Hasse-schmidt Derivations On Grassmann Algebras: With Applications To Vertex Operators (impa Monographs)
by Letterio Gatto /
2016 / English / PDF
2.3 MB Download
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.
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.
