Related papers: Pairs of commuting nilpotent matrices, and Hilbert…

The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions $(P,Q)$, where $Q={\mathcal Q}(P)$ is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent…

Given an nxn nilpotent matrix over an algebraically closed field K, we prove some properties of the set of all the nxn nilpotent matrices over K which commute with it. Then we give a proof of the irreducibility of the variety of all the…

An $n\times n$ nilpotent matrix $B$ is determined up to conjugacy by a partition $P_B$ of $n$, its Jordan type given by the sizes of its Jordan blocks. The Jordan type $\mathfrak D(P)$ of a nilpotent matrix in the dense orbit of the…

Let $k$ be an infinite field. Fix a Jordan nilpotent $n$ by $n$ matrix $B = J_P$ with entries in $k$ and associated Jordan type $P$. Let $Q(P)$ be the Jordan type of a generic nilpotent matrix commuting with $B$. In this paper, we use the…

Let $B$ be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition $\lambda$. Then it is known that its nilpotent commutator $N_B$ is an irreducible variety and that there is a unique partition $\mu$ such…

It is well-known that a nilpotent n by n matrix B is determined up to conjugacy by a partition of n formed by the sizes of the Jordan blocks of B. We call this partition the Jordan type of B. We obtain partial results on the following…

We study the variety of n by n matrices with commutator of rank at most one. We describe its irreducible components; two of them correspond to the pairs of commuting matrices, and n-2 components of smaller dimension corresponding to the…

I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t-tuple of linear operators. Moreover, it contains…

In this work, we consider a family of Gelfand pairs $(K \ltimes N, N)$ (in short $(K,N)$) where $N$ is a two step nilpotent Lie group, and $K$ is the group of orthogonal automorphisms of $N$. This family has a nice analytic property: almost…

Given a classical symmetric pair, $(G,K)$, with $\mathfrak g = Lie(G)$, we provide descriptions of the Hilbert series of the algebra of $K$-invariant vectors in the associated graded algebra of $\mathcal U(\mathfrak g)$ viewed as a…

Let $B$ be an $n \times n$ nilpotent matrix with entries in an infinite field $\k$. Assume that $B$ is in Jordan canonical form with the associated Jordan block partition $P$. In this paper, we study a poset $\mathcal{D}_P$ associated to…

Let H and K be infinite dimensional Hilbert spaces, while B(H) and B(K) denote the algebras of all linear bounded operators on H and K, respectively. We characterize the forms of additive mappings from B(H) into B(K) that preserve the…

The authors here show that the partition $P_{k,l}(Q)$ in the table $\mathcal T(Q)$ of partitions having maximal nilpotent commutator a given stable partition $Q$, defined in [IKVZ2], is identical to the analogous partition $P_{k,l}^Q$…

We explore aspects of dilation theory in the finite dimensional case and show that for a commuting $n$-tuple of operators $T=(T_1,...,T_n) $ acting on some finite dimensional Hilbert space $H$ and a compact set $X\subset \mathbb{C}^n$ the…

In this article, we give a complete characterization of the bijective maps which commute with the mean transform under Jordan product. The main result is the following : Let $H,K$ be two complex Hilbert spaces and $\Phi :B(H) \to B(K)$ be a…

The aim of this paper is to study the stable birational type of $Hilb^n_X$, the Hilbert scheme of degree $n$ points on a surface $X$. More precisely, it addresses the question for which pairs of positive integers $(n,n')$ the variety…

We prove the Box Conjecture for pairs of commuting nilpotent matrices, as formulated by Iarrobino et al [28]. This describes the Jordan type of the dense orbit in the nilpotent commutator of a given nilpotent matrix. Our main tool is the…

Let $N$ be a connected and simply connected nilpotent Lie group, and let $K$ be a subgroup of the automorphism group of $N$. We say that the pair $(K,N)$ is a nilpotent Gelfand pair if $L^1_K(N)$ is an abelian algebra under convolution. In…

For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M]. This strengthens the K_2 part of the main result of [G5] in two ways: the coefficient field of characteristic 0 is…

Let $\mathcal{H}_{a,b}^n$ denote the component of the Hilbert scheme whose general point parameterizes an $a$-plane union a $b$-plane meeting transversely in $\mathbf{P}^n$. We show that $\mathcal{H}_{a,b}^n$ is smooth and isomorphic to…