We consider the two-sided Arnoldi method and propose a two-sided Krylov–Schurtype restarting method. We discuss the restart for standard Rayleigh–Ritz extraction as well as harmonic Rayleigh–Ritz extraction. Additionally, we provide error bounds for Ritz values and Ritz vectors in the context of oblique projections and present generalizations of, e.g., the Bauer–Fike theorem and Saad’s theorem....

Pλ(y; θ), (1) where δ := (n− 1, n− 2, . . . , 1, 0) and λ are partitions, aδ = ∏ 1≤i<j≤n(yi − yj) is the Vandermonde determinant and u is a free parameter. Under a general result, S. Sahi proves [8, Theorem 5.2] the existence of a unique polynomial P ∗ μ(y; θ), now known as shifted Jack polynomials, satisfying a certain vanishing condition. In the special case θ = 1, Okounkov and Olshanski [6,7...

We prove Okounkov’s conjecture, a conjecture of Fomin-FultonLi-Poon, and a special case of Lascoux-Leclerc-Thibon’s conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. We include an alternative derivation of this result directly from Haiman’s work on Schur positive immanants. Our results imply an intriguing log-concavity propert...

The Pólya-Schur theory describes the class of hyperbolicity preservers, i.e., the class of linear operators acting on univariate polynomials and preserving real-rootedness. We attempt to develop an analog of Pólya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i...

In 1981, G. D. James proved two theorems about the decomposition matrices of Schur algebras involving the removal of the first row or column from a Young diagram. He established corresponding results for the symmetric group using the Schur functor. We apply James’ techniques to prove that row removal induces an injection on the corresponding Ext between simple modules for the Schur algebra. We ...

This is an expository article on the singularities of nilpotent orbit closures in simple Lie algebras over the complex numbers. It is slanted towards aspects that are relevant for representation theory, including Ma ei's theorem relating Slodowy slices to Nakajima quiver varieties in type A. There is one new observation: the results of Juteau and Mautner, combined with Ma ei's theorem, give a g...

The Pólya-Schur theory describes the class of hyperbolicity preservers, i.e., the linear operators on univariate polynomials preserving realrootedness. We attempt to develop an analog of Pólya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e., the minimal dist...

We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of ǫ, t...

We prove Okounkov’s conjecture, a conjecture of Fomin-FultonLi-Poon, and a special case of Lascoux-Leclerc-Thibon’s conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. An alternative proof of this result is provided. We also give an intriguing log-concavity property of Schur functions. 1. Schur positivity conjectures The ring of...

and Applied Analysis 3 holds in Di ∩ Dj , then f a1, a2, . . . , an ≥ ≤ f A a , A a , . . . , A a 1.7 for all a a1, a2, . . . , an ∈ D, with equality if only if a1 a2 · · · an. Proof. If n 2, then Theorem 1.2 follows from Lemma 1.1 and l |a1 − a2|/2. We assume that n ≥ 3 in the next discussion. Without loss of generality, we only prove the case of ∂f/∂xi > ∂f/∂xj with i / j. If a1 a2 · · · an, ...