scholarly journals A Basis for the Graded Identities of the Matrix Algebra of Order Two over a Finite Field of Characteristic p≠2

2002 ◽  
Vol 8 (4) ◽  
pp. 597-609 ◽  
Author(s):  
Plamen Koshlukov ◽  
Sérgio S. Azevedo
Keyword(s):  
Finite Field ◽  
Matrix Algebra ◽  
Characteristic P ◽  
Graded Identities ◽  
The Matrix

scholarly journals Inclusion Matrices and the MDS Conjecture

10.37236/5713 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Ameera Chowdhury
Keyword(s):  
Finite Field ◽  
Nonnegative Integer ◽  
Characteristic P ◽  
Algebraic Properties ◽  
The Matrix ◽  
Mds Conjecture ◽  
Almost All ◽  
Inclusion Matrix

Let $\mathbb{F}_{q}$ be a finite field of order $q$ with characteristic $p$. An arc is an ordered family of at least $k$ vectors in $\mathbb{F}_{q}^{k}$ in which every subfamily of size $k$ is a basis of $\mathbb{F}_{q}^{k}$. The MDS conjecture, which was posed by Segre in 1955, states that if $k \leq q$, then an arc in $\mathbb{F}_{q}^{k}$ has size at most $q+1$, unless $q$ is even and $k=3$ or $k=q-1$, in which case it has size at most $q+2$. We propose a conjecture which would imply that the MDS conjecture is true for almost all values of $k$ when $q$ is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when $k \leq p$, and if $q$ is not prime, for $k \leq 2p-2$. To accomplish this, given an arc $G \subset \mathbb{F}_{q}^{k}$ and a nonnegative integer $n$, we construct a matrix $M_{G}^{\uparrow n}$, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix $M_{G}^{\uparrow n}$ to properties of the arc $G$ and may provide new tools in the computational classification of large arcs.

scholarly journals On types of matrices and centralizers of matrices and permutations

2014 ◽  
Vol 17 (5) ◽  
Author(s):  
John R. Britnell ◽  
Mark Wildon
Keyword(s):  
Finite Field ◽  
Alternating Groups ◽  
Type Invariant ◽  
The Matrix ◽  
Generalized Type

AbstractIt is known 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 captures more information; using this invariant the result on centralizers is extended to arbitrary fields. The converse is also proved: thus two matrices have conjugate centralizers if and only if they have the same generalized type. The paper ends with the analogous results for symmetric and alternating groups.

A New Spectral Approach in the Matrix Algebra: C-Determinant Pseudospectrum

2021 ◽  
Vol 65 (7) ◽  
pp. 1-7
Author(s):  
Aymen Ammar ◽  
Aref Jeribi ◽  
Kamel Mahfoudhi
Keyword(s):  
Matrix Algebra ◽  
Spectral Approach ◽  
The Matrix

Johnson pseudo-contractibility of second duals of certain Banach algebras

2019 ◽  
pp. 2150012
Author(s):  
A. Sahami ◽  
E. Ghaderi ◽  
S. M. Kazemi Torbaghan ◽  
B. Olfatian Gillan
Keyword(s):  
Tensor Product ◽  
Matrix Algebra ◽  
Banach Algebras ◽  
Amenable Group ◽  
Semigroup Algebra ◽  
Archimedean Semigroup ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
The Matrix ◽  
Second Dual

In this paper, we study Johnson pseudo-contractibility of second dual of some Banach algebras. We show that the semigroup algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is a finite amenable group, where [Formula: see text] is an archimedean semigroup. We also show that the matrix algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is finite. We study Johnson pseudo-contractibility of certain projective tensor product second duals Banach algebras.

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