Particle Content of the $(\boldsymbol k, 3)$-configurations

Abstract

For all k, we construct a bijection between the set of sequences of non-negative integers a = (ai )i\_∈ℤ≥0 satisfying ai + ai+1 + ai+2 ≤ k and the set of rigged partitions (λ, ρ). Here λ = (λ1, . . . , λ\_n ) is a partition satisfying k ≥ λ1 ≥ · · · ≥ λ\_n\_ ≥ 1 and ρ = (ρ1, . . . , ρn ) ∈ ℤ\_n\_≥0 is such that ρj ≥ ρ\_\_j+1 if λ\_j\_ = λ\_j\_+1. One can think of λ as the particle content of the configuration a and ρj as the energy level of the j-th particle, which has the weight λ\_j\_. The total energy ∑\_i\_ i\_\_ai is written as the sum of the two-body interaction term ∑\_j\_\<j' A\_λ\_j,λ\_j'\_ and the free part ∑\_j\_ ρj. The bijection implies a fermionic formula for the one-dimensional configuration sums ∑a q\_∑\_i i\_\_ai. 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 ai = 0 for all i > N.