Stewart’s recently introduced Krylov-Schur algorithm is a modification of the implicitly restarted Arnoldi algorithm which employs reordered Schur decompositions to perform restarts and deflations in a numerically reliable manner. This paper describes a variant of the Krylov-Schur algorithm suitable for addressing eigenvalue problems associated with products of large and sparse matrices. It per...

We prove that the duality operator preserves the Frobenius–Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius–Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius–Schur indicators of certain real-valued, i...

This paper introduces a generalized Schur algorithm in the Krein space with an indefinite inner product. Concepts such as Caratheodory classes and Schur classes used in the classical Schur algorithm cannot be applied in the Krein space since the positivedefiniteness corresponds merely to the nonsingularity in the Krein space. We note also that these problems appear when fast algorithms for subo...

Introduction 1 1. Representation Theory of Finite Groups 2 1.1. Preliminaries 2 1.2. Group Algebra 4 1.3. Character Theory 5 2. Irreducible Representations of the Symmetric Group 8 2.1. Specht Modules 8 2.2. Dimension Formulas 11 2.3. The RSK-Correspondence 12 3. Schur-Weyl Duality 13 3.1. Representations of Lie Groups and Lie Algebras 13 3.2. Schur-Weyl Duality for GL(V ) 15 3.3. Schur Functor...

Several Schur complement-based preconditioners have been proposed for solving (generalized) saddlepoint problems. We consider probing-based methods for approximating those Schur complements in the preconditioners of the type proposed by [Murphy, Golub and Wathen ’00], [de Sturler and Liesen ’03] and [Siefert and de Sturler ’04]. This approach can be applied in similar preconditioners as well. W...

The Bernstein operators allow to build recursively the Schur functions. We present a recursion formula for k-Schur functions at t = 1 based on combinatorial operators that generalize the Bernstein operators. The recursion leads immediately to a combinatorial interpretation for the expansion coefficients of k-Schur functions at t = 1 in terms of homogeneous symmetric functions.

This paper concerns the Schur stability of polynomials. Firstly necessary conditions are presented for the Schur stability of complex polynomials with fixed coefficients based on the discrete Hermite-Biehler theorem. Then the result is applied to obtain necessary conditions for the robust Schur stability of real interval polynomials.

In this paper, we obtain some formulas for compound matrices of generalized Schur complements of matrices. Further, we give some Löwner partial orders for compound matrices of Schur complements of positive semidefinite Hermitian matrices, and obtain some estimates for eigenvalues of Schur complements of sums of positive semidefinite Hermitian matrices.

In [HiMu] the authors, in their analysis on Schur rings, pointed out that it is not known whether there exists a non-Schurian p-Schur ring over an elementary abelian p-group of rank 3. In this paper we prove that every p-Schur ring over an elementary abelian p-group of rank 3 is in fact Schurian.

We prove Stanley’s conjecture that, if δn is the staircase shape, then the skew Schur functions sδn/μ are non-negative sums of Schur P -functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function sδn/δn−2 , we discuss connections with Eulerian numbers and alternating permutations.