We study the class C of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, k-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur fu...

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 introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula. Résumé. Nous introduisons une nouvelle base des fonctions symé...

An experimental design is said to be Schur optimal, if it is optimal with respect to the class of all Schur isotonic criteria, which includesKiefer’s criteria of p-optimality, distance optimality criteria andmany others. In the paper we formulate an easily verifiable necessary and sufficient condition for Schur optimality in the set of all approximate designs of a linear regression experiment w...

The result on the Geršgorin disc separation from the origin for strictly diagonally dominant matrices and their Schur complements in (Liu and Zhang in SIAM J. Matrix Anal. Appl. 27(3):665-674, 2005) is extended to nonstrictly diagonally dominant matrices and their Schur complements, showing that under some conditions the separation of the Schur complement of a nonstrictly diagonally dominant ma...

Flagged Schur functions are generalizations of Schur functions. They appear in the work of Lascoux and Schutzenberger [2] in their study of Schubert polynomials. Gessel [ 1 ] has shown that flagged Schur functions can be expressed both as a determinant in the complete homogeneous symmetric functions and in terms of column-strict tableaux just as can ordinary Schur functions (Jacobi-Trudi identi...

Symmetric functions are vital to the study of combinatorics because they provide valuable information about partitions and permutations, topics which constitute the core of the subject. The significance of symmetric function theory is manifest by its connections to other branches of mathematics, including group theory, representation theory, Lie algebras, and algebraic geometry. One important b...

It is well known, see [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. Schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant , as well as its diagonal-Schur complement. Also, if a matrix is an H-matrix, then its Schur complement and diagonal-Schur complement are H-matrices, too, see [J. Liu, Y. Huang, Some properties on...

Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of “shift” parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expressed as the partition function of a particular statistical-mechanical system (six-vertex model). The proof is based on the Yang-Baxter equation. There...

We introduce an analogue of the q-Schur algebra associated to Coxeter systems of type b A n?1. We give two constructions of this algebra. The rst construction realizes the algebra as a certain endomorphism algebra arising from an aane Hecke algebra of type b A r?1 , where n r. This generalizes the original q-Schur algebra as deened by Dipper and James, and the new algebra contains the ordinary ...