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.

scholarly journals On Absolute Points of Correlations of $\mathrm{PG}(2,q^n)$

10.37236/8920 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Jozefien D'haeseleer ◽  
Nicola Durante
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Final Section ◽  
Dimensional Vector ◽  
Sesquilinear Forms ◽  
Dimensional Vector Space ◽  
Sesquilinear Form ◽  
The Absolute ◽  
Dimensional Projective Space

Let $V$ be a  $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of  the $d$-dimensional projective space $\mathrm{PG}(V)$.  Everything is known in this case for both degenerate and non-degenerate reflexive forms if  $\mathbb{F}$  is either  ${\mathbb R}$, ${\mathbb C}$ or a finite field  ${\mathbb F}_q$.   In this paper we consider  degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of $\mathrm{PG}(2,q^n)$ whose set of absolute points are, besides cones,  the (possibly degenerate) $C_F^m$-sets studied by Donati and Durante in 2014. In the final section we collect some  results from the huge work of B.C. Kestenband  regarding what is known for the set of  the absolute  points  of correlations in $\mathrm{PG}(2,q^n)$ induced  by a  non-degenerate, non-reflexive sesquilinear form of $V=\mathbb{F}_{q^n}^3$.

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.

Bivectors over a Finite Field

1981 ◽  
Vol 24 (4) ◽  
pp. 489-490
Author(s):  
J. A. MacDougall
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Symmetric Matrices ◽  
Dimensional Vector ◽  
Dimensional Vector Space

AbstractLet U be an n -dimensional vector space over a finite field of q elements. The number of elements of Λ2U of each irreducible length is found using the isomorphism of Λ2U with Hn, the space of n x n skew-symmetric matrices, and results due to Carlitz and MacWilliams on the number of skew-symmetric matrices of any given rank.

scholarly journals On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

ON THE POSSIBILITY OF GROUP-THEORETIC DESCRIPTION OF AN EQUIVALENCE RELATION CONNECTED TO THE PROBLEM OF COVERING SUBSETS IN FINITE FIELDS WITH COSETS OF LINEAR SUBSPACES

2019 ◽  
Vol 53 (1 (248)) ◽  
pp. 23-27
Author(s):  
D.S. Sargsyan
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Equivalence Relation ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Linear Subspaces ◽  
Theoretic Description

Let $ F^{n}_{q} $ be an $ n $-dimensional vector space over a finite field $ F_q $ . Let $ C(F^{n}_{q} ) $ denote the set of all cosets of linear subspaces in $ F^{n}_{q} $. Cosets $ H_1, H_2, \ldots H_s $ are called exclusive if $ H_i \nsubseteq H_j $, $ 1 \mathclose{\leq} i \mathclose{<} j \mathclose{\leq} s $. A permutation $ f $ of $ C(F^{n}_{q} ) $ is called a $ C $-permutation, if for any exclusive cosets $ H, H_1, H_2, \ldots H_s $ such that $ H \subseteq H_1 \cup H_2 \cup \cdots \cup H_s $ we have:i) cosets $ f(H), f(H_1), f(H_2), \ldots, f(H_s) $ are exclusive;ii) cosets $ f^{-1}(H), f^{-1}(H_1), f^{-1}(H_2), \ldots, f^{-1}(H_s) $ are exclusive;iii) $ f(H) \subseteq f(H_1) \cup f(H_2) \cup \cdots \cup f(H_s) $;vi) $ f^{-1}(H) \subseteq f^{-1}(H_1) \cup f^{-1}(H_2) \cup \cdots \cup f^{-1}(H_s) $.In this paper we show that the set of all $ C $-permutations of $ C(F^{n}_{q} ) $ is the General Semiaffine Group of degree $ n $ over $ F_q $.

scholarly journals On the Number of Orthogonal Systems in Vector Spaces over Finite Fields

10.37236/907 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Graph Theoretic ◽  
Theoretic Proof ◽  
Orthogonal Systems

Iosevich and Senger (2008) showed that if a subset of the $d$-dimensional vector space over a finite field is large enough, then it contains many $k$-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of this result.

scholarly journals Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
You Gao ◽  
XinZhi Fu
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Vector ◽  
Orthogonal Groups ◽  
Reverse Inclusion ◽  
Dimensional Vector Space ◽  
Geometric Lattices

Let𝔽q(2ν+δ+l)be a(2ν+δ+l)-dimensional vector space over the finite field𝔽q. In this paper we assume that𝔽qis a finite field of odd characteristic, andO2ν+δ+l,  Δ(𝔽q)the singular orthogonal groups of degree2ν+δ+lover𝔽q. Letℳbe any orbit of subspaces underO2ν+δ+l,  Δ(𝔽q). Denote byℒthe set of subspaces which are intersections of subspaces inℳ, where we make the convention that the intersection of an empty set of subspaces of𝔽q(2ν+δ+l)is assumed to be𝔽q(2ν+δ+l). By orderingℒby ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these latticesℒare geometric lattices.

On two conjectures on the subspace inclusion 6 graph of a vector space

2018 ◽  
Vol 17 (10) ◽  
pp. 1850189 ◽  
Author(s):  
Dein Wong ◽  
Xinlei Wang ◽  
Chunguang Xia
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Cayley Graph ◽  
Domination Number ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Vertex Set

The subspace inclusion graph on a vector space [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set consists of nontrivial proper subspaces of [Formula: see text] and two vertices are adjacent if one is properly contained in another. In a recent paper, Das posed the following four conjectures on the subspace inclusion graph [Formula: see text]: If [Formula: see text] is a [Formula: see text]-dimensional vector space over a finite field [Formula: see text] with [Formula: see text] elements, then: (1) The domination number of [Formula: see text] is [Formula: see text]. (2) [Formula: see text] is distance regular. (3) [Formula: see text] is Hamiltonian. (4) [Formula: see text] is a Cayley graph. In the present paper, we prove the first two conjectures: If [Formula: see text] is a [Formula: see text]-dimensional vector space over a finite field [Formula: see text] with [Formula: see text] elements, then: (1) The domination number of [Formula: see text] is [Formula: see text]. (2) [Formula: see text] is distance regular.

scholarly journals Involutions and commutators in orthogonal groups

1998 ◽  
Vol 65 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Frieder Knüppel ◽  
Gerd Thomsen
Keyword(s):  
Vector Space ◽  
Orthogonal Group ◽  
Commutator Subgroup ◽  
Dimensional Vector ◽  
Orthogonal Groups ◽  
Commutative Field ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Characteristic 2

AbstractSuppose we are given a regular symmetric bilinear from on a finite-dimensional vector space V over a commutative field K of characteristic ≠ 2. We want to write given elements of the commutator subgroup ω(V) (of the orthogonal group O(V)) and also of the kernel of the spinorial norm ker(Θ) as (short) products of involutions and as products of commutators

On the Decomposition of a Representation of SOn When Restricted to SOn-1

1992 ◽  
Vol 44 (5) ◽  
pp. 974-1002 ◽  
Author(s):  
Benedict H. Gross ◽  
Dipendra Prasad
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Local Field ◽  
Orthogonal Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Special Orthogonal Group ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Linear Maps

Let k be a local field, with char(k) ≠ 2. A quadratic space V over k is a finite dimensional vector space together with a non-degenerate quadratic form Q: V → k.The special orthogonal group SO(V) consists of all linear maps T: V → V which satisfy:Q(Tv) = Q(v) for all ν and det T = 1.

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