Equivalences of LLT polynomials via lattice paths

TL;DR

This paper establishes a bijection between vertex model configurations for LLT polynomials with swapped boundary conditions, leading to new equivalences and linear relations among these symmetric polynomials.

Contribution

It introduces an algorithm for a bijection between vertex model configurations with swapped boundary conditions, proving when LLT polynomials are equivalent up to a factor.

Findings

Bijection between configurations with swapped boundary conditions

Conditions for weight-preserving bijections and polynomial equivalences

Systematic determination of linear relations among LLT polynomials

Abstract

The LLT polynomials $\mathcal{L}_{\mathbf{\beta}/\mathbf{\gamma}} (X;t)$ are a family of symmetric polynomials indexed by a tuple of (possibly skew-)partitions $\mathbf{\beta}/\mathbf{\gamma}= (\beta^{(1)}/\gamma^{(1)},\ldots,\beta^{(k)}/\gamma^{(k)})$. It has recently been shown that these polynomials can be seen as the partition function of a certain vertex model whose boundary conditions are determined by $\mathbf{\beta}/\mathbf{\gamma}$. In this paper we describe an algorithm which gives a bijection between the configurations of the vertex model with boundary condition $\mathbf{\beta}/\mathbf{\gamma} = (\beta^{(1)}/\gamma^{(1)},\beta^{(2)}/\gamma^{(2)})$ and those with boundary condition $(\mathbf{\beta}/\mathbf{\gamma})_{swap} = (\beta^{(2)}/\gamma^{(2)},\beta^{(1)}/\gamma^{(1)})$. We prove a sufficient condition for when this bijection is weight-preserving up to an overall factor of $t$, which in turn implies that the corresponding LLT polynomials are equal up to the same overall factor. Using these techniques, we are also able to systematically determine linear relations within families of LLT polynomials.