We define a generic multiplication in quantised Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantised Schur algebras, defined in [1], a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied in [10]. We also prove that the subalgebra of the new algebra gi...

In this paper, the Schur-convexity, the Schur-geometric-convexity and the Schur-harmonicconvexity of dual form of the complete symmetric function are investigated. As consequences, some new inequalities are established via majorilization theory. Mathematics subject classification (2010): 26B25, 05E05.

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition ν, we denote by νc its complement in a square partition (mm). We conjecture a Schur-positivity criterion for symmetric functions of the form sμ′sμc − sν′sνc , where ν is a partitio...

The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in the cyclic group case any pure Schur ring over R is the tensor product of a pure cyclotomic ring and Schur rings of rank 2 over non-fields. Moreover, it is s...

for a ∈ mn, the schur multiplier of a is defined as s a(x) =a ◦ x for all x ∈ mn and the spectral norm of s a can be stateas ∥s a∥ = supx,0 ∥a ∥x ◦x ∥ ∥. the other norm on s a can be definedas ∥s a∥ω = supx,0 ω(ω s( ax (x ) )) = supx,0 ωω (a (x ◦x ) ), where ω(a) standsfor the numerical radius of a. in this paper, we focus on therelation between the norm of schur multiplier of product of matric...

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type A by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual k-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs...

We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover this class includes examples which are useful to describe the eigenvalues of...

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...

Symplectic and orthogonal Schur functions can be defined combinatorially in a manner similar to the classical Schur functions. This paper demonstrates that they can also be expressed as determinants. These determinants are generated using planar decompositions of tableaux into strips and the equivalence of these determinants to symplectic or orthogonal Schur functions is established by Gessel-V...