# Particle Content of the $(k,3)$-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 $a=(a_{i})_{i∈Z_{≥0}}$ satisfying $a_{i}+a_{i+1}+a_{i+2}≤k$ and the set of rigged partitions $(λ,ρ)$. Here $λ=(λ_{1},...,λ_{n})$ is a partition satisfying $k≥λ_{1}≥⋅⋅⋅≥λ_{n}≥1$ and $ρ=(ρ_{1},...,ρ_{n})∈Z_{≥0}$ is such that $ρ_{j}≥ρ_{j+1}$ if $λ_{j}=λ_{j+1}$. One can think of $λ$ as the particle content of the conﬁguration $a$ and $ρ_{j}$ as the energy level of the $j$-th particle, which has the weight $λ_{j}$. The total energy $∑_{i}ia_{i}$ is written as the sum of the two-body interaction term $∑_{j<j_{′}}A_{λ_{j},λ_{j}}$ and the free part $∑_{j}ρ_{j}$. The bijection implies a fermionic formula for the one-dimensional conﬁguration sums $∑_{a}q_{∑_{i}ia_{i}}$. 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 $i>N$.

## Cite this article

Evgeny Mukhin, Boris Feigin, M. Jimbo, Tetsuji Miwa, Yoshihiro Takeyama, Particle Content of the $(k,3)$-conﬁgurations. Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, pp. 163–220

DOI 10.2977/PRIMS/1145475969