All Questions
Tagged with matrices gr.group-theory
123 questions
8
votes
1
answer
361
views
Invertible matrix with group ring coefficient
Before asking the question I do need
some notations.
$G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$
$R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings.
$Mat_{n}(R)$ the ...
- 223
7
votes
2
answers
200
views
When is a linear isomorphism of $M_n(\mathbb{C})$ given by unitary conjugation?
Let $M_n(\mathbb{C})$ represent the space of $n \times n$ matrices over $\mathbb{C}$. We will think of it as a $\mathbb{C}$-vector space.
Notice that if $A \in M_n(\mathbb{C})$ is invertible, then the ...
- 431
9
votes
2
answers
646
views
Are these two methods for constructing Hadamard matrices known?
These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers:
Context:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this ...
- 3,054
1
vote
1
answer
214
views
Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
- 187
0
votes
1
answer
171
views
Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 823
4
votes
1
answer
214
views
Diameter of the unimodular group with Gauss moves
$\DeclareMathOperator\GL{GL}$Consider the unimodular group $\GL_n(\mathbb{Z})$, consisting of integral matrices $A \in \mathbb{Z}^{n \times n}$ such that that $\det(A) =\pm 1$.
It is well known that ...
- 327
2
votes
0
answers
137
views
Decompose a rational matrix as an integer matrix and an inverse of integer matrix
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
- 823
1
vote
1
answer
252
views
Smith normal form and last invariant factor of certain matrices
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight.
Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
- 823
2
votes
0
answers
90
views
decidability special case of column generation problem
I have the following problem:
Input: sub-spaces $V_1, \dots, V_d$ of $\mathbb{Z}^{d}$
Question: are there $v_i \in V_i$ such that the matrix $(v_1, \dots, v_d)$ has determinant $\pm 1$ (equivalently, ...
- 55
2
votes
1
answer
207
views
A subgroup of $\mathrm{SL}_n(\mathbb{Z}/p\mathbb{Z})$
Let $p$ be an odd prime, and consider the group
$$\{U\in \operatorname{SL}_n(\mathbb{Z}/p\mathbb{Z}) : U^{t}U=I \bmod p \}\subseteq \operatorname{SL}_n(\mathbb{Z}/p\mathbb{Z}).$$
I wonder what is the ...
- 253
5
votes
1
answer
200
views
Abelianisation of certain congruence subgroups in GL_2(Z)
$\DeclareMathOperator\SL{SL}\newcommand{\ab}{\mathrm{ab}}$Denote by $$\Gamma(m) = \left\{ \left( \begin{array}{cc} 1 +ma_{11} & m a_{12} \\ ma_{21} & 1 +ma_{22} \end{array} \right) \in \SL_2(\...
1
vote
0
answers
70
views
Another matrices for a semigroup with intermediate growth
Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...
- 883
3
votes
1
answer
111
views
Invariant subgroups of $\mathbb{Z}^m$ and commutativity
$\DeclareMathOperator\Im{Im}$Given a unimodular matrix $A$ (of finite order).
Every finite index subgroup of $\mathbb{Z}^m$ can be written as $\Im B = B\mathbb{Z}^m$ for some square integral matrix $B$...
- 61
1
vote
0
answers
75
views
Is the element in the connected component?
I posted this question at stack exchange, got two upvotes but no answer. If it doesn't belong here, please let me know.
In the algebraic group $G$ = PGL$_{8}$($\mathbb{C}$), there are two involutions $...
- 501
3
votes
0
answers
132
views
the growth rate of poly-$\mathbb{Z}$ group
I am interested in the growth rate of the poly-$\mathbb{Z}$ group. Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\...
- 823
1
vote
0
answers
74
views
Associating octonionic matrices
Is it possible for three square octonionic matrices to associate multiplicatively even though some or all of their entries do not? If so, is it possible to construct matrix groups in this way?
- 2,773
2
votes
1
answer
512
views
Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ [closed]
This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help).
Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i....
- 823
5
votes
3
answers
673
views
Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix
Let $M $ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...
- 823
0
votes
0
answers
92
views
Classification of elements $GL(d, \mathbb{R})$
Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here.
Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
- 1,043
15
votes
1
answer
821
views
Conjugated subgroups in $\mathsf{GL}(m+n,\mathbb{Z})$ implies conjugated subgroups in $\mathsf{GL}(n,\mathbb{Z})$?
