video recording (YouTube) | notes

4/14/2021 at 2:00 pm PT Sergei Korotkikh Local relations and q-moments of height functions of stochastic vertex models

I will talk about the stochastic six-vertex model from the integrable probability point of view. Namely, I will describe a new way to use the model's solvability by focusing on a specific recurrence relation (called local relation) on q-deformed moments of the height function. This relation leads to a short but non-constructive proof of the integral expressions for these moments. I will also sketch recent results about the colored stochastic six-vertex model, obtained using this technique.

Based on joint work with Alexey Bufetov.

video recording (YouTube) | notes

4/07/2021 at 2:00 pm PT Jason Saied Alcove walk formula for SSV polynomials

SSV polynomials are a new family of polynomials discovered by Sahi, Stokman, and Venkateswaran, generalizing both Macdonald polynomials and metaplectic Iwahori Whittaker functions. Similarly to both of these families, SSV polynomials satisfy a recursion coming from a Hecke algebra representation. We will use this recursion to give a combinatorial formula for SSV polynomials in terms of alcove walks, generalizing Ram and Yip's formula for Macdonald polynomials.

video recording (YouTube) | notes

Relevant papers: A combinatorial formula for SSV polynomials, Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials, A combinatorial formula for Macdonald polynomials, Metaplectic Iwahori Whittaker functions and supersymmetric lattice models

3/24/2021 at 2:00 pm PT Valentin Buciumas Solvable lattice models and special functions appearing in algebraic geometry

In this talk I will survey some applications of solvable lattice models to the study of special functions appearing in algebraic geometry, focusing on double Grothendieck polynomials. I will discuss some applications, like deriving combinatorial formulas for such functions. Time permitting, I will also mention structure coefficients.

video recording (YouTube) | notes

Relevant papers: ICERM slides on Schubert Calculus, Double Grothendieck polynomials and colored lattice models, Frozen Pipes: Lattice Models for Grothendieck Polynomials, Littlewood-Richardson coefficients for Grothendieck polynomials

3/17/2021 at 2:00 pm PT Kohei Motegi Izergin-Korepin analysis on wavefunctions

The Izergin-Korepin analysis is originally a method to determine the exact forms of the domain wall boundary partition functions of the six-vertex model, which was originated in the works by Korepin and Izergin. In this talk, I will present the Izergin-Korepin analysis on the wavefunctions which are a larger class of partition functions. I will first explain the prototype case in detail, and then explain the case with reflecting boundary and show some applications.

video recording (YouTube) | notes

Relevant papers: Izergin-Korepin analysis, Quantum inverse scattering method, Metaplectic Whittaker Functions and Crystals of Type B

3/10/2021 at 2:00 pm PT Slava Naprienko How to Come Up with Solvable Lattice Models?

Values of matrix coefficients of p-adic groups can be written in terms of solvable lattice models. But the usual argument for that is ad hoc -- you first know the model and then show that the partition functions match the values of matrix coefficients. In my talk, I'll show how one can start with the p-adic side and naturally come up with solvable lattice models. Since these models are closely related to ones introduced in probability theory, I hope that it will give a new way to come up with models for integrable probability. My results extend Peter J. McNamara's approach from 0907.2675.

video recording (YouTube) | notes

Relevant papers: Metaplectic Whittaker Functions and Crystal Bases, Metaplectic Iwahori Whittaker Functions and Colored Crystal Bases (available by request: naprienko@stanford.edu)

3/3/2021 at 2:00 pm PT Alexey Bufetov Interacting particle systems and random walks on Hecke algebras

Multi-species versions of several interacting particle systems, including ASEP, q-TAZRP, and k-exclusion processes, can be interpreted as random walks on Hecke algebras. In the talk I will discuss this connection and its probabilistic applications.

video recording (YouTube) | notes

Relevant papers: Interacting particle systems and random walks on Hecke algebras, Color-position symmetry in interacting particle systems

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.

video recording (YouTube) | notes

Relevant 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.

video recording | notes

Relevant papers: Colored five-vertex models and Demazure atoms, Colored Vertex Models and Iwahori Whittaker Functions, Metaplectic Iwahori Whittaker functions and supersymmetric lattice models

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.

video recording (YouTube) | notes

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.

video recording (YouTube) | notes

Relevant 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.

video recording (YouTube) | notes

Relevant papers: Limit shapes and fluctuations, limit shape and local statistics, framework for evaluating q-moments

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.

video recording (YouTube) | notes

Relevant papers: Tokuyama model for Spherical Whittaker Function, Colored Vertex Models and Iwahori Whittaker Functions, Metaplectic Iwahori Whittaker functions and Supersymmetric Lattice Models