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IMPA Monographs

Hasse-Schmidt Derivations on Grassmann Algebras

With Applications to Vertex Operators

Authors: Gatto, Letterio, Salehyan, Parham

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  • Offers a comprehensive approach to advanced topics such as linear ODEs and generalized Wronskians, Schubert calculus for ordinary Grassmannians and vertex operators arising from the representation theory of infinite-dimensional Lie algebras
  • Examines topics within a common interdisciplinary framework provided by the notions of linear recurrent sequences and Hasse-Schmidt derivations on a Grassmann algebra
  • Provides a self-contained presentation of pioneering research material starting from elementary observations
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  • ISBN 978-3-319-31842-4
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  • ISBN 978-3-319-31841-7
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Softcover $59.99
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  • ISBN 978-3-319-81134-5
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About this book

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 authors

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 quantum), algebraic curves and moduli (families of Weierstrauss points and jets of line bundles on Gorenstein curves) and integrable systems from the algebro-geometrical point of view.
Parham Salehyan graduated from Sharif University of Technology, Iran, and received his PhD in mathematics from the IMPA, Brazil, in 2003. He holds a permanent position at the Department of Mathematics of São Paulo State University - Unesp, São José do Rio Preto Campus. His research interests lie in algebraic geometry, mainly in the theory of Weierstrauss points on families of curves and, more recently, in the algebro-combinatorial aspects related to Schubert calculus.


Reviews

“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)


Table of contents (7 chapters)

Table of contents (7 chapters)

Buy this book

eBook $44.99
$84.99 (listprice)
price for USA in USD (gross)
valid through July 31, 2020
  • ISBN 978-3-319-31842-4
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $59.99
$109.99 (listprice)
price for USA in USD
valid through July 31, 2020
  • ISBN 978-3-319-31841-7
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
Softcover $59.99
$109.99 (listprice)
price for USA in USD
valid through July 31, 2020
  • ISBN 978-3-319-81134-5
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Hasse-Schmidt Derivations on Grassmann Algebras
Book Subtitle
With Applications to Vertex Operators
Authors
Series Title
IMPA Monographs
Series Volume
4
Copyright
2016
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-31842-4
DOI
10.1007/978-3-319-31842-4
Hardcover ISBN
978-3-319-31841-7
Softcover ISBN
978-3-319-81134-5
Edition Number
1
Number of Pages
XXIV, 195
Number of Illustrations
16 b/w illustrations
Topics

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