Assume that $(N,L)$, is a pair of finite dimensional nilpotent Lie algebras, in which $L$ is non-abelian and $N$ is an ideal in $L$ and also $mathcal{M}(N,L)$ is the Schur multiplier of the pair $(N,L)$. Motivated by characterization of the pairs $(N,L)$ of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

Poirier and Reutenauer defined a Hopf algebra on the Z-span of all standard Young tableaux in [10], which is later studied in [4, 11]. The Robinson-Schensted-Knuth insertion was used to relate the bialgebra to Schur functions. Schur function is a class of symmetric functions that can be determined by the summation of all semistandard Young tableaux of shape . With the help of the PR-bialgebra, ...

Most practical constructions of lattice codes with high coding gains are multilevel constructions where each level corresponds to an underlying code component. Construction D, Construction D, and Forney’s code formula are classical constructions that produce such lattices explicitly from a family of nested binary linear codes. In this paper, we investigate these three closely related constructi...

We extend the preorder on k-tuples of dominant weights of a simple complex Lie algebra g of classical type adding up to a fixed weight λ defined by Chari, Sagaki and the author [Posets, tensor products and Schur positivity, Algebra and Number Theory, to appear]. We show that the induced extended partial order on the equivalence classes has a unique minimal and a unique maximal element. For k = ...

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 note, we investigate a kind of double centralizer property for general linear supergroups. For the super space $V=\mathbb{K}^{m\mid n}$ over an algebraically closed field $\mathbb{K}$ whose characteristic is not equal to $2$, consider its $\mathbb{Z}_2$-homogeneous one-dimensional extension $\underline V=V\oplus\mathbb{K}v$, and natural action supergroup $\tilde G:=\text{GL}(V)\times \t...

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...

Kronecker or inner tensor products of representations of symmetric groups (and many other groups) have been studied for a long time. But even for the symmetric groups no reasonable formula for decomposing Kronecker products of two irreducible complex representations into irreducible components is available (cf. [7, 5]). An equivalent problem is to decompose the inner product of the correspondin...