We determine which nilpotent orbits in E6 have closures which are normal varieties and which do not. At the same time we are able to verify a conjecture in [14] concerning functions on nonspecial nilpotent orbits for E6.

let $r$ be a ring with unity. the undirected nilpotent graph of $r$, denoted by $gamma_n(r)$, is a graph with vertex set ~$z_n(r)^* = {0neq x in r | xy in n(r) for some y in r^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in n(r)$, or equivalently, $yx in n(r)$, where $n(r)$ denoted the nilpotent elements of $r$. recently, it has been proved that if $r$ is a left ar...

We determine which nilpotent orbits in E6 have closures which are normal varieties and which do not. At the same time we are able to verify a conjecture in [14] concerning functions on nonspecial nilpotent orbits for E6.

We give a definition of nilpotent association schemes as a generalization of nilpotent groups and investigate their basic properties. Moreover, for a group-like scheme, we characterize the nilpotency by its character products.

Let G be a free product of two groups with amalgamated subgroup, π be either the set of all prime numbers or the one-element set {p} for some prime number p. Denote by Σ the family of all cyclic subgroups of group G, which are separable in the class of all finite π-groups. Obviously, cyclic subgroups of the free factors, which aren’t separable in these factors by the family of all normal subgro...

Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D, E. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynkin type as n. We prove a formula for the product of two elements of the dual of this semicanonical basis, ...

In this work, we present a new classification of nilpotent orbits in a real reductive Lie algebra g under the action of its adjoint group. Our classification generalizes the Bala-Carter classification of the nilpotent orbits of complex semisimple Lie algebras. Our theory takes full advantage of the work of Kostant and Rallis on pC , the “complex symmetric space associated with g”. The Kostant-S...

Abstract. Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper we initiate an investigation into which zero-nonzero patterns are potentially nilpotent over F, with a special emphasis on the case that F = Zp is a finite field. As part of this investigation, w...

We give a new characterization of Lusztig’s canonical quotient, a finite group attached to each special nilpotent orbit of a complex semisimple Lie algebra. This group plays an important role in the classification of unipotent representations of finite groups of Lie type. We also define a duality map. To each pair of a nilpotent orbit and a conjugacy class in its fundamental group, the map assi...