The symmetric functions catalog

An overview of symmetric functions and related topics



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.

The 6 admissible local labelings 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 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 [CFYZZ21], the 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 is unique in a certain sense.

For metaplectic Whittaker functions, see [BBBG20].

Lattice models for Grothendieck polynomials, [BFHTW20BS20].