# Particle Content of the (<em>k</em>, 3)-conﬁgurations

### Evgeny Mukhin

Indiana University Purdue University Indianapolis, United States### Boris Feigin

Independent University of Moscow, Russian Federation### M. Jimbo

University of Tokyo, Japan### Tetsuji Miwa

Kyoto University, Japan### Yoshihiro 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