Yangian Gelfand-Zetlin bases, \({\mathfrak gl}_N\)-Jack polynomials and computation of dynamical correlation functions in the spin Calogero-Sutherland model
type: Item
instance of: scholarly article
title: Yangian Gelfand-Zetlin bases, \({\mathfrak gl}_N\)-Jack polynomials and computation of dynamical correlation functions in the spin Calogero-Sutherland model
zbMATH Open document ID: 0962.17012
author: Denis Uglov
published in: Communications in Mathematical Physics
publication date: 1998-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/9702020
review text: We consider the \({\mathfrak {gl}}_N\)-invariant Calogero-Sutherland models with \(N= 1,2,3,\dots\) in the framework of symmetric polynomals. In this framework it becomes apparent that all these models are manifestations of the same entity, which is the commuting family of Macdonald operators. Macdonald operators depend on two parameters \(q\) and \(t\). The Hamiltonian of the \({\mathfrak {gl}}_N\)-invariant Calogero-Sutherland model belongs to a degeneration of this family in the limit when both \(q\) and \(t\) approach an \(N\)th elementary root of unity. This is a generalization of the well-known situation in the case of the scalar Calogero-Sutherland model \((N=1)\). In the limit the commuting family of Macdonald operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the \({\mathfrak {gl}}_N\)-invariant Calogero-Sutherland model. The limits of Macdonald polynomials which we call \({\mathfrak {gl}}_N\)-Jack polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The \({\mathfrak {gl}}_N\)-Jack polynomials describe the orthogonal eigenbasis of the \({\mathfrak {gl}}_N\)-invariant Calogero-Sutherland model in exactly the same way as Jack polynomials describe the orthogonal eigenbasis of the scalar model \((N=1)\). For each known property of Macdonald polynomials there is a corresponding property of \({\mathfrak {gl}}_N\)-Jack polynomials. As a simplest application of these properties we compute two-point dynamical spin-density and density correlation functions in the \({\mathfrak {gl}}_2\)-invariant Calogero-Sutherland model at integer values of the coupling constant.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 33D80
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 05E35
Mathematics Subject Classification ID: 17B81
Mathematics Subject Classification ID: 81V70
zbMATH DE Number: 1213871
zbMATH Keywords: Jack polynomials
zbMATH Keywords: Calogero-Sutherland models
zbMATH Keywords: symmetric polynomals
zbMATH Keywords: Macdonald operators
zbMATH Keywords: Macdonald polynomials
zbMATH Keywords: Yangian Gelfand-Zetlin bases
zbMATH Keywords: Yangian action
MaRDI profile type: Publication
OpenAlex ID: W3105660635
arXiv ID: hep-th/9702020
DOI: 10.1007/S002200050283
description: scientific article