DESCRIPTION
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.
CONTENT
1 Prologue
1.1 The KdV and the KP Equations
1.2 Vertex Operators and Affine Lie Algebras
1.3 Vertex Operators and Vertex Algebras
1.4 The KP Hierarchy via Vertex Operators
1.5 Notes and References
2 Generic Linear Recurrence Sequences
2.1 Sequences in Modules over Algebras
2.2 Generic Polynomials, Partitions and Schur Determinants
2.3 Generic Linear Recurrence Sequences
2.4 Cauchy Problems for Linear Recurrence Sequences
2.5 Formal Laplace Transform and Linear ODEs
2.6 Generalized Wronskians
2.7 Notes and References
2.8 Exercices
3 Algebras and Derivations
3.1 Tensor and Exterior Algebra of a Module
3.2 Exterior Algebra of a Free A-Module
3.3 Exterior Algebras Versus Clifford Algebras
3.4 Derivations on an Exterior Algebra
3.6 Notes and References
3.7 Exercises
4 Hasse-Schmidt Derivations on exterior Algebras
4.1 Main definitions and Properties
4.2 the Trace Operator Polynomials
4.3 The Exponentioal of an Endomorphism
4.4 Notes and References
4.5 Exercises
5 Schubert Derivations
5.1 Generalities on Schubert Derivations
5.2 Pieri Formula for Schubert Derivations
5.3 The Grassmannian and Its Plüker Embedding
5.4 Relationship with Schubert Calculus
5.5 Examples
5.6 Module Structures Induced by Shubert Derivations, I
5.7 Module Structures Induced by Schubert Derivations, II
5.8 Giambelli Formula
5.9 Application to Modules of Finite Rank
5.10 Notes and References
5.11 Exercises
6 Decomposable Tensors in Esterior Powers
6.1 A Criterion for Decomposability
6.2 A Schubert Derivation with a Stability Property
6.3 On Vertex-Like Operators
6.4 Plücker Equations for Grassmann Cones
6.5 On the Infinite Exterior Power I
6.6 Notes and References
6.7 Exercises
7 Vertex Operators via Generic LRS
7.1 Bosonic and Fermionic Fock Spaces of Finite Order
7.2 The Finite Order Boson-Fermion Correspondence
7.3 On the Infinite Exterior Power II
7.4 On the Finite-Dimensional Heisenberg Algebra
7.5 Schubert Derivations on the Fermionic Modules
7.6 Extending the Boson-Fermion Correspondence
7.7 Computing Truncated Vertex Operators
7.8 Vertex Operators in the Classical Boson-Fermion Correspondence
7.9 Notes and References
1.10 Exercises
References
Index
ABOUT THE AUTHORS
Letterio Gatto
Parham Salehyan
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