scholarly journals Automorphisms of Drinfeld half-spaces over a finite field

2013 ◽  
Vol 149 (7) ◽  
pp. 1211-1224 ◽  
Author(s):  
Bertrand Rémy ◽  
Amaury Thuillier ◽  
Annette Werner
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Vector Space ◽  
Half Space ◽  
Linear Group ◽  
Analytic Geometry ◽  
Base Field ◽  
Projective Linear Group

AbstractWe show that the automorphism group of Drinfeld’s half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

CONSTRUCTING THE GROUP PRESERVING A SYSTEM OF FORMS

2008 ◽  
Vol 18 (02) ◽  
pp. 227-241 ◽  
Author(s):  
PETER A. BROOKSBANK ◽  
E. A. O'BRIEN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Linear Group ◽  
Jacobson Radical ◽  
Matrix Algebra ◽  
General Linear Group ◽  
Efficient Algorithms ◽  
Sesquilinear Forms ◽  
Group Of Units ◽  
Practical Algorithm

We present a practical algorithm to construct the subgroup of the general linear group that preserves a system of bilinear or sesquilinear forms on a vector space defined over a finite field. Components include efficient algorithms to construct the Jacobson radical and the group of units of a matrix algebra.

scholarly journals On the dimension of $H^{*}((\mathbb Z_2)^{\times t}, \mathbb Z_2)$ as a module over Steenrod ring

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Positive Integer ◽  
Tensor Product ◽  
Finite Field ◽  
Vector Space ◽  
Linear Group ◽  
Computer Algebra ◽  
General Linear Group ◽  
Polynomial Algebra ◽  
Minimal Set ◽  
Trivial Subgroup

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $(\mathbb P^{\otimes t})^{G_t}$ as a module over Steenrod ring, $\mathscr A_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t, \mathbb Z_2).$ Equivalently, we want to find a basis of the $\mathbb Z_2$-vector space $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each degree $n\geq 0.$ The problem is proved surprisingly difficult and has been not yet known for $t\geq 5.$ In the present paper, we consider the trivial subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators for $(\mathbb P^{\otimes 5})^{G_5}$ in degree $5(2^{1} - 1) + 13.2^{1}$ and for $(\mathbb P^{\otimes 6})^{G_6}$ in "generic" degree $n = 5(2^{d+4}-1) + 47.2^{d+4}$ with a positive integer $d.$ An efficient approach to studying $(\mathbb P^{\otimes 5})^{G_5}$ in this case has been provided. In addition, we introduce an algorithm on the MAGMA computer algebra for the calculation of this space. This study is a continuation of our recent works in \cite{D.P2, D.P4}.

scholarly journals On Generalized Galois Cyclic Orbit Flag Codes

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 217
Author(s):  
Clementa Alonso-González ◽  
Miguel Ángel Navarro-Pérez
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Linear Group ◽  
General Linear Group ◽  
Cyclic Subgroup ◽  
General Linear ◽  
The Relationship

Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.

Erdös-Ko-Rado theorem in some linear groups and some projective special linear group

2014 ◽  
Vol 51 (1) ◽  
pp. 83-91
Author(s):  
Milad Ahanjideh ◽  
Neda Ahanjideh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Prime Number ◽  
Linear Group ◽  
Projective Line ◽  
Special Linear Group ◽  
Linear Groups ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
Derangement Graph

Let V be the 2-dimensional column vector space over a finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document} (where q is necessarily a power of a prime number) and let ℙq be the projective line over \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. In this paper, it is shown that GL2(q), for q ≠ 3, and SL2(q) acting on V − {0} have the strict EKR property and GL2(3) has the EKR property, but it does not have the strict EKR property. Also, we show that GLn(q) acting on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {\mathbb{F}_q } \right)^n - \left\{ 0 \right\}$$ \end{document} has the EKR property and the derangement graph of PSL2(q) acting on ℙq, where q ≡ −1 (mod 4), has a clique of size q + 1.

scholarly journals G-codes, self-dual G-codes and reversible G-codes over the ring ${\mathscr{B}}_{j,k}$

2021 ◽  
Author(s):  
S. T. Dougherty ◽  
Joe Gildea ◽  
Adrian Korban ◽  
Serap Şahinkaya
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Structure ◽  
Gray Map ◽  
Base Field ◽  
Frobenius Ring ◽  
New Family ◽  
The Family

AbstractIn this work, we study a new family of rings, ${\mathscr{B}}_{j,k}$ B j , k , whose base field is the finite field ${\mathbb {F}}_{p^{r}}$ F p r . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over ${\mathscr{B}}_{j,k}$ B j , k to a code over ${\mathscr{B}}_{l,m}$ B l , m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible $G^{2^{j+k}}$ G 2 j + k -code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

scholarly journals On orbits of the automorphism group on an affine toric variety

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Ivan Arzhantsev ◽  
Ivan Bazhov
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Toric Variety ◽  
Connected Component ◽  
Linear Action

AbstractLet X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\bar X \to X$$ of X as a quotient of a vector space $$\bar X$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

Sign in / Sign up

Export Citation Format

Share Document