Hasse-Schmidt Derivations on Grassmann Algebras

1. Front Matter

   Pages i-xxiv

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2. Prologue

   • Letterio Gatto, Parham Salehyan

   Pages 1-22

3. Generic Linear Recurrence Sequences

   • Letterio Gatto, Parham Salehyan

   Pages 23-49

4. Algebras and Derivations

   • Letterio Gatto, Parham Salehyan

   Pages 51-62

5. Hasse–Schmidt Derivations on Exterior Algebras

   • Letterio Gatto, Parham Salehyan

   Pages 63-80

6. Schubert Derivations

   • Letterio Gatto, Parham Salehyan

   Pages 81-120

7. Decomposable Tensors in Exterior Powers

   • Letterio Gatto, Parham Salehyan

   Pages 121-146

8. Vertex Operators via Generic LRS

   • Letterio Gatto, Parham Salehyan

   Pages 147-183

9. Back Matter

   Pages 185-195

   PDF

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.

• Grassmann algebras
• Schubert calculus
• Vertex operators
• Symmetric functions
• KP hierarchy
• Hasse-Schmidt derivations
• Exterior algebra
• matrix theory
• ordinary differential equations

“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 highlightingfurther 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)

• Mathematical Sciences, Politecnico di Torino , Torino, Italy

  Letterio Gatto

• São José do Rio Preto Campus, São Paulo State University São José do Rio Preto Campus, São José do Rio Preto, Brazil

  Parham Salehyan

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.

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