Supersymmetric elements in divided powers algebras

Central elements in the distribution algebra of a general linear supergroup and supersymmetric elements

In this paper we investigate the image of the center $Z$ of the distribution algebra $Dist(GL(m|n))$ of the general linear supergroup over a ground field of positive characteristic under the Harish-Chandra morphism $h:Z \to Dist(T)$ obtained by the restriction of the natural map $Dist(GL(m|n))\to Dist(T)$. We define supersymmetric elements in $Dist(T)$ and show that each image $h(c)$ for $c\in Z$ is supersymmetric. The central part of the paper is devoted to a description of a minimal set of generators of the algebra of supersymmetric elements over Frobenius kernels $T_r$.

Divided power algebras over an operad

The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group, the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.

Supersymmetric polynomials and the center of the walled Brauer algebra

We study a commuting family of elements of the walled Brauer algebra $B_{r,s}(δ)$, called the Jucys-Murphy elements, and show that the supersymmetric polynomials in these elements belong to the center of the walled Brauer algebra. When $B_{r,s}(δ)$ is semisimple, we show that those supersymmetric polynomials generate the center. Under the same assumption,we define a maximal commutative subalgebra of $B_{r,s}(δ)$, called the \emph{Gelfand-Zetlin subalgebra}, and show that it is generated by the Jucys-Murphy elements. As an application, we construct a complete set of primitive orthogonal idempot…

Levels of Ultrafilters with Extension Divisibilities

To work more accurately with elements of the semigroup of the Stone Cech compactification of the discrete semigroup of natural numbers N under multiplication. We divided these elements into ultrafilters which are on finite levels and ultrafilters which are not on finite levels. For the ultrafilters that are on finite levels we prove that any element is irreducible or product of irreducible elements and all elements on higher levels are extension divided by some elements on lower levels. We characterize ultrafilters that are not on finite levels and the effect of extension divisibility on the u…

群代数中支撑大小为$3$的零因子与$GF(2)$上被不可约多项式整除的三项式

Zero divisors of support size $3$ in group algebras and trinomials divided by irreducible polynomials over $GF(2)$

本文研究无挠群群代数中支撑大小为3的零因子存在性问题，探讨了有理群代数及含$p$个元素的域上群代数中该类零因子的可能性，为解决零因子不存在猜想提供了新进展。

Generators of supersymmetric polynomials in positive characteristic

Kantor and Trishin described the algebra of polynomial invariants of the adjoint representation of the Lie supergalgebra $gl(m|n)$ and a related algebra $A_s$ of what they called pseudosymmetric polynomials over an algebraically closed field $K$ of characteristic zero. The algebra $A_s$ was investigated earlier by Stembridge who called the elements of $A_s$ supersymmetric polynomials and determined generators of $A_s$. The case of positive characteristic $p$ has been recently investigated by La Scala and Zubkov. They formulated two conjectures describing generators of polynomial invariants of …