In my research I came up with the following question:
Question: Let $H_1$ and $H_2$ be finite abelian subgroups of $\mathsf{GL}(n,\mathbb{Z})$. Define $$ H_1'=\left\{\begin{pmatrix} I_m &0\\0&...
5
votes
1
answer
175
views
Growth of the word norm for elementary matrices in $\rm SL_3 (\mathbb{Z})$
This is a reference request, since the answer is probably well known, but I could not find it.
Given a finitely generated group $\Gamma$ with a generating set $S$, define the word norm $l = l_S : \...
- 1,163
4
votes
1
answer
204
views
Making Hermitian matrices almost commute
Consider two Hermitian matrices $A, B \in \mathbb{C}^{n \times n}$. I'm interested in finding another Hermitian matrix $A'$ that is close to $A$ and almost commutes with $B$. More precisely, I'd like ...
- 209
2
votes
1
answer
187
views
A closed subgroup $G$ of $\operatorname{GL}_2 \mathbb{Z}_\ell$ which surjects onto $\operatorname{GL}_2 \mathbb{F}_\ell$
Let $\ell \ge 5$ be a prime and $G$ be a closed subgroup of $\operatorname{GL}_2 \mathbb{Z}_\ell$ whose image in $\operatorname{GL}_2 \mathbb{F}_\ell$ is $\operatorname{GL}_2 \mathbb{F}_\ell$.
Then $G ...
- 185
4
votes
0
answers
180
views
Zariski density for certain subsemigroups
$\DeclareMathOperator\GL{GL}$Suppose that we have a Zariski-dense subgroup $\Gamma$ of $\GL(d,\mathbb{R})$. Let $\delta$ be the abscissa of convergence of the series
$$
\sum_{x \in \Gamma} e^{-s \log\|...
- 89
9
votes
1
answer
228
views
On the coefficients that appear in finite groups of matrices with integer entries
Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}_n(\mathbb{Z})$. It is known that for sufficiently large $n$, ...
- 193
12
votes
2
answers
493
views
A specific coset decomposition of $\mathrm{GL}_n(\mathbb{C})$
Disclaimer: I am a theoretical chemist (not a mathematician). I have tried asking this question at Math SE with no luck (https://math.stackexchange.com/questions/4080696/a-specific-coset-decomposition-...
- 121
28
votes
1
answer
2k
views
Integer matrices which are not a power
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$In a group $G$, an element $g$ is said to be primitive if there is no $h \in G$ and integer $n >1$ such that $g = h^n$. (For clarification, I ...
5
votes
1
answer
420
views
Analogue of the special orthogonal group for singular quadratic forms
The special orthogonal group $SO(n)$ is the subgroup of the special linear group $SL(n)$ of $n\times n$ matrices with determinant one that preserve a non-degenerate symmetric bilinear form. If such a ...
user82886
4
votes
1
answer
776
views
Finding isomorphism between $\mathbb{Z}^2\ltimes_{A,B} \mathbb{Z}^4$ and $\mathbb{Z}^2\ltimes_{A,C} \mathbb{Z}^4$
Let $A,B(a,b,c,d)\in\mathsf{GL}(4,\mathbb{Z})$ be given by $$A=\begin{pmatrix} I_2 & \begin{pmatrix} 0&0\\0&1 \end{pmatrix} \\0& -I_2\end{pmatrix},\quad B(a,b,c,d)=\begin{pmatrix} -2a-...
3
votes
1
answer
302
views
Deciding isometry of unimodular lattices by Gram matrices
Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
- 9,501
1
vote
0
answers
214
views
When is the product of regular matrices regular
$\DeclareMathOperator{\GL}{\operatorname{GL}}$We say that a matrix $g \in \GL_n(F)$ is regular if it has a centraliser of minimal dimension, or equivalently, if the minimal and characteristic ...
- 121
9
votes
2
answers
530
views
A "subtle" isomorphism testing problem: $\mathbb{Z}\ltimes_{A} \mathbb{Z}^5\cong \mathbb{Z}\ltimes_{B}\mathbb{Z}^5$ or not?
EDIT: I've made a mistake with the matrices. Now it is corrected.
A couple of days ago I asked this question. There, answerers gave me excellent hints to solve that case and others too. But I've found ...
14
votes
3
answers
780
views
Deciding if $\mathbb{Z}\ltimes_A \mathbb{Z}^5$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^5$ are isomorphic or not
I asked this in this MSE question but I didn't get answers. I think maybe here someone can help me.
