Solvable Lattice Models

• This class met during the Fall of 2023.

Solvable lattice models arose in statistical mechanics, where they were used to model systems such as ice and (via a relationship with quantum mechanics) Heisenberg spin chains. But beyond this origin in physics, solvable lattice models have emerged as a powerful multi-purpose tool. They have applications to algebraic combinatorics and related areas in algebraic geometry, to representations of p-adic groups, integrable probability and other areas. So this topic will be of interest beyond its origins in physics.

Baxter noticed that there is some machinery behind solvability based on the Yang-Baxter equation, and others, particularly Drinfeld, showed how the representation theory of quantum groups explains the Yang-Baxter equation. Exactly the same machinery underlies knot invariants such as the Jones polynomial.

We will try to strike a balance between examining the underlying theory, which comes from the representation theory of quantum groups, and looking at examples and applications.

We will discuss the instructive example of the six-vertex model in detail. There are two main kinds of integrable six-vertex models, corresponding to quantum affine $\mathfrak{gl}(2)$ and the superalgebra $\mathfrak{gl}(1|1)$. These are the field-free and free-fermionic six vertex models. Their theories are instructively different. The field free case is connected with realistic physical systems and also led to Kuperberg's famous proof of the alternating sign matrix conjecture. On the other hand, the free-fermionic case is the prototype of many applications in algebraic combinatorics. We will study both, and we will also study related bosonic models (including stochastic models). We will also look at the eight-vertex model. We will then study colored variants which are related to quantum affine $\mathfrak{gl}(n)$ and $\mathfrak{gl}(m|n)$. There we will see representations of affine Hecke algebras through Demazure-Lusztig operators. These same representations appear in other areas of mathematics, implying surprising connections.

Lecture Notes

Here are all the lectures as a single file:

• Solvable Lattice Models

Here are the individual lectures.

• Lecture 1. Introduction: from Ice to the six vertex model. Two examples of the Yang-Baxter equation.
• Lecture 2. Commuting row transfer matrices, following Baxter.
• Lecture 3. Gelfand Tsetlin patterns and six-vertex model. Vector space version of the Yang-Baxter equation.
• Lecture 4. Digression explaining that braided monoidal categories are the source of solutions to the Yang-Baxter equation. Parametrized Yang-Baxter equations. The field-free YBE revisited.
• Lecture 5. Tokuyama models I. These are free-fermionic models whose partition functions are related to Schur functions. We will use these to prove the equivalence of two definitions of the Schur function.
• Lecture 6. Tokuyama models II. By evaluating the partition functions of the Tokuyama models when $q=1$ and $q=0$ and comparing, we obtain the equivalence of two definitions of the Schur function.
• Lecture 7. The general free-fermionic Yang-Baxter equation with parameter group $GL(2)\times GL(1)$. Adding column parameters.
• Lecture 8. Bosonic Models I. Colored models I: the open model, and its Demazure recursion.
• Lecture 9. The open model II: the ground state. Some Lie theory, and more Demazure operators.
• Note: Lectures 8 and 9 were edited Friday October 27: the operator previously denoted $\delta_i$ is now $\delta_i^\circ$.
• Lecture 10. Bruhat order and the relationship between $\partial_w$ and $\partial_w^\circ$. Revised on October 31.
• Lecture 11. Open and Closed models, including a couple of open questions.
• Lecture 12. Hecke algebras, Demazure-Lusztig operators, and colored models depending on a parameter $q$. Hopf algebras.
• Lecture 13. Yang-Baxter equations and quantum groups. $U_q(\mathfrak{sl}_2)$.
• Lecture 14. Survey of some topics. Affine Lie groups, Weyl groups and Hecke algebras. PBW Theorem.
• Lecture 15. Verma modules and bosonic models.
• Lecture 16. Fusion. Colored bosonic models. Explanation in terms of Verma modules.
• Lecture 17. Heisenberg spin chains. The XXZ Hamiltonian and the six-vertex model.
• Lecture 18. Comparision of bosonic and fermionic models. Lie superalgebras and their Kac modules
• Lecture 19. The Fermionic Fock space. The Heisenberg Lie algebra. Row transfer matrices as vertex operators.
• Lecture 20. Fermionic operators. Conclusion of proofs from Lecture 19. Delta Ice and U-Turn models.

Exercises

• Exercises 1 about the open and closed models. Some exercises were added on Friday October 27.
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Code

• field-free1.sage. This sage program verifies the Yang-Baxter equation for the field free case (Lecture 4)
• tokuyama.sage. This sage program verifies the Yang-Baxter equation for Tokuyama models (Lecture 5)
• free-fermionic1.sage. This sage program verifies the Yang-Baxter equation for the general parametrized Yang-Baxter equation with parameter group $GL(2)\times GL(1)$ (Lecture 7)
• open.sage. This sage program verifies the Yang-Baxter equation for the open models (Lecture 8)
• sl2rmatrix.sage. This verifies the $U_q(\mathfrak{sl}_2)$ Yang-Baxter equation in Lecture 13, Proposition 4.3.
• sl2param.sage. This verifies the parametrized Yang-Baxter equation in Lecture 13, Proposition 5.1.

San Francisco

In January 2024, there will be a Special Session on Solvable Lattice Models at the Joint Mathematical Meetings.

• Program Schedule

Links

• Baxter: Exactly Solved Models in Statistical Mechanics at Baxter's home page.
• Brubaker, Bump and Friedberg: Schur Polynomials and the Yang-Baxter equation
• Kuperberg: Another Proof of the Alternating Sign Matrix Conjecture
• Naprienko: Integrability of the Six-Vertex model and the Yang-Baxter Groupoid