Particle Content of the $(\mathbf{k},3)$-configurations

Abstract

For all $k$, we construct a bijection between the set of sequences of non-negative integers $\mathbf{a}=(a_i{)}_{i\in {\mathbf{Z}}_{\ge 0}}$ satisfying $a_i+a_{i+1}+a_{i+2}\le k$ and the set of rigged partitions $(\lambda ,\rho )$. Here $\lambda =({\lambda }_1,...,{\lambda }_n)$ is a partition satisfying $k\ge {\lambda }_1\ge \cdot \cdot \cdot \ge {\lambda }_n\ge 1$ and $\rho =({\rho }_1,...,{\rho }_n)\in {\mathbf{Z}}_{\ge 0}^n$ is such that ${\rho }_j\ge {\rho }_{j+1}$ if ${\lambda }_j={\lambda }_{j+1}$. One can think of $\lambda$ as the particle content of the configuration $\mathbf{a}$ and ${\rho }_j$ as the energy level of the $j$-th particle, which has the weight ${\lambda }_j$. The total energy ${\sum }_{i}i{a}_i$ is written as the sum of the two-body interaction term ${\sum }_{j<j^{\prime }}{A}_{{\lambda }_j,{\lambda }_j^{\prime }}$ and the free part ${\sum }_j{\rho }_j$. The bijection implies a fermionic formula for the one-dimensional configuration sums ${\sum }_{\mathbf{a}}{q}^{{\sum }_{i}i{a}_i}$. We also derive the polynomial identities which describe the configuration sums corresponding to the configurations with prescribed values for $a_0$ and $a_1$, and such that $a_i=0$ for all $i>N$.