I have the two following groups
$G_A=\mathbb{Z}\ltimes_A \mathbb{Z}^5$, where $A=\begin{pmatrix} 1&...
2
votes
2
answers
265
views
Commuting nilpotent matrices and conjugation isomorphisms
Trying to study isomorphism classes of certain commutative Artinian $\mathbb{C}$-algebras I was lead to the following problem about matrices.
Suppose you have a (non-zero) nilpotent matrix $A\in M_n(\...
- 375
2
votes
1
answer
308
views
The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$
I asked this question in MSE few days ago but there was no response.
Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\...
- 379
5
votes
1
answer
729
views
The normalizer of block diagonal matrices
Let $G=\mathrm U_n$ or $\mathrm{GL}_n(\mathbf C)$ and $H$ the subgroup of block diagonal matrices respecting a partition $n=n_1+\dots+n_k$. Is the normalizer $N=N_G(H)$ computed anywhere in the ...
- 31.5k
0
votes
1
answer
104
views
Isomorphism between two groups [closed]
Let $\mathbf{F}_q$ be finite field of order $q$, where $q$ is an odd-prime (or power of an odd prime). Is there an isomorphism between the following subgroups of unipotent $3\times 3$ matrices over $\...
- 175
5
votes
0
answers
203
views
Number of elements in $\mathrm{GL}(n,p)$ with maximal order
I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.
I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
8
votes
0
answers
268
views
Membership problem in general linear group
This is surely a very well known problem, but I could not find an answer on MO or on Google, so here I am.
Given some finitely generated free subgroup $H$ of $\operatorname{GL}_n(\mathbb{Z}[t,t^{-1}])...
- 469
0
votes
1
answer
450
views
Conjugacy in the quaternion group
Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
- 1,703
13
votes
2
answers
1k
views
Abelianization of general linear group of a polynomial ring
For $K$ a field, is it known what the abelianization of $GL_2(K[X])$ is?
- 965
1
vote
0
answers
175
views
Abelian subgroup of maximal order
Let $\mathbf{F}_q$ be a finite field of order $q$ ($q$, an odd prime, or a power of the same). I know that for the matrix algebra $M(n,\mathbf{F}_q)$ (or $gl(n,q)$), the maximal dimension for a ...
- 175
12
votes
2
answers
983
views
Common basis for permutation matrices
How can I check whether there exists a common basis with respect to which two matrices 𝐴 and 𝐵 are permutation matrices?
More explicitly, let $A$ and $B$ be two unitary matrices whose eigenvalues ...
- 295
12
votes
3
answers
1k
views
Two equivalent irreducible representations given by integer matrices
Let $G$ be a finite group, and $\rho_1, \rho_2: G\to GL_n(\mathbb C)$ be two representations. Suppose that $\rho_1$ and $\rho_2$ are equivalent (i.e. conjugate over $\mathbb C$), and suppose that ...
- 14.3k
1
vote
0
answers
179
views
Matrix factorizations over $GL_2$ of a real quadratic ring of integers
tl;dr: The groups $GL_2(K)$, or $SL_2(K)$, where $K = \mathbb{C,R}$ admits several factorizations (the polar decomposition,
the KAN decomposition, the Schur triangular form, etc). Those
...
- 1,090
13
votes
2
answers
414
views
Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?
Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
- 23.3k
0
votes
0
answers
207
views
Sylow subgroups of orthogonal group
According to Wikipedia (current revision) the cardinality of $O(n,q)$ depends on the properties of the field we're working over. These are the results:
We have the following formulas for the order ...
- 109
6
votes
0
answers
259
views
Diameter of finite rational matrix groups
Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$.
For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
- 165
6
votes
1
answer
500
views
Rank and frequency of permutations
(a) Let $[n] = \{1,\dotsc,n\}$, and let $\pi:[n]\to [n]$ be a permutation. Define an $n$-by-$n$ matrix $A=A(\pi)$ as follows: $A_{i,j}=1$ if $j>i$ and $\pi(j)>\pi(i)$, $A_{i,j}=-1$ If $j<i$ ...
- 20.2k
10
votes
3
answers
680
views
Direct product of free groups in $\mathrm{SL}_3(\mathbb{Z})$
Let $\mathbb{F}_2$ be the free group on two generators. Does $\mathbb{F}_2 \times \mathbb{F}_2$ embed as a subgroup of $\mathrm{SL}_3(\mathbb{Z})$?
- 1,769