Geometric Invariant Theory and Generalized Eigenvalue Problem II

Geometric Invariant Theory and Generalized Eigenvalue Problem II

Let G be a connected reductive subgroup of a complex connected reductive group Ĝ. Fix maximal tori and Borel subgroups of G and Ĝ. Consider the cone LR◦(Ĝ,G) generated by the pairs (ν, ν̂) of strictly dominant characters such that Vν is a submodule of Vν̂ . The main result of this article is a bijective parametrisation of the faces of LR◦(Ĝ,G). We also explain when such a face is contained in ano...

Geometric Invariant Theory and Generalized Eigenvalue Problem

Let G be a connected reductive subgroup of a complex connected reductive group Ĝ. Fix maximal tori and Borel subgroups of G and Ĝ. Consider the cone LR(G, Ĝ) generated by the pairs (ν, ν̂) of dominant characters such that Vν is a submodule of Vν̂ (with usual notation). Here we give a minimal set of inequalities describing LR(G, Ĝ) as a part of the dominant chamber. In way, we obtain results about...

Gershgorin Theory for the Generalized Eigenvalue Problem Ax — \ Bx

A generalization of Gershgorin's theorem is developed for the eigenvalue problem Ax = XBx and is applied to obtain perturbation bounds for multiple eigenvalues. The results are interpreted in terms of the chordal metric on the Riemann sphere, which is especially convenient for treating infinite eigenvalues.

The Definite Generalized Eigenvalue Problem: A New Perturbation Theory

Let (A, B) be a definite pair of n × n Hermitian matrices. That is, |x∗Ax| + |x∗Bx| 6= 0 for all non-zero vectors x ∈ C. It is possible to find an n × n non-singular matrix X with unit columns such that X∗(A + iB)X = diag(α1 + iβ1, . . . , αn + iβn) where αj and βj are real numbers. We call the pairs (αj, βj) normalized generalized eigenvalues of the definite pair (A, B). These pairs have not b...

Basic geometric invariant theory