From solutions to linear ordinary differential equations to vertex operators
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. Laksov}, Acta Math. 132, 153--162 (1974; Zbl 0295.14023)] was rephrased by \textit{D. Laksov} and \textit{A. Thorup} in terms of such a formalism, see [Indiana Univ. Math. J. 61, No. 3, 1253--1312 (2012; Zbl 1318.14050)]. Going back to the article under review, it begins by carefully recalling the natural isomorphisms between the vector space generated by the generalized Wronskians, in the sense of \textit{L. Gatto} and \textit{I. Scherbak} [in: Contributions to algebraic geometry. Impanga lecture notes. Based on the Impanga conference on algebraic geometry, Banach Center, Bȩdlewo, Poland, July 4--10, 2010. Zürich: European Mathematical Society (EMS). 257--295 (2012; Zbl 1256.14049)] and the \(r\)-th exterior power of the module of solutions to a linear ODE. Working with exterior powers all at once, i.e. with exterior algebras, one is supplied with contraction and wedging operators, allowing to move from linear ODEs of a given order to one of order one more or one less. The author uses these natural and classic operators, to define some vector-space homomorphisms \(B_r\to B_{r+1} \) and \(B_r\to B_{r-1}\), where \(B_r\) stands for \(\mathbb{Q}\)-vector space generated by the indeterminate coefficients of a generic linear ODE of order \(r\). A nice expression, holding for all \(r\geq 1\), including the ostensibly (but not) trivial case \(r=1\), is computed. The expression for \(r=\infty\) recovers, as a particular case, the shape one learns on introductory books on vertex operators like, e.g., the celebrated by \textit{V. Kac} [Vertex algebras for beginners. 2nd ed. Providence, RI: American Mathematical Society (1998; Zbl 0924.17023)]. The article is divided into 7 sections, whose first is a very informative introduction, concerned with clearly explained motivations, background and goals. Section 2 is devoted to preliminaries intended to introduce the readers not aware of the subject to the adopted formalism. Section 3 is about the so called \textit{Schubert derivations}, while Section 4 is about the construction of what the author correctly calls a \textit{finite-type boson-fermion correspondence}. The main theorems are collected in Section 6, where the shape of these novel expressions (because defined for finite dimensional contexts) describing the boson-fermion correspondences are computed independently on \(r\), including the extremal case \(r=\infty\). This is detailedly shown in Section 7. It is fair to emphasizes that while the vertex operator expression which Section 7 is concerned with is very well known, that for finite \(r\) in Section 6 are new. This is beautiful indeed, because they look the same as those holding for \(r=\infty\), which at the same time shows a relationship of the subject with Schubert Calculus. The paper ends with a comprehensive reference list, mainly of papers by the reviewer who has massively worked on the subject for many years.
scientific article; zbMATH DE number 7629792