a famous theorem of schur states that for a group $g$ finiteness of $g/z(g)$‎ ‎implies the finiteness of $g'.$ the converse of schur's theorem is an interesting problem which has been considered by some‎ ‎authors‎. ‎recently‎, ‎podoski and szegedy proved the truth of the converse of schur's theorem for capable groups‎. ‎they also established an explicit bound for the index of the...

This paper provides a survey for the latest developments in the theory of affine q-Schur algebras and Schur–Weyl duality between affine quantum gln and affine type A Hecke algebras. More precisely, we will establish, on the one side, an isomorphism between the double Ringel–Hall algebra D△(n) of a cyclic quiver △(n) and the quantum loop algebra of gln, and establish, on the other side, explicit...

A classic result of I. Schur [9] asserts that for every r 2 and for n sufficiently large, if the set [n]=[1, 2, ..., n] is partitioned into r classes, then at least one of the classes contains a solution to the equation x+ y=z. Any such solution with x{y will be called a Schur triple. Let us say that A [n] has the Schur property if for every partition (or 2-coloring) of A=R _ B (for red and blu...

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...

We consider a filtration of the symmetric function space given by Λ (k) t , the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k. We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur functions in the subspaces Λ (k) t . We prove several properties for the k-Schur functions including that they form ...

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit q = t = 0 and q = t→∞, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition fo...

σλ = |c T λi+j−i(Q− Fl+i−λi)|1≤i,j≤k. These determinants are variations of Schur polynomials, which we will call double Schur polynomials and denote sλ(x|y), where the two sets of variables are x = (x1, . . . , xk) and y = (y1, . . . , yn). (Here k ≤ n, and the length of λ is at most k.) Setting the y variables to 0, one recovers the ordinary Schur polynomials: sλ(x|0) = sλ(x). In fact, sλ(x|y)...

In the framework of BIBO stability tests for one-dimensional (1-D) linear systems, the Schur-Cohn stability test has the appealing property of being a recursive algorithm. This is a consequence of the simultaneously algebric and analytic aspect of the Schur coefficients, which can be also regarded as reflection coefficients. In the multidimensional setting, this dual aspect gives rise to two ex...

In this paper, the Schur convexity is generalized to Schur f -convexity, which contains the Schur geometrical convexity, harmonic convexity and so on. When f : R+ →R is defined by f (x) = (xm−1)/m if m = 0 and f (x) = lnx if m = 0 , the necessary and sufficient conditions for f -convexity (is called Schur m -power convexity) of Daróczy means are given, which improve, generalize and unify Shi et...