All Questions
Tagged with linear-algebra gr.group-theory
193 questions
-3
votes
0
answers
139
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
4
votes
2
answers
206
views
Normalizers of the principal congruence subgroups in $\mathrm{GL}(n,\mathbf Q)$
A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence ...
- 43
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
201
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
14
votes
2
answers
851
views
Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...
- 143
1
vote
0
answers
189
views
The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 173
5
votes
1
answer
349
views
Commutator subgroup of $\mathrm{GL}_n(R)$ when $R$ is a PID with unity
Hua and Reiner in their paper titled "Automorphisms of the unimodular group" have established what will be the commutator subgroup of $\mathrm{GL}_n(\mathbb{Z})$, $\forall n\geq 0$. I have ...
- 51
2
votes
0
answers
101
views
On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
- 667
9
votes
3
answers
350
views
$G$-module structure of the relation module for a presentation of a finite group $G$
Let $F_n$ be a free group of rank $n\ge 2$, and $F_n\rightarrow G$ a surjection with $G$ finite. Let $R$ be the kernel. From this, we get an action of $G$ on the abelianization $R/R'$ (a free abelian ...
- 7,527
1
vote
1
answer
206
views
Reflections on subspaces of $\text{codim} > 1$
Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$.
Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\...
- 1,813
3
votes
0
answers
70
views
Admissibility of Ulm's invariants
Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define
$$G_{\alpha}=pG_{\beta}.$$
If $\alpha$ is a limit ...
- 31
4
votes
1
answer
211
views
Nonempty intersection of cosets of finite-index subgroups
$\DeclareMathOperator\lcm{lcm}$This question is crossposted from MSE.
Let $H_1,\dots,H_{n+2}$ be cosets of finite-index subgroups of $\mathbb{Z}^n$ and suppose for all $i=1,\dots,n+2$, $\bigcap_{j\neq ...
- 10.6k
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
7
votes
1
answer
633
views
Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
- 823
4
votes
0
answers
219
views
Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$
EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect.
Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
- 7,300
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
1
vote
0
answers
69
views
Convolutions of (m)-associahedra and (m)-noncrossing partition polynomials--combinatorial proofs?
I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II).
(Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...
- 10.5k
15
votes
3
answers
1k
views
Group of matrices in which every matrix is similar to unitary
$\DeclareMathOperator\GL{GL}$Let $G$ be a subgroup of $\GL_n(\mathbb{C})$ such that for every $g \in G$ there exists $c \in \GL_n(\mathbb{C})$ for which $cgc^{-1}$ is unitary (or, which is the same, $...
3
votes
0
answers
71
views
Automorphisms of matrix algebras and Picard group
This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE).
Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
- 1,766
1
vote
0
answers
206
views
About the question "Tannaka–Krein duality"
I saw this post recently: Tannaka–Krein duality
I have this question please: in the following which I report here:
The problem is with surjectivity: let us denote $\mathcal{G}:=\mathcal{G}(\mathcal{R}...
- 29
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
4
votes
0
answers
108
views
Doubly stochastic matrices that remain doubly stochastic after conjugating by the character table of a finite abelian group
I am curious if anything is known about the following.
Let $\Gamma$ be a finite abelian group, and let $\chi$ be its character table, normalized so that it is a unitary matrix. E.g., if $\Gamma$ is $\...
- 2,339
0
votes
0
answers
172
views
A characterisation of full subgroups of $\mathrm{GL}_n(\mathbf{F}_p)$
Let $p\geq 5$ be a prime and $\mathbf{F}_p$ a finite field of characteristic $p$. A subgroup of ${\rm GL}_n(\mathbf{F}_p)$ is full if it contains ${\rm{SL}}_n(\mathbf{F}_p)$. When $n=2$, we have the ...
- 667
4
votes
1
answer
277
views
Is there a good notion of kernels of quadratic forms on abelian groups?
Let $G$ be an abelian group and let $q:G \to \mathbb{Q/Z}$ be a quadratic form, i.e. $q(a)=q(-a)$ and $b(x,y)=q(x+y)-q(x)-q(y)$ is a bihomomorphism. On vector spaces, when people speak about the ...
- 1,813
4
votes
0
answers
112
views
Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.1k
4
votes
0
answers
107
views
Two questions about $\mathbf{PSL}(V)$ with $V$ a vector space over a division ring
Let $V$ be a (possibly infinite-dimensional) vector space over a division ring $d$, and consider the projective special linear group $\mathbf{PSL}(V)$. We suppose that if the dimension of $V$ would be ...
- 4,547
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
1
vote
1
answer
185
views
A system of linear equations with way too many unknowns — constructing a bivariate distribution from marginals and "the diagonal"
Suppose we are given information about distributions of random permutations $\sigma, \tau : \Omega \to S_n$ as follows:
$$p^1_{k,l} = \mathbb P(\sigma(k) = l), p^2_{k',l'} = \mathbb P(\tau(k) = l), p^{...
