Related papers: Nilpotent matrices having a given Jordan type as m…

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element $s$ in a dual group such that $s^2=1$. We prove that there is a bijection…

We consider the groups G which arise from real semisimple Jordan algebras via the Tits-Koecher-Kantor construction. Such a G is characterized by the fact that it admits a parabolic subgroup P=LN which is conjugate to its opposite, and for…

The problem of classifying tuples of nilpotent matrices over a field under simultaneous conjugation is considered "hopeless". However, for any given matrix order over a finite field, the number of concerned orbits is always finite. This…

In this paper, we give a conjecture, which generalises Euler's partition theorem involving odd parts and different parts for all moduli. We prove this conjecture for two family partitions. We give $q$-difference equations for the related…

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the…

Let G be a transitive group of permutations of a finite set X, and suppose that some element of G has at most two orbits on X. We prove that any two maximal chains of groups between G and a point-stabilizer of G have the same length, and…

Suppose that $G$ is a finite solvable group and $V$ is a finite, faithful and completely reducible $G$-module. Let $N$ be a nilpotent subgroup of $G$, then there exits $v \in V$ such that $|\bC_N(v)| \leq (|N|/p)^{1/p}$, where $p$ is the…

The paper is devoted to the description of the varieties of complex 5-dimensional nilpotent Jordan superalgebras. We find all representatives for the isomorphism classes, using the Jordan normal form, results of simultaneous matrix…

Let $X$ be an open subset of $\Bbb C^N$, and let $A$ be an $n\times n$ matrix of holomorphic functions on $X$. We call a point $\xi\in X$ $\mathbf{Jordan}$ $\mathbf{stable}$ for $A$ if $\xi$ is not a splitting point of the eigenvalues of…

Let \theta be an involution of the semisimple Lie algebra g and g=k+p be the associated Cartan decomposition. The nilpotent commuting variety of (g,\theta) consists in pairs of nilpotent elements (x,y) of p such that [x,y]=0. It is…

We study the structure of the nilpotent commutator $\nb$ of a nilpotent matrix $B$. We show that $\nb$ intersects all nilpotent orbits for conjugation if and only if $B$ is a square--zero matrix. We describe nonempty intersections of $\nb$…

Jespers and Sun conjectured that if a finite group $G$ has the property ND, i.e. for every nilpotent element $n$ in the integral group ring $\mathbb{Z}G$ and every primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in…

Let $RG$ be the group ring of an arbitrary group $G$ over an associative non-commutative ring $R$ with identity. In this paper, we have obtained the necessary and sufficient conditions under which $RG$ is Jordan nilpotent of index $4$.

When $p$ and $q$ are coprime odd integers no less than 3, Olsson proved that the $q$-bar-core of a $p$-bar-core is again a $p$-bar-core. We establish a generalisation of this theorem: that the $p$-bar-weight of the $q$-bar-core of a bar…

We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting…

In this paper we determine, for all $r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting $r$-tuples of nilpotent elements of $\mathfrak{gl}_n$. Our main result is that in characteristic $\neq…

In this paper, we study the non trivial idempotents of the $2 \times 2$ matrix ring over the polynomial ring $\mathbb{Z}_{pqr}[x]$ for distinct primes $p, q $ and $r$ greater than $3$. We have classified all the idempotents of this matrix…

Kato developed an exotic Deligne-Langlands correspondence using a geometric model for the multiparameter affine Hecke algebra of type C, based on his exotic nilpotent cone. Achar-Henderson and Springer showed that this exotic nilpotent is…

Recently, V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra \g. Our aim is to contribute to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pair (e_1,e_2)…

The Jordan type $P_{A,\ell}$ of a linear form $\ell$ acting on a graded Artinian algebra $A$ over a field $\sf k$ is the partition describing the Jordan block decomposition of the multiplication map $m_\ell$, which is nilpotent. The Jordan…