Differential Invariants of Prehomogeneous Vector Spaces
Christian Barz
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Christian Barz, Differential Invariants of Prehomogeneous Vector Spaces (2019), Logos Verlag, Berlin, ISBN: 9783832588755
Beschreibung / Abstract
Differential invariants of prehomogeneous vector spaces studies in detail two differential invariants of a discriminant divisor of a prehomogeneous vector space. The Bernstein-Sato polynomial and the spectrum, which encode the monodromy and Hodge theoretic informations of an associated Gauss-Manin system.
The theoretical results are applied to discriminants in the representation spaces of the Dynkin quivers An, Dn, E6, E7 and three non classical series of quiver representations.
The theoretical results are applied to discriminants in the representation spaces of the Dynkin quivers An, Dn, E6, E7 and three non classical series of quiver representations.
Inhaltsverzeichnis
- BEGINN
- Introduction
- 1 Prehomogeneous vector spaces
- 1.1 Prehomogeneous vector spaces and relative invariants
- 1.2 The contragredient representation
- 1.3 Reductive prehomogeneous vector spaces and the Bernstein-Sato polynomial
- 1.4 The differential of the orbit map
- 2 The spectrum and the Birkhoff problem
- 2.1 The spectrum of a meromorphic connection with a lattice
- 2.2 The Birkhoff problem
- 3 Logarithmic vector fields and the logarithmic de Rham complex
- 3.1 Logarithmic forms and vector fields
- 3.2 The relative de Rham complex
- 4 Functions on prehomogeneous discriminants and their Milnor fibrations
- 4.1 R-finiteness of linear functions on discriminants
- 4.2 R-finiteness for reductive groups
- 4.3 Restrictions to the Milnor fiber
- 5 The Gauss-Manin System associated to a linear section of the Milnor fibration
- 5.1 The localized Gauss-Manin system
- 5.2 The partial Fourier Laplace transform of the Gauss-Manin system
- 6 The Bernstein-Sato polynomial
- 6.1 The Bernstein-Sato polynomial of the V-filtration
- 6.2 A comparision of the Bernstein-Sato polynomial with the spectral polynomial
- 7 Quivers and their representation theory
- 7.1 Quiver representations - an algebraic point of view
- 7.2 Quiver representations - a geometric point of view
- 7.3 Equations for the fundamental semi-invariants
- 7.4 Example - Finding equations for E6
- 8 Calculations for classical Singularities
- 8.1 The exponent condition and the invariant subspace conjecture
- 8.2 Semi-invariants and spectrum for An
- 8.3 Semi-invariants and spectrum for Dn
- 8.4 Semi-invariants and spectrum for E6
- 8.5 Semi-invariants and spectrum for E7
- 8.6 The serie Bn;l
- 9 The spectrum of the star quiver
- 9.1 A solution to the Birkhoff problem
- 9.2 Specializing the linear function
- 9.3 The spectrum of the star quiver
- 9.4 The exponent condition and the invariant subspace conjecture for the star quiver
- 10 The serie ExtkStarn
- 10.1 Constructing new Series
- 10.2 The fundamental semi-invariants of ExtkStarn
- 10.3 Calculations for ExtkStarn
- Bibliography