Particle Content of the (<em>k</em>, 3)-configurations
Evgeny Mukhin
Indiana University Purdue University Indianapolis, United StatesBoris Feigin
Independent University of Moscow, Russian FederationM. Jimbo
University of Tokyo, JapanTetsuji Miwa
Kyoto University, JapanYoshihiro Takeyama
Graduate School of Pure and Applied Sciences, Ibaraki, Japan

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.
Cite this article
Evgeny Mukhin, Boris Feigin, M. Jimbo, Tetsuji Miwa, Yoshihiro Takeyama, Particle Content of the (<em>k</em>, 3)-configurations. Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, pp. 163–220
DOI 10.2977/PRIMS/1145475969