Related papers: Extremal rays in the Hermitian eigenvalue problem …

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of $n\times n$ Hermitian matrices, given the eigenvalues of the summands. The regular faces of the cones $\Gamma_n(s)$ controlling this problem have been…

Let $G$ be a simple connected complex Lie group. The additive eigencone of $G$ is a polyhedral cone containing the set of solutions to the additive eigenvalue problem, a generalization of the Hermitian eigenvalue problem. The additive…

We give an inductive procedure for finding the extremal rays of the equivariant Littlewood-Richardson cone, which is closely related to the solution space to S. Friedland's majorized Hermitian eigenvalue problem. In so doing, we solve the…

We examine the extremal rays of the cone of dominant weights $(\mu, \widehat\mu)$ for groups $G\subseteq \widehat G$ for which there exists $N \gg0$ such that $$ \left(V(N\mu)\otimes V(N\widehat \mu)\right)^G\ne (0). $$ We exhibit formulas…

The classical Hermitian eigenvalue problem addresses the following question: What are the possible eigenvalues of the sum A+B of two Hermitian matrices A and B, provided we fix the eigenvalues of A and B. A systematic study of this problem…

We classify the finite-dimensional rational representations $V$ of the exceptional algebraic groups $G$ with $\mathfrak g={\sf Lie}(G)$ such that the symmetric invariants of the semi-direct product $\mathfrak g\ltimes V$, where $V$ is an…

For finite-dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm. In case the cone is simplicial a similar statement applies to…

Let $H$ be a connected reductive subgroup of a complex connected reductive group $G$. Fix maximal tori and Borel subgroups of $H$ and $G$. Consider the pairs $(V,V')$ of irreducible representations of $H$ and $G$ such that $V$ is a…

We give a minimal list of inequalities characterizing the possible eigenvalues of a set of Hermitian matrices with positive semidefinite sum of bounded rank. This answers a question of A. Barvinok.

Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite…

Let G be a connected semisimple complex algebraic group and let P be a parabolic subgroup. In this paper we define a new (commutative and associative) product on the cohomology of the homogenous spaces G/P and use this to give a more…

This paper studies the asymptotic behavior of eigenvalues of random abelian G-circulant matrices, that is, matrices whose structure is related to a finite abelian group G in a way that naturally generalizes the relationship between…

Let $G$ be a connected reductive subgroup of a complex connected reductive group $\hat{G}$. Fix maximal tori and Borel subgroups of $G$ and $\hat{G}$. Consider the cone $LR^\circ(\hat{G},G)$ generated by the pairs $(\nu,\hat{\nu})$ of…

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These…

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any unitary irreducible…

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.

Euclidean random matrices arise in a wide range of physical systems where interactions are determined by spatial configurations, including disordered media and cooperative phenomena in atomic ensembles. Unlike classical random matrix…

An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular…

Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of $G$. The inverse eigenvalue…

The objective of this paper is to understand the superlinear convergence behavior of the GMRES method when the coefficient matrix has clustered eigenvalues. In order to understand the phenomenon, we analyze the convergence using the…