Related papers: Conjugacy of Integral Matrices over Algebraic Exte…

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal…

A known result in random matrix theory states the following: Given a random Wigner matrix $X$ which belongs to the Gaussian Orthogonal Ensemble (GOE), then such matrix $X$ has an invariant distribution under orthogonal conjugations. The…

Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of…

Let $R$ be a commutative ring with unity. The essential ideal graph $\mathcal{E}_R$ of $R$ is a graph in which the vertex set comprises of set of all nonzero proper ideals of $R$ and two vertices $I$ and $K$ are adjacent if and only if…

We prove that the conjugacy problem in the first Grigorchuck group $\Gamma$ can be solved in linear time. Furthermore, the problem to decide if a list of elements $w_1,\ldots,w_k\in\Gamma$ contains a pair of conjugate elements can be solved…

We consider the {\em Deligne-Simpson problem}: {\em Give necessary and sufficient conditions for the choice of the conjugacy classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$, $j=1,..., p+1$, so that there exist irreducible…

We present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. The result exhibits a striking duality…

We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$, depending on a matrix $a\in M_{p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary…

Over an arbitrary field $\mathbb{F}$, Harbourne conjectured that $$I^{(N (r-1)+1)} \subseteq I^r$$ for all $r>0$ and all homogeneous ideals $I$ in $S = \mathbb{F} [\mathbb{P}^N] = \mathbb{F} [x_0, \ldots, x_N]$. The conjecture has been…

We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller's groups. We show that for "almost all", in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is…

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gr\"obner bases. Ideal lattices are ideals in the residue class ring, $\mathbb{Z}[x]/\langle f \rangle$ (here $f$ is a monic…

Let G be a graph with adjacency matrix A(G). Consider the matrix IA(G)=(I | A(G)), where I is the identity matrix, and let M(IA(G)) be the binary matroid represented by IA(G). Then suitably parametrized versions of the Tutte polynomial of…

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of…

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and…

The classical Gleason-Kahane-\.{Z}elazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $\Lambda(\mathbf 1)=1$, is multiplicative, that is, $\Lambda(ab)=\Lambda(a)\Lambda(b)$ for…

It is a result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process…

We establish the universality of the singular numbers in random matrix products over $\mathrm{GL}_n(\mathbb{Q}_p)$ as the number of products approaches infinity, with a fixed $n\ge 1$. We demonstrate that, under a broad class of…

Let $\Gamma$ be a simple finite graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$. Let $\mathcal{R}$ be an equivalence relation on $V(\Gamma)$. The $\mathcal{R}$-super $\Gamma$ graph $\Gamma^{\mathcal{R}}$ is a simple graph with…

We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A, which have entries in the ring $\mathbb Z[v,v^{-1}]$. These matrices may also be interpreted as Gram matrices of the Shapovalov form on…

Parametrization of $4\times 4$-matrices $G$ of the complex linear group $GL(4,C)$ in terms of four complex 4-vector parameters $(k,m,n,l)$ is investigated. Additional restrictions separating some subgroups of $GL(4,C)$ are given explicitly.…