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.

We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence length $n-r$ index a subset set all $r$th Kronecker powers $n \times n$ permutation matrices which is basis for linear span set. Thanks to known Schur-Weyl duality, this gives new centralizer algebra partition acting on tensor power vector space. give some related results doubly stochastic in...

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...

Introduced by Okounkov and Reshetikhin in 2003, the Schur Process has been shown to be a determinantal point process, so that each of its correlation functions are determinants of minors of one correlation kernel matrix. In previous papers, this was derived using determinantal expressions of the skew-Schur functions; in this paper, we obtain this result in a different way, using the fact that t...

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C∗-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we es...