A random process related to a random walk on upper triangular matrices over a finite field

2014 ◽  
Vol 89 ◽  
pp. 77-80
Author(s):  
Martin Hildebrand
Keyword(s):  
Random Walk ◽  
Finite Field ◽  
Random Process ◽  
Triangular Matrices ◽  
Upper Triangular Matrices

Automorphisms of the total graph over upper triangular matrices

2019 ◽  
Vol 19 (08) ◽  
pp. 2050161
Author(s):  
Long Wang ◽  
Xianwen Fang ◽  
Fenglei Tian
Keyword(s):  
Finite Field ◽  
Total Graph ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisors ◽  
Singular Matrices

Let [Formula: see text] be a finite field, [Formula: see text] the ring of all [Formula: see text] upper triangular matrices over [Formula: see text], [Formula: see text] the set of all zero-divisors of [Formula: see text], i.e. [Formula: see text] consists of all [Formula: see text] upper triangular singular matrices over [Formula: see text]. The total graph of [Formula: see text], denoted by [Formula: see text], is a graph with all elements of [Formula: see text] as vertices, and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we determine all automorphisms of the total graph [Formula: see text] of [Formula: see text].

Automorphism group and fixing number of the orthogonality graph based on rank one upper triangular matrices

2020 ◽  
pp. 2250023
Author(s):  
Shikun Ou ◽  
Yanqi Fan ◽  
Fenglei Tian
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Canonical Form ◽  
Undirected Graph ◽  
Induced Subgraph ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisors ◽  
Rank One ◽  
Vertex Set

The orthogonality graph [Formula: see text] of a ring [Formula: see text] is the undirected graph with vertex set consisting of all nonzero two-sided zero divisors of [Formula: see text], in which for two vertices [Formula: see text] and [Formula: see text] (needless distinct), [Formula: see text] is an edge if and only if [Formula: see text]. Let [Formula: see text], [Formula: see text] be the set of all [Formula: see text] matrices over a finite field [Formula: see text], and [Formula: see text] the subset of [Formula: see text] consisting of all rank one upper triangular matrices. In this paper, we describe the full automorphism group, and using the technique of generalized equivalent canonical form of matrices, we compute the fixing number of [Formula: see text], the induced subgraph of [Formula: see text] with vertex set [Formula: see text].

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.

scholarly journals ON THE KROHN–RHODES COMPLEXITY OF SEMIGROUPS OF UPPER TRIANGULAR MATRICES

2007 ◽  
Vol 17 (01) ◽  
pp. 187-201 ◽  
Author(s):  
MARK KAMBITES
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Finite Fields ◽  
Finite Semigroup ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Triangular Representation

We consider the Krohn–Rhodes complexity of certain semigroups of upper triangular matrices over finite fields. We show that for any n > 1 and finite field k, the semigroups of all n × n upper triangular matrices over k and of all n × n unitriangular matrices over k have complexity n - 1. A consequence is that the complexity c > 1 of a finite semigroup places a lower bound of c + 1 on the dimension of any faithful triangular representation of that semigroup over a finite field.

scholarly journals Restricting Supercharacters of the Finite Group of Unipotent Uppertriangular Matrices

10.37236/112 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Nathaniel Thiem ◽  
Vidya Venkateswaran
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Field ◽  
Symmetric Group ◽  
Irreducible Representations ◽  
Combinatorial Theory ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Supercharacter Theory

It is well-known that understanding the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies the supercharacter theory of a family of subgroups that interpolate between $U_{n-1}$ and $U_n$. We supply several combinatorial indexing sets for the supercharacters, supercharacter formulas for these indexing sets, and a combinatorial rule for restricting supercharacters from one group to another. A consequence of this analysis is a Pieri-like restriction rule from $U_n$ to $U_{n-1}$ that can be described on set-partitions (analogous to the corresponding symmetric group rule on partitions).

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