Related papers: Bound on the Jordan type of a generic nilpotent ma…

The conjugacy classes of nilpotent $n\times n$ matrices can be parametrised by partitions $\lambda$ of $n$, and for a nilpotent $\eta$ in the class parametrised by $\lambda$, the variety $F_\eta$ of $\eta$-stable flags has its irreducible…

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 $\mathfrak{M}(\mathbb{D}, m, n, P)$ be the ring of all $m \times n$ matrices over a division ring $\mathbb{D}$, with the product given by $A \bullet B=A P B$, where $P$ is a fixed $n \times m$ matrix over $\mathbb{D}$. When $2\leq m, n…

It is known that that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general…

Let ${\mathcal B}_u$ be the Springer fiber over a nilpotent endomorphism $u\in End(\mathbb{C}^n)$. Let $J(u)$ be the Jordan form of $u$ regarded as a partition of $n$. The irreducible components of ${\mathcal B}_u$ are all of the same…

We develop an abstract framework for studying the strong form of Malle's conjecture for nilpotent groups $G$ in their regular representation. This framework is then used to prove the strong form of Malle's conjecture for any nilpotent group…

We prove a generalization of the Jordan canonical form theorem for a class of bounded linear operators on complex separable Hilbert spaces.

It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup…

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…

This paper deals with singularities of closures of $2$-nilpotent Borel conjugacy classes in either a $\text{GL}_n$-conjugacy class or in the nilpotent cone of $\text{GL}_n$. In the latter case we construct a resolution of singularities, in…

It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting…

We show that, for $n \ge 3$, the mapping on $M_n(\mathbb{C})$ which sends a matrix to its diagonalizable part in its Jordan-Chevalley decomposition, is {\bf norm-unbounded} on any neighbourhood of the zero matrix. Let $X$ be a Stonean…

The Jordan algebra of the symmetric matrices of order two over a field $K$ has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is…

In this note, we prove that any Jordan derivation on the generalized matrix ring $T_n(R,M)$ is a derivation. This extends some well-known results of this branch due to Bre\v{s}ar et al. in the cited literature.

For any natural numbers n,r, we construct an algebra P_n^r via generators and quadratic relations, and show that it deforms the W-algebra of gl_{nr} with respect to a nilpotent with Jordan block decomposition r+...+r. We introduce a…

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors F_i from finitely generated kE-modules of constant Jordan type to vector bundles over projective space of dimension r-1. The…

We establish bounds on the mixing times of conjugacy-invariant random walks on finite nilpotent groups in terms of the mixing times of their projections onto the abelianization. This comparison framework shows that, in several natural cases…

The Jacobson Coordinatization Theorem describes the structure of unitary Jordan algebras containing the algebra $H_n(F)$ of symmetric nxn matrices over a field F with the same identity element, for $n\geq 3$. In this paper we extend the…

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 G be a simple algebraic group over the complex numbers. Let N be the cone of nilpotent elements in the Lie algebra of G. Let K_{G x C^*}(N) denote the Grothendieck group of the category of G x C^*-equivariant coherent sheaves on N. In…