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 derive several more general statements using a recent result of Rhoades and Skandera. 1. Schur positivity conjectures The ring of symmetric functions has a linear basis of Schur functions sλ labelled by partitions λ = (λ1 ≥ λ2 ≥ · · · ≥ 0). A sym...

We define a new family F̃w(X) of generating functions for w ∈ S̃n which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions such as their symmetry and conjecture certain positivity properties. As an application, we relate these functions to the k-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. ...

this paper gives a short survey of some of the known results generalizing the theorem, credited to i. schur, that if the central factor group is finite then the derived subgroup is also finite.

Characterization of Schur functions in terms of their Taylor coefficients is due to C. Carathéodory and I. Schur. We discuss the boundary analogue of this problem.

Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.

We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables.

To rely on authenticity of Image we can use Image Watermarking. In this Paper a new robust Image watermarking algorithm is proposed, which is based on Schur decomposition. Schur decomposition gives good result in terms of performance, robustness, imperceptibility and speed etc Schur decomposition technique shows a good performance against the various attacks such as JPEG compression, cropping, ...

For x = (x1, x2, · · · , xn) ∈ R+, the symmetric function φn(x, r) is defined by φn(x, r) = φn(x1, x2, · · · , xn; r) = ∏ 1≤i1<i2···<ir≤n  r ∑ j=1 1 + xij xij  , where r = 1, 2, · · · , n, and i1, i2, · · · , in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of φn(x, r) are discussed. As applications, some inequalitie...