Related papers: On matrices commuting with their Frobenius
We will give a short proof of the fact that if the algebraic closure of a field $\mathbb F$ is a finite extension, then for $n\geq 3$ the commuting graph $\Gamma(M_n(\mathbb F))$ is connected and its diameter is four.
This work is devoted to study of algebraicty modulo p of Siegel's G-functions. Our goal is to emphasize the relevance of the notion of strong Frobenius structure, clasically studied in the theory of the p-adic diffenrential equations, for…
A tuple (Z_1,...,Z_p) of matrices of size r is said to be a commuting extension of a tuple (A_1,...,A_p) of matrices of size n <r if the Z_i pairwise commute and each A_i sits in the upper left corner of a block decomposition of Z_i. This…
To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource. It allows one, in principle, to…
Let $\|\!\cdot\!\|_p$ denote the Schatten $p$-norm of matrices and $\|\!\cdot\!\|_F$ the Frobenius norm. For a square matrix $X$, let $|X|$ denote its absolute value. In 2010, Eun-Young Lee posed the problem of determining the smallest…
A Frobenius manifold is a manifold with a flat metric and a Frobenius algebra structure on tangent spaces at points of the manifold such that the structure constants of multiplication are given by third derivatives of a potential function…
We show that the universal measuring coalgebras between Frobenius algebras turn the category of Frobenius algebras into a Hopf category (in the sense of Batista-Caenepeel-Vercruysse), and the universal comeasuring algebras between Frobenius…
We consider a class of $q$-hypergeometric equations describing the quantum difference equation for the cotangent bundles over projective spaces $X=T^{*}\mathbb{P}^{n-1}$ . We show that over $\mathbb{Q}_p$ these equations are equipped with…
In this paper, with the help of the theory of matrices and finite fields we generalize Zolotarev's theorem to an arbitrary finite dimensional vector space over $\mathbb{F}_q$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements.
Given a ring $R$ with center $Z(R)$, we say a linear map $f:R\rightarrow R$ is commuting if $[f(x),x]=0$ for all $x\in R$. Such a map has a standard form if there exists $\lambda\in R$ and additive $\mu:R\rightarrow Z(R)$ such that…
We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture…
A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key…
The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the…
In this paper, we introduce a particular class of matrices. We study the concept of a matrix to be \emph{balanced}. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix…
We study a quantum system of $p$ commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state. In contrast it is possible to…
Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller…
Motivic local systems over a curve in finite characteristic form a countable set endowed with an action of the absolute Galois group of rational numbers commuting with the Frobenius map. I will discuss three series of conjectures about such…
We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by establishing…
This paper investigates the conditions under which the centralizer algebra $S_n(c,R)$ of a matrix $ c\in M_n(R)$ is a (separable) Frobenius extension of the base algebra $R$. For an algebra $R$ over an integral domain $\mathbb{k}$, we…
Let $m,n$ be positive integers. For all $m\times n$ complex matrices $A, C$ and an $n\times m$ matrix $B$, we define a generalized commutator as $ABC-CBA$. We estimate the Frobenius norm of it, and finally get the inequality, which is a…