Related papers: On varieties of commuting nilpotent matrices

A nilpotent Lie algebra n_{n,1} with an (n-1) dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with n_{n,1} as their nilradical are obtained. Their dimension is at most n+2. The generalized Casimir invariants…

We prove that if $A$ is a string algebra then there are not three irreducible morphisms between indecomposable $A$-modules such that its composition belongs to $\Re^{6} \backslash \Re^{7}$, whenever the compositions of two of them are not…

An orthogonal n-frame is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames in a d-dimensional quadratic vector space is an algebraic variety V(d,n). In this paper, we investigate the variety V(d,n) as well…

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $\mathfrak p$ be the…

We prove that non-commutative martingale transforms are of weak type $(1,1)$. More precisely, there is an absolute constant $C$ such that if $\M$ is a semi-finite von Neumann algebra and $(\M_n)_{n=1}^\infty$ is an increasing filtration of…

Let A,B be complex n,n complex matrices such that AB-BA and A commute. We show that, if n=2 then A,B are simultaneously triangularizable and if n>=3 then there exists such a couple A,B such that the pair (A,B) has not property L of…

We study the decidability of three well-known problems related to integer matrix multiplication: Mortality (M), Zero in the Left-Upper Corner (Z), and Zero in the Right-Upper Corner (R). Let d and k be positive integers. Define M(k, d x d)…

Let $G$ be a finite $p$-solvable group, where $p$ is an odd prime. We establish a connection between extendible irreducible characters of subgroups of $G$ that lie under monomial characters of $G$ and nilpotent subgroups of $G$. We also…

Let $k$ be an algebraically closed field of characteristic 0 and let $H_G(d,N)$ be the open locus of the Hilbert scheme $H(d,N)$ corresponding to Gorenstein subschemes of degree $d$ in the projective N-space. We proved in a previous paper…

Let ${{\overline{p}}_{3}}(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\overline{p}}_{3}(n)$ modulo small powers of 2, such as…

A unimodular $2\times 2$ matrix with entries in a commutative $R$ is called extendable (resp.\ simply extendable) if it extends to an invertible $3\times 3$ matrix (resp.\ invertible $3\times 3$ matrix whose $(3,3)$ entry is $0$). We obtain…

We obtain a series of new results on the problem of irreducibility of commuting varieties associated with symmetric pairs or, in other words, $Z_2$-graded simple Lie algebras. In particular, we present many examples of reducible commuting…

When the plane is pie-sliced in $n\leq 4$ parts (with nonempty interior and common vertex at the origin) our main result provides a sufficient condition for any map $L$, that is continuous and piecewise linear relatively to this slicing, to…

Let M_{7,n} be the (coarse) moduli space of smooth curves of genus 7 with n marked points defined over the complex field. We denote by M^1_{7,n;4} the locus of points inside M_{7,n} representing curves carrying a g^1_4. It is classically…

We study the multiplication operation of square matrices over lattices. If the underlying lattice is distributive, then matrices form a semigroup; we investigate idempotent and nilpotent elements and the maximal subgroups of this matrix…

Let A be a finite dimensional central division algebra over a local non-archimedean field F. Fix any parabolic subgroup P of GL(n,A) and a Levi factor M of P. Let \pi be an irreducible unitary representation of M and \phi (not necessarily…

Let $A \in \mathbb{Z}^{m \times n}$ be an integral matrix and $a$, $b$, $c \in \mathbb{Z}$ satisfy $a \geq b \geq c \geq 0$. The question is to recognize whether $A$ is $\{a,b,c\}$-modular, i.e., whether the set of $n \times n$…

For the class of $d\times d$ matrices $B=[b_{i,j}]$ with complex nonzero entries satisfying $\sum_{i=1}^{d}|b_{i,j}|=1$, we provide the conditions for the convergence of power matrices $B^n$ to a nonzero limit matrix. In particular, for…

We determine the regular irreducible components of the variety mod(A,d), where A=kQ/I is a string algebra and I is generated by a set of paths of length two. Our case is among the first examples of descriptions of irreducible components,…

Let $p$ be a prime and $b(x)$ be an irreducible polynomial of degree $k$ over $\mathbb{F}_p$. Let $d\geq 1$ be an integer. Consider the following question: Is $b(x^d)$ irreducible? We derive necessary conditions for $b(x^d)$ to be…