On Modular Vector Invariant Fields

2020 ◽  
Vol 27 (04) ◽  
pp. 749-752
Author(s):  
Ying Han ◽  
Runxuan Zhang
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
Dual Space ◽  
General Linear Group ◽  
Homogeneous Polynomials ◽  
Invariant Polynomials ◽  
Standard Representation ◽  
Direct Sum ◽  
Invariant Ring ◽  
Vector Invariant

Let [Formula: see text] be a finite field of any characteristic and [Formula: see text] be the general linear group over [Formula: see text]. Suppose W denotes the standard representation of [Formula: see text], and [Formula: see text] acts diagonally on the direct sum of W and its dual space W∗. Let G be any subgroup of [Formula: see text]. Suppose the invariant field [Formula: see text], where [Formula: see text] in [Formula: see text] are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials [Formula: see text] in the invariant ring [Formula: see text] such that the invariant field [Formula: see text] is generated by [Formula: see text] over [Formula: see text].

scholarly journals Relations between modular invariants of a vector and a covector in dimension two

2020 ◽  
pp. 1-8
Author(s):  
Yin Chen
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Two Dimensional ◽  
Modular Invariant ◽  
Invariant Ring ◽  
Generating Relations ◽  
Modular Invariants

Abstract We exhibit a set of generating relations for the modular invariant ring of a vector and a covector for the two-dimensional general linear group over a finite field.

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.

Intersection graphs of general linear groups

2020 ◽  
pp. 2150039
Author(s):  
Mai Hoang Bien ◽  
Do Hoang Viet
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
General Linear Group ◽  
Undirected Graph ◽  
Intersection Graph ◽  
General Linear ◽  
Linear Groups ◽  
Intersection Graphs ◽  
General Linear Groups ◽  
Vertex Set

Let [Formula: see text] be a field and [Formula: see text] the general linear group of degree [Formula: see text] over [Formula: see text]. The intersection graph [Formula: see text] of [Formula: see text] is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of [Formula: see text]. Two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] is a finite field containing at least three elements, then the diameter [Formula: see text] is [Formula: see text] or [Formula: see text]. We also classify [Formula: see text] according to [Formula: see text]. In case [Formula: see text] is infinite, we prove that [Formula: see text] is one-ended of diameter 2 and its unique end is thick.

scholarly journals Abelian p-subgroups of the general linear group

1970 ◽  
Vol 11 (3) ◽  
pp. 257-259 ◽  
Author(s):  
J. T. Goozeff
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
Finite Fields ◽  
Abelian Subgroup ◽  
General Linear Group ◽  
General Linear ◽  
Normal Abelian Subgroup ◽  
Abelian Subgroups

A. J. Weir [1] has found the maximal normal abelian subgroups of the Sylow p-subgroups of the general linear group over a finite field of characteristic p, and a theorem of J. L. Alperin [2] shows that the Sylow p-subgroups of the general linear group over finite fields of characteristic different from p have a unique largest normal abelian subgroup and that no other abelian subgroup has order as great.

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):  
Tensor Product ◽  
Finite Field ◽  
Linear Group ◽  
General Linear Group ◽  
Polynomial Algebra ◽  
General Linear ◽  
Minimal Set ◽  
Equivalent Problem ◽  
Monomial Basis ◽  
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 $GL(t, \mathbb Z_2).$ An equivalent problem is to find a monomial basis of the space of "unhit" elements, $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each $t$ and degree $n\geq 0.$ The structure of this tensor product is proved surprisingly difficult and has been not yet known for $t\geq 5,$ even for the trivial subgroup $G_t = \{e\}.$ In the present paper, we consider the subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators of $(\mathbb P^{\otimes t})^{G_t}$ in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [23, 25].

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.

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