Springer Hasse-Schmidt Derivations on Grassmann Algebras: With Applications to Vertex Operators: 4 (IMPA Monographs, 4)

This book provides a comprehensive advanced multi-linear algebra course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra (an analogue of the Taylor expansion for real-valued functions), and shows how this notion provides a natural framework for many ostensibly unrelated subjects: traces of an endomorphism and the Cayley-Hamilton theorem, generic linear ODEs and their Wronskians, the exponential of a matrix with indeterminate entries (Putzer's method revisited), universal decomposition of a polynomial in the product of two monic polynomials of fixed smaller degree, Schubert calculus for Grassmannian varieties, and vertex operators obtained with the help of Schubert calculus tools (Giambelli's formula). Significant emphasis is placed on the characterization of decomposable tensors of an exterior power of a free abelian group of possibly infinite rank, which then leads to the celebrated Hirota bilinear form of the Kadomtsev-Petviashvili (KP) hierarchy describing the Plücker embedding of an infinite-dimensional Grassmannian. By gathering ostensibly disparate issues together under a unified perspective, the book reveals how even the most advanced topics can be discovered at the elementary level. Review “It provides a comprehensive advanced multi-linear algebra course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra … . It covers a wealth of important material in a concise, nevertheless instructive manner, and as such it may serve as an excellent guide to further, more advanced and detailed reading in this fundamental area of contemporary mathematics.” (Ahmed Lesfari, Mathematical Reviews, June, 2017) “It is entirely self-contained, and at the same time advanced in that it touches on many different areas in the fields of differential equations and mathematical physics and has further notes and references at the end of every chapter, as well as exercises highlighting further connections. … This book will be welcomed not only by scholars interested in generalized global coordinate-free settings, but also by students wishing to become acquainted with advanced areas of multilinear algebra and their applications.” (Rabe von Randow, zbMATH 1350.15001, 2017) From the Back Cover This book provides a comprehensive advanced multi-linear algebra course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra (an analogue of the Taylor expansion for real-valued functions), and shows how this notion provides a natural framework for many ostensibly unrelated subjects: traces of an endomorphism and the Cayley-Hamilton theorem, generic linear ODEs and their Wronskians, the exponential of a matrix with indeterminate entries (Putzer's method revisited), universal decomposition of a polynomial in the product of two monic polynomials of fixed smaller degree, Schubert calculus for Grassmannian varieties, and vertex operators obtained with the help of Schubert calculus tools (Giambelli's formula). Significant emphasis is placed on the characterization of decomposable tensors of an exterior power of a free abelian group of possibly infinite rank, which then leads to the celebrated Hirota bilinear form of the Kadomtsev-Petviashvili (KP) hierarchy describing the Plücker embedding of an infinite-dimensional Grassmannian. By gathering ostensibly disparate issues together under a unified perspective, the book reveals how even the most advanced topics can be discovered at the elementary level. About the Author Letterio Gatto received his PhD in mathematics from the University of Torino in 1993, and since then has held permanent positions at the Department of Mathematical Sciences of the Politecnico di Torino. He is currently associate professor at the Politecnico, where he offers courses on linear algebra and geometry for students in Engineering. His research interests range from Schubert calculus (classical, equivariant and qua