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#Introduction

From [Kar22]: A lattice model is based on an edge-labeled graph (usually a rectangular grid) with some local constraints. Each vertex is assigned a weight, depending on its adjacent edges. If every vertex satisfies the constraints, we say that this assignment is an admissible state. The goal is to conclude some sort of global behavior.

The partition function of a lattice model, is the sum over all admissible states: \[Z = \sum_{S \in \mathrm{ states}} \prod_{v \in V} \mathrm{weight}_S(v).\]

Depending on the conditions, various functions and identities may be recovered. For example, we can obtain the Schur functions, the dual Cauchy identity, etc.

From now on, we assume that we are in the rectangular grid setting (each vertex has four adjacent neighbors) unless stated otherwise.

#Vertex models

The terms five-vertex model, six-vertex model or eight-vertex model, refer to the number of admissible local labelings around a vertex. For example, the figure below illustrates the six labelings in the six-vertex model.

An admissible local labeling for the 6-vertex model.

An admissible local labeling for the 6-vertex model.

An admissible local labeling for the 6-vertex model.

An admissible local labeling for the 6-vertex model.

An admissible local labeling for the 6-vertex model.

An admissible local labeling for the 6-vertex model.

Each such state is then assigned a weight, so in the \(n\)-vertex model, there are \(n\) possible weights associated with a vertex. This weight is sometimes referred to as the Boltzmann weight, and the matrix which records the weights is called the R-matrix.

It is usually interesting to compute the partition function with some fixed boundary condition or wall condition that is, labels on the edges exiting the rectangular grid.

#The Yang–Baxter equation

In order to discuss the Yang–Baxter equation, the edges in the graph can be seen as forming strands, or braidings.

The Yang–Baxter equation is a relation that describes how certain quantities are preserved as strands are permuted. This is useful for proving symmetries.

Models which satisfy the Yang–Baxter equation are called integrable or exactly solvable.

#Symmetric functions via lattice models

In [CFYZ+21], the authors construct a 6-vertex model for (supersymmetric) LLT polynomials and they prove a Cauchy identity for the spin LLT polynomials \(\LLTG^{(k)}_{\lambda/\mu}(\xvec;q).\) The same model is considered in [Har21], and it is shown that the partition function is more or less unique in a certain sense.

For metaplectic Whittaker functions, see [BBBG20]. For a broad introduction to the interplay between integrability and combinatorics, see [Zin24].

For lattice models for Grothendieck polynomials, see [BFHT+20, BS20].

A 5-vertex lattice model for Demazure character is given in [Yan25]. A few years earlier, 5-vertex model for Demazure atoms was described in [BBBG21].

K. Matveev constructs positivity-preserving operators from the stochastic six-vertex model and applies them to Hall–Littlewood positivity [Mat23]. The same framework is related to \(t\)-deformed Schensted insertions.

A. Kuniba, M. Okado, and T. Scrimshaw construct a \(t\)-oscillator weighted five-vertex model for the stationary states of the multispecies ASEP on a ring [KOS24]. The model gives a partition-function interpretation of the multiline queue construction and clarifies its relation with corner transfer matrices.

In [GWZ25], the authors construct a lattice model for a new family of symmetric functions, which unifies \(q\)-Whittaker polynomials, inhomogeneous \(q\)-Whittaker polynomials, Grothendieck polynomials and their duals.

A. Aggarwal, A. Borodin, and M. Wheeler construct colored fermionic vertex models whose partition functions give symmetric functions with Cauchy and branching identities [ABW21]. Their specializations include LLT and factorial LLT polynomials, while a cylindrical version gives partition functions for symmetric and nonsymmetric Macdonald polynomials.

L. Johnston, E. Nguyen, and A. Schilling realize RSK and uncrowding directly in a five-vertex model and construct the associated crystal structure [JNS26]. This connects the vertex-model partition functions to tableaux and Grothendieck-polynomial combinatorics.

A. Garbali, J. d. Gier, W. Mead, and M. Wheeler define symmetric functions from the stochastic six-vertex model in half-space with integrable boundary weights [GGMW24]. Double-row commutation relations give a skew Cauchy identity, and a degeneration gives a Pfaffian partition-function formula generalizing G. Kuperberg’s formula for symmetry classes of alternating sign matrices.

A. Aggarwal, A. Borodin, L. Petrov, and M. Wheeler define two families of symmetric rational functions as partition functions of the fully inhomogeneous free-fermion six-vertex model [ABPW23]. These functions generalize Schur, factorial Schur, and supersymmetric Schur polynomials, and satisfy Cauchy-type identities from the Yang–Baxter equation.

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