Representation of symmetric functions as Gram determinants
type: Item
instance of: scholarly article
title: Representation of symmetric functions as Gram determinants
zbMATH Open document ID: 0923.05049
author: Luis Verde-Star
published in: Advances in Mathematics
publication date: 1999-10-28T00:00:00Z
review text: Let \({\mathfrak Q}\) be the complex vector space of all rational functions in a complex variable \(z\). Using an alternating bilinear from \(\langle , \rangle\) introduced earlier [see Adv. Math. 116, No. 2, 377-388 (1995; Zbl 0845.16038)], the author defines a symmetric bilinear form \(S\) on \({\mathfrak Q}^{n+1}\) as follows: if \({\mathbf u}= (u_0,\dots, u_n)\), \({\mathbf v}= (v_0,\dots, v_n)\), then \(S({\mathbf u},{\mathbf v})= [\langle u_j, Rv_k\rangle]\), where \(R\) is the involution on \({\mathfrak Q}\) that sends \(f(z)\) to \(z^{-1}f(z^{-1})\). If, in particular, \(u_j= z^j/w(z)\) \((j= 0,\dots, n)\), where \(w(z)= (z- x_0)\cdots(z- x_n)\), then \(S({\mathbf u},{\mathbf v})\) is a symmetric function of the \(x_i\) for each choice of \({\mathbf v}\). Various classical symmetric functions, such as the \(S\)-functions, arise in this way. As an illustration of the method, two functional relations satisfied by \(S({\mathbf u},{\mathbf v})\) are shown to specialize to the Jacobi-Trudi formula for \(S\)-functions and Cauchy identity for \(\Pi(1- x_iy_j)^{-1}\). Clearly many variations can be played on the author's theme and he makes brief mention of a few of them.
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 1241238
zbMATH Keywords: rational functions
zbMATH Keywords: Gram determinant
zbMATH Keywords: symmetric function
zbMATH Keywords: Jacobi-Trudi formula
zbMATH Keywords: \(S\)-functions
reviewed by: G. E. Wall
MaRDI profile type: Publication
full work available at URL: aima.1998.1765
OpenAlex ID: W2038122197
cites work: Q3947818
cites work: Q3748279
cites work: Biorthogonal Polynomial Bases and Vandermonde‐like Matrices
cites work: A Hopf algebra structure on rational functions
cites work: Divided Differences and Linearly Recursive Sequences
cites work: An algebraic approach to convolutions and transform methods
DOI: 10.1006/AIMA.1998.1765
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description: scientific article; zbMATH DE number 1241238