From solutions to linear ordinary differential equations to vertex operators
type: Item
instance of: scholarly article
title: From solutions to linear ordinary differential equations to vertex operators
author: Parham Salehyan
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2022-12-08T00:00:00Z
review text: The main lesson one learns from the paper under review is that there is no analysis at all behind linear Ordinary Differential Equations (ODEs), as long as one uses constant coefficients only. This probably was, and is, very well known as common folklore, but more officially and clearly emphasized in the paper by \textit{L. Gatto} and \textit{D. Laksov} [J. Algebra Appl. 15, No. 6, Article ID 1650109, 23 p. (2016; Zbl 1358.13027)], where one shows that there is a \(1-1\) correspondence between solutions to linear ODEs and solutions to linear recurrent relations with constant coefficients. Moreover, the isomorphism between such spaces of solutions is provided by a purely algebraic version of the classical \textit{Laplace transform}. The other quite known observation is that the celebrated vertex operators occurring in the boson-fermion correspondence, as explained by \textit{V. G. Kac} and \textit{A. K. Raina} [Bombay lectures on highest weight representations of infinite dimensional Lie algebras. Hackensack, NJ: World Scientific (1987; Zbl 0668.17012)], is nothing else than a way to phrase identities coming from the theory of symmetric functions, as one learns from the foundational book by \textit{I. G. Macdonald} [Symmetric functions and Hall polynomials. With contributions by A. V. Zelevinsky. Reprint of the 1998 2nd edition. Oxford: Oxford University Press (2015; Zbl 1332.05002)]. As a matter of fact, once one reads with little care the amusing paper under review, one recognizes that the Ariadne thread bringing the algebraic formalism of \textit{generic} linear ODEs (i.e. with indeterminate coefficients) to vertex operators is precisely the theory of symmetric functions, whose relationship with exterior algebras was heavily exploited by the reviewer in 2005 and, later, used by Laksov and other authors, focusing to a single exterior power, motivated by Schubert Calculus. Even the celebrated Kempf-Laksov determinantal formula, as in [\textit{G. Kempf} and \textit{D…
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 34A35
zbMATH DE Number: 7629792
zbMATH Keywords: linear ODEs
zbMATH Keywords: Boson-Fermion correspondence
zbMATH Keywords: vertex operators
reviewed by: Letterio Gatto
MaRDI profile type: Publication
full work available at URL: s00574-022-00317-5
OpenAlex ID: W4311608019
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DOI: 10.1007/S00574-022-00317-5
description: scientific article; zbMATH DE number 7629792