Intertwining operators and polynomials associated with the symmetric group
type: Item
instance of: scholarly article
title: Intertwining operators and polynomials associated with the symmetric group
zbMATH Open document ID: 0912.33007
DOI: 10.1007/BF01367762
author: Charles F. Dunkl
published in: Monatshefte für Mathematik
publication date: 1999-05-10T00:00:00Z
full work available at URL: https://eudml.org/doc/178831
review text: Let \(T_i{:=}{\partial \over \partial x_i}+k\sum _{j\neq i} {1-\sigma _{ij} \over {x_i-x_j}}\), \(i=1,2,\dots ,N\), where \(k\) is a fixed parameter and \(\sigma _{ij}\) is the transposition of \(x_i\) and \(x_j\). These operators generate an algebra which is used in studying analytic structures invariant under permutation of coordinates. Let \(\{ p_\alpha \}\) be a family of functions indexed by \(m\)-tuples of non-negative integers \(\alpha =(\alpha _1,\alpha _2,\dots ,\alpha _m)\), \(m\leq N\). Let \(V\) be a linear map (an intertwining operator) preserving the degree of homogeneity for which \(T_iV=V {\partial \over \partial x_i}\). The author shows that \(T_ip_\alpha =0\) for \(i>m\) and gives the expression for \(V(x_1^{\alpha _1} \cdots x_m^{\alpha _m})\). The operator has triangular matrix representation. There exists an inner product structure on the span of \(\{ p_\alpha \}\) and a convenient set of self-adjoint operators \(\{ T_i\rho _i\} \), where \(\rho _i p_\alpha = p_{(\alpha _1,\dots ,\alpha _i+1,\dots ,\alpha _m)}\). This structure has a bi-orthogonal relationship with the Jack polynomials in \(m\) variables. For any \(a=1,2,\dots \) such that \(\text{ gcd} (N-m+1,a)< (N-m+1)/m\) and \(m\leq N/2\), the author constructs a space of polynomials annihilated by each \(T_i\) for \(k=-a/(N-m+1)\) and on which the symmetric group \(S_N\) acts according to the irreducible representation given by \((N-m,m)\).
reviewed by: A. U. Klimyk
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 33C70
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 1236106
zbMATH Keywords: intertwining operator
zbMATH Keywords: differential-difference operator
zbMATH Keywords: symmetric group
zbMATH Keywords: Jack polynomials
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1236106