Particle Content of the (<em>k</em>, 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.