Advertisement

Hasse-Schmidt Derivations on Grassmann Algebras

With Applications to Vertex Operators

  • Letterio Gatto
  • Parham Salehyan

Part of the IMPA Monographs book series (IMPA, volume 4)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Letterio Gatto, Parham Salehyan
    Pages 1-22
  3. Letterio Gatto, Parham Salehyan
    Pages 23-49
  4. Letterio Gatto, Parham Salehyan
    Pages 51-62
  5. Letterio Gatto, Parham Salehyan
    Pages 63-80
  6. Letterio Gatto, Parham Salehyan
    Pages 81-120
  7. Letterio Gatto, Parham Salehyan
    Pages 121-146
  8. Letterio Gatto, Parham Salehyan
    Pages 147-183
  9. Back Matter
    Pages 185-195

About this book

Introduction

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.

Keywords

Grassmann algebras Schubert calculus Vertex operators Symmetric functions KP hierarchy Hasse-Schmidt derivations Exterior algebra

Authors and affiliations

  • Letterio Gatto
    • 1
  • Parham Salehyan
    • 2
  1. 1.Mathematical SciencesPolitecnico di Torino TorinoItaly
  2. 2.São José do Rio Preto CampusSão Paulo State University São José do Rio Preto CampusSão José do Rio PretoBrazil

Bibliographic information