Particle content of the \((k,3)\)-configurations
type: Item
instance of: scholarly article
title: Particle content of the \((k,3)\)-configurations
zbMATH Open document ID: 1062.05010
author: Yoshihiro Takeyama
author: B. Feigin
author: M. Jimbo
author: T. Miwa
author: E. Mukhin
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2004-11-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0212348
review text: Summary: For all \(k\), we construct a bijection between the set of sequences of non-negative integers \({\mathbf a}=(a_i)_{i\in\mathbb Z_{\geq 0}}\) satisfying \(a_i+ a_{i+1}+a_{i+2}\leq k\) and the set of rigged partitions \((\lambda,\rho)\). Here \(\lambda=(\lambda_1,\dots, \lambda_n)\) is a partition satisfying \(k\geq \lambda_1\geq\cdots \geq \lambda_n\geq 1\) and \(\rho=(\rho_1,\dots,\rho_n)\in \mathbb{Z}^n_{\geq 0}\) is such that \(\rho_j\geq\rho_{j+1}\) if \(\lambda_j=\lambda_{j+1}\). One can think of \(\lambda\) as the particle content of the configuration \({\mathbf a}\) and \(\rho_j\) as the energy level of the \(j\)th particle, which has the weight \(\lambda_j\). The total energy \(\sum_iia_i\) is written as the sum of the two-body interaction term \(\sum_{j<j'}\) \(A_{\lambda_j, \lambda_j'}\) and the free part \(\sum_j \rho_j\). The bijection implies a fermionic formula for the one-dimensional configuration sums \(\sum_{\mathbf a}q^{\sum_iia_i}\). 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 \(i>N\).
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 82B23
zbMATH DE Number: 2113005
zbMATH Keywords: rigged partitions
zbMATH Keywords: polynomial identities
zbMATH Keywords: configuration sums
MaRDI profile type: Publication
OpenAlex ID: W2074033457
arXiv ID: math/0212348
cites work: Q4146776
DOI: 10.2977/PRIMS/1145475969
description: scientific article