Equivalences of LLT polynomials via lattice paths

The LLT polynomials ${\mathcal{L}}_{\mathbf{\beta }\mathrm{/}\mathbf{\gamma }}(X;t)$ are a family of symmetric polynomials indexed by a tuple of (possibly skew-)partitions $\mathbf{\beta }\mathrm{/}\mathbf{\gamma }=({\beta }^{(1)}\mathrm{/}{\gamma }^{(1)},\dots ,{\beta }^{(k)}\mathrm{/}{\gamma }^{(k)})$. 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 $\mathbf{\beta }\mathrm{/}\mathbf{\gamma }$. In this paper we describe an algorithm which gives a bijection between the configurations of the vertex model with boundary condition $\mathbf{\beta }\mathrm{/}\mathbf{\gamma }=({\beta }^{(1)}\mathrm{/}{\gamma }^{(1)},{\beta }^{(2)}\mathrm{/}{\gamma }^{(2)})$ and those with boundary condition $(\mathbf{\beta }\mathrm{/}\mathbf{\gamma }{)}_{swap}=({\beta }^{(2)}\mathrm{/}{\gamma }^{(2)},{\beta }^{(1)}\mathrm{/}{\gamma }^{(1)})$. 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.