- 677
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
2
votes
1
answer
389
views
Existence of regular semisimple elements in linear group over local field
Let $ L $ be a finite extension of $p$-adic numbers $ \mathbb{Q}_p $. Let $ \text{GL}_{n}(L) $ denote the general linear group $ \text{GL}_{n}(L) $ over $L$ equipped with the topology induced from the ...
- 863
5
votes
1
answer
523
views
Is there a non-split algebraic torus (over a finite field) satisfying the following properties?
Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties?
$T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori ...
- 1,833
12
votes
1
answer
2k
views
Is there an eigenvalue of modulus larger than 1?
Given a matrix $A\in \operatorname{SL}_d(\mathbb{Z})$ (the special linear group) satisfying the two conditions: (1) no eigenvalue of $A$ is a root of unity, (2) the characteristic polynomial of $A$ is ...
- 43.2k
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
3
votes
0
answers
232
views
Difficulty about Jordan decomposition, (and also an ambiguity about the quadratic forms in indecomposable Jordan components of quadratic modules)
I am trying to understand a concept through solving some exercises, but I can't solve one of them, and I need a hint and guide.
I asked my questions in the boxes (See the end of this question). (I ...
3
votes
1
answer
495
views
Computing conjugacy between two elements of $\mathrm{SL}_2(\mathbb{Z})$
The conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$ are well characterized (see, e.g., this question). Assuming two matrices $A, B \in \mathrm{SL}_2(\mathbb{Z})$ are conjugate, is there a way to ...
- 39
1
vote
0
answers
91
views
Diophantine equation about the automorphism group of lattice by constraints
Fixed $\sigma_x=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right)$ and $K=\left(
\begin{array}{ccc}
3 & 32 & -64 \\
1 & 32 & -32 \\
-2 & -32 & 64 \\
\...
- 149
0
votes
0
answers
94
views
Large subgroups of infinite-dimensional vector spaces
Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$.
Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...
- 4,547
7
votes
1
answer
326
views
Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$
Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
1
vote
0
answers
32
views
Diameter of general linear group wrt monomials and fixed root subgroup
Let $G=GL_n(F)$ where $F$ is a field.
Let $S\subset G$ be the collection of monomial matrices in $G$ union a fixed root subgroup $U_{\alpha}$ of $G$. I.e.
$$
U_{\alpha}:=\{I_n+\lambda E_{i,j} : \...
- 121
3
votes
1
answer
237
views
invariant subspaces of general linear groups for finite fields
Let $K$ be a finite field, let $n\ge 1$ be an integer and let $G=\mathrm{GL}(n,K)$ be acting linearly on a finite dimensional $K$-vector space $V$. Although $G$ is a reductive group, it is not ...
- 7,722
0
votes
0
answers
99
views
Unimodular matrices fixing $(1, 1, \cdots, 1)$
What is known about the subgroup of $GL(n, \mathbb Z)$ fixing (under left multiplication) the vector ${(1, 1, \cdots, 1)}^T$ ('T' denotes transposition). I'm particularly interested in the case $n = 5$...
- 356
4
votes
1
answer
298
views
Characterizations of groups whose general linear representations are all trivial
Let $G$ be a group. Suppose for any general linear representation $\rho:G\to\mathrm{GL}(n)$,
$\rho$ must be trivial.
Question: Are there any characterizations or equivalent conditions for $G$?
Thanks ...
- 1,990
4
votes
2
answers
433
views
What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?
An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia.
I ...
- 2,689
1
vote
0
answers
144
views
Simultaneous similarity classes of pairs in $\mathrm{GL}_{n}(\Bbb Z / p\Bbb Z)$?
$\DeclareMathOperator{\GL}{\operatorname{GL}}$Let $G$ be an elementary abelian $p$-group of rank $2$. Let $\alpha, \beta :G\rightarrow \GL_{n}(\Bbb Z / p\Bbb Z)$ be two injective homomorphisms. The ...
- 181
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-...
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
8
votes
1
answer
305
views
Groups that act transitively on $\mathrm{Gr}(k,\Bbb R^n)$ but not transitively on $\mathrm{Gr}(k+1,\Bbb R^n)$
Is it known for which $n, k\in\Bbb N$ there exists a matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^n)$ that
acts transitively on $\mathrm{Gr}(k,n)$, i.e., on the $k$-dimensional subspaces of $\Bbb ...
- 13.6k
3
votes
2
answers
412
views
Indecomposable integral representations of a group of order 2 "by hand"
This question is a duplicate of
that 2010 MO question.
I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order $2$.
Clearly, ...
- 14.1k
2
votes
0
answers
75
views
Constructing representations of a topological group from characteristic polynomials of a generating set
Given a topological group $G$ and a subset $S$ of $G$ that topologically generates it, what are the conditions under which an $n$-dimensional continuous linear representation of $G$ over an ...
- 21