scholarly journals A Plethysm Formula on the Characteristic Map of Induced Linear Characters from $U_n(\mathbb F_q)$ to $GL(\mathbb F_q)$

10.37236/2364 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Zhi Chen
Keyword(s):  
Finite Field ◽  
Recurrence Relation ◽  
Linear Group ◽  
General Linear Group ◽  
Symmetric Functions ◽  
General Linear ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Characteristic Map

This paper gives a plethysm formula on the characteristic map of the induced linear characters from the unipotent upper-triangular matrices $U_n(\mathbb F_q)$ to $GL_n(\mathbb F_q)$, the general linear group over finite field $\mathbb F_q$. The result turns out to be a multiple of a twisted version of the Hall-Littlewood symmetric functions $\tilde{P}_n[Y;q]$. A recurrence relation is also given which makes it easy to carry out the computation.

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.

Conjugacy classes of torsion in GL_n(Z)

2015 ◽  
Vol 30 ◽  
Author(s):  
Qingjie Yang
Keyword(s):  
Linear Group ◽  
Equivalence Relation ◽  
General Linear Group ◽  
Conjugacy Classes ◽  
Complete List ◽  
General Linear ◽  
Triangular Matrices ◽  
Torsion Element ◽  
Ring Of Integers

The problem of integral similarity of block-triangular matrices over the ring of integers is connected to that of finding representatives of the classes of an equivalence relation on general integer matrices. A complete list of representatives of conjugacy classes of torsion in the 4 × 4 general linear group over ring of integers is given. There are 45 distinct such classes and each torsion element has order of 1, 2, 3, 4, 5, 6, 8, 10 or 12.

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 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.

scholarly journals Finding hidden Borel subgroups of the general linear group

2012 ◽  
Vol 12 (7&8) ◽  
pp. 661-669
Author(s):  
Gabor Ivanyos
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Quantum Algorithm ◽  
General Linear ◽  
Base Field ◽  
Triangular Matrices ◽  
Hidden Subgroup Problem ◽  
Subgroup Problem ◽  
Lower Triangular ◽  
Lower Triangular Matrices

We present a quantum algorithm for solving the hidden subgroup problem in the general linear group over a finite field where the hidden subgroup is promised to be a conjugate of the group of the invertible lower triangular matrices. The complexity of the algorithm is polynomial when size of the base field is not much smaller than the degree.

ROOK PLACEMENTS AND CLASSIFICATION OF PARTITION VARIETIES B\ Mλ

2001 ◽  
Vol 03 (04) ◽  
pp. 495-500 ◽  
Author(s):  
KEQUAN DING
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Projective Variety ◽  
Quotient Space ◽  
Borel Subgroup ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Rook Placements ◽  
Ferrers Board

Let M be the set of m by n complex matrices of rank m. Let λ = (λ1,…,λm) be a partition with λi ≥ λi + 1 for 1 ≤ i ≤ m - 1 and λ1 = n. A Ferrers board Fλ is a right justified subarray in a m by n matrix with the length of the ith row being λi, and define Mλ={a ∈ M|ai,j = 0 if (i,j) ∉ Fλ}. Let B be the Borel subgroup of the general linear group GLm (C) consisting of upper triangular matrices. Define B\Mλ={Ba|a∈ Mλ}. The quotient space B\Mλ is a projective variety called a partition variety associated to λ. In this note, we classify partition varieties B\Mλ according to their homology and cohomology groups (up to isomorphisms).

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.

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