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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Prologue

Abstract
This chapter lays down a non-technical and expository pathway to two non-linear PDEs: the Korteweg–de Vries (KdV) equation, modelling solitary waves, and the Kadomtsev–Petviashvili (KP) equation, a generalization of the KdV originally motivated by applications to plasma physics. Although the chapter’s contents might not seem related to the rest of the book in a straightforward manner, the KdV and KP equations are, surprisingly, linked to a number of subjects that any mathematician has had early close encounters with. To name but a few, cubic plane curves, linear recurrence sequences, linear ODEs and (generalized) Wronskians associated with fundamental systems of solutions, exterior algebras of free modules, Plücker embeddings of finite-dimensional Grassmannians and Schubert calculus. In sum, these two equations provide us with an excuse to return to the early stages of our scholarly education and shed a new light on it. The chapter culminates with the appearance, as a kind of deus ex machina, of the bosonic expression of the vertex operators occurring in the so-called vertex algebra of free charged fermions. These operators act on a polynomial ring in infinitely many indeterminates and encode the full system of PDEs known under the name of KP hierarchy, which arises as compatibility conditions for another system of infinitely many PDEs (expressed in Lax form, see [101, 140] or [78, p. 73]).
Letterio Gatto, Parham Salehyan

Chapter 2. Generic Linear Recurrence Sequences

Abstract
Let A be a commutative ring with unit and M any A-module. An M-valued linear recurrence sequence (LRS) generalizes the sequence of powers of the roots of a given monic polynomial with A-coefficients. A generic LRS is a linear recurrence sequence with indeterminate coefficients. A main character of this chapter, as of the entire book, is the ring \(B_{r}:= \mathbb{Z}[e_{1},\ldots,e_{r}]\), thought of as a free polynomial algebra generated by the coefficients of a generic LRS of order r ≥ 1. The name of the indeterminates, (e 1, , e r ), is reminiscent of the elementary symmetric polynomials in r variables. The ring B r will later be bestowed the more ambitious task of approximating the bosonic Fock representation of the oscillator Heisenberg algebra.
Letterio Gatto, Parham Salehyan

Chapter 3. Algebras and Derivations

Abstract
To render the text self-contained and also to set the notation, Section 3.1 recalls the basic notion of tensor and exterior algebra of a module. Readers may refer to [7, p. 24], [99, Ch. XVI, §7] and [99, Ch. XIX] for more complete explanations. Additional combinatorial details are worked out in Section 3.2 to describe the exterior algebra of a free module. Clifford algebras are introduced in Section 3.3 in the customary way, as quotients of the tensor algebra. Derivations on general Grassmann algebras are defined exactly as for other kinds of algebras (like commutative or non-associative): we overview them in Section 3.4 to emphasize that iterating derivations produces higher-order Leibniz rules, the main formal devices characterizing Hasse–Schmidt derivations. The definition of the latter, however, is postponed to the beginning of Chapter 4.​1 Throughout this chapter, unless otherwise stated, M will denote a module over a commutative ring A with unit, and all constructions involving M will be understood in the category of A-modules.
Letterio Gatto, Parham Salehyan

Chapter 4. Hasse–Schmidt Derivations on Exterior Algebras

Abstract
We are going to keep the same notation of Chapter 3, unless otherwise specified. The main goal of this chapter is to introduce the pivotal algebraic notion of Hasse–Schmidt (HS) derivation on an exterior algebra and develop its basic formalism, with emphasis on what we call integration by parts. The latter is a key tool and will be used heavily throughout. The original motivation comes from Schubert calculus (see, e.g. Chapter 5), a subject concerned with the intersection theory on Grassmann varieties, quickly revised in Section 5.​4 HS-derivations on Grassmann algebras also provide a natural framework to state and prove a generalization (as in [51]) of the classical celebrated Cayley–Hamilton theorem: any endomorphism is a root of its characteristic polynomial. This is shown in Section 4.2 and will be used in Chapter 5 to construct a structure of free B r -module out of any free abelian group of rank n  ≥ r. That construction will enable us to describe the cone of decomposable tensors in an exterior power by means of the same compact and elegant formula we advertised in the introduction. Section 4.3 offers a few simple applications to matrix exponentials and first integrals of linear ODEs with constant coefficients, by means of the same language of Section 2.​5
Letterio Gatto, Parham Salehyan

Chapter 5. Schubert Derivations

Abstract
Let A be a commutative ring with unit, p ∈ A[X] a monic polynomial of degree n > 0, M p the quotient of A[X] modulo (p) and \( b_{i}:= X^{i} + (p) \). The protagonist of this chapter is a distinguished HS-derivation on \( \bigwedge M_{p} \), called Schubert derivation. It is the unique HS-derivation \( \sigma _{+}(z):=\sum _{i\geq 0}\sigma _{i}z^{i} \) such that \( \sigma _{i}b_{j} = b_{i+j} \). The subscript ‘+’ of \( \sigma \) in the notation is to mark the difference from its relative \( \sigma _{-}(z):=\sum _{i\geq 0}\sigma _{-i}z^{-i} \), to be investigated in Chapter 6 Its connection with Schubert calculus, summarized in Section 5.4, where a few notions of intersection theory are outlined as well, motivates notation and terminology: it is based on Pieri- and Giambelli-type formulas enjoyed by the coefficients of the Schubert derivation. Pieri’s rule is treated in Section 5.2, in its quantum version as well, while Giambelli’s is proved in Section 5.8, basing on a flexible determinantal formula due to Laksov and Thorup as in [96, 97].
Letterio Gatto, Parham Salehyan

Chapter 6. Decomposable Tensors in Exterior Powers

Abstract
The geometrical picture presented in this chapter is essentially a reworking, within a finite-dimensional context, of the idea of writing Plücker equations for the infinite Grassmannian parametrizing the solutions of the KP hierarchy, classically due to Sato, Date, Jimbo, Kashiwara and Miwa [18, 19, 133, 134]. All this can be told in a purely algebraic way from the point of view of vertex operators [5, 73, 78, 79].
Letterio Gatto, Parham Salehyan

Chapter 7. Vertex Operators via Generic LRS

Abstract
This closing part of the book is concerned with the identification of the free abelian group M 0, studied in Chapters 5 and 6, with the \(\mathbb{Z}\)-module spanned by generic LRSs of finite order. The latter will be embedded in the larger free abelian group \(\mathbb{U}_{r}:=\bigoplus _{i\in \mathbb{Z}}\mathbb{Z} \cdot u_{i}\), where \((u_{i})_{i\in \mathbb{Z}}\) is the fundamental sequence of formal power series introduced in Section 2.3.1. Certain abelian subgroups F i r of semi-infinite exterior products of elements of \(\mathbb{U}_{r}\) are interpreted as approximations of fermionic Fock spaces in the sense of [78]. The index r keeps track that everything is constructed by means of formal power series with B r -coefficients, while the index i is called charge , in compliance with the terminology of standard reference texts on the subject. The vertex operators of the Prologue re-enter the game in the last section of this chapter, where the results of Chapter 6 are recast in terms of the natural basis of the LRSs of order r, encountered in Chapter 2
Letterio Gatto, Parham Salehyan

Backmatter

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