Questions tagged [general-linear-group]
The general linear group of order $n$, $GL_n(\mathbb{F})$, over a field $\mathbb{F}$ (usually $\mathbb{R}$ or $\mathbb{C}$) is the group of $n\times n$ invertible matrices over $\mathbb{F}$. The operation is the usual matrix multiplicatoin.
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$U_2$ homeomorphic to $S^3 \times S^1$, how? [duplicate]
Recently, I have been reading Linear Groups, chapter 9, from Michael Artin, Algebra book.
There is a question in the exercise (Chapter 9, Exercise section 3, no.3.2), given as follows
Prove that $U_2$ ...
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Action of general linear group on points of affine space
I'm not sure if my GAP code is correct. The exercise is to define action of general linear group on the points of 5-dimensional affine space over the smallest field.
gap> Afina:=AffineSpace(5,2);
...
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Irreducible actions of groups
Let $p$ be a prime and let $A$ be a vector space of finite dimension $n$ over the field of order $p$. Does $Aut(A)$ contains a group of the form $R\ltimes S$, where $R$ and $S$ are cyclic, $(|R|,|S|)=...
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Is the Valentiner Group isomorphic to PGL(3, 4)?
In this answer to a question from a while back, it says that the Valentiner group is isomorphic to $PGL_3(\mathbb{F}_4)$. However, when I implement in sage:
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Kernel of the natural map $\mathrm{GL}_2(\mathbb{Z})\to\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$
I want to study the kernel of the natural map $\mathrm{GL}_2(\mathbb{Z})\to\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$ given by reducing mod $n$ each entry of the matrix, let us call this group $G$. I ...
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Product of elementary matrices in $\mathrm{GL}_n(\mathbb{Z}_2)$ equal to the identity matrix.
For $i\neq j$ define the matrix $E_{i,j}$ in $\mathrm{GL}_n(\mathbb{Z}_2)$ to be the matrix that difers from the identity by a $1$ in the position $(i,j)$. I want to check if there exists matrices $...
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Understanding the Terminology of Closed Subsets in $GL(n, \mathbb{C})$
In the context of the general linear group $GL(n, \mathbb{C})$, we use specific terminology to describe closed subsets. I'm paying attention to two definitions, which I report here and which come from ...
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Find a basis that belongs to two arbitrary complete flags
The question is as below:
We say a basis of $V_n(F)$ belongs to a complete flag ($V_1$,...,$V_n$) if each $V_i$ contains exactly one element of the basis that is not in $V_{i-1}$.
(a)Show that if ($...
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Find the $\mathbb{Z}/2\mathbb{Z}$ rank of $\mathrm{GL}_n(\mathbb{R})$.
Given a group $G$ we define its $\mathbb{Z}/2\mathbb{Z}$ rank, denoted as $\mathrm{rk}_{\mathbb{Z}/2\mathbb{Z}}(G)$ to be the maximum $n$ such that $\left(\mathbb{Z}/2\mathbb{Z}\right)^n\...
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Every finite subgroup of $GL_n(Q)$ is conjugated to a subgroup of $GL_n(R)$.
Claim: let $R$ be a principal ideal domain and $Q$ be its quotient field. Then every finite subgroup of $GL_n(Q)$ is conjugated to a subgroup of $GL_n(R)$.
So I want to solve this exercise but I ...
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Is each representation induced by a simpler representation
Let $G$ be a finite group, $V$ be a real vector space of dimension $N$, and $N > |G|$.
Can you find a representation $\sigma: G \to GL(\text{End}(V))$ such that $\sigma$ is not induced by a ...
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Reference request: To find a proof of the fact that cuspidal characters vanish on non-primary conjugacy classes of finite general linear groups
Let $\chi$ be a cuspidal character of finite general linear group $GL(n,q)$. We call a conjugacy class of $GL(n,q)$ to be non-primary if the characteristic polynomial of its elements is divisible by ...
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Rational matrix whose power is an integer matrix
Problem For arbitrary matrix $A\in M_n(\mathbb Q)$ such that $\det (xI-A)\in \mathbb Z[x]$ and $\det A=\pm1$, is there any $k\in \mathbb N_+$ such that $A^k\in M_n(\mathbb Z)$?
This problem is ...
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Schur’s lemma over $\mathbb{F}_p$
I’m studying modular representation theory, and I got really stuck with the seemingly innocent statement.
Consider $\mathrm{GL}_{2}(\mathbb{F}_{p})$ and its center $Z$, which is just a set of all ...
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Quotient group structure of general linear group
I want to find all quotient group structure in $GL(\mathbb{R}^n)$, which is all n*n invertible matrix on real number. The quotient group structure in $GL(\mathbb{R}^n)$ means to find normal subgroup $...
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Does every spherical sector of a circle in $\mathrm{M}_n(\mathbb{C})$ contain an invertible matrix?
Some context :
(Notations :
For the rest of the post, I will identify $\mathrm{M}_n(\mathbb{C})$ with $\mathbb{C}^{n^2}$, and I will note $\mathcal{S}^{n^2-1}$ the unit sphere of $\mathbb{C}^{n^2}$)
I ...
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References for Maltsev's Theorem on $GL_n(F) \equiv GL_m(K)$ iff $K \equiv F$ and $m=n$.
I have recently found Maltsev's theorem: for $F$ and $K$ algebraically closed fields, $GL_n(F) \equiv GL_m(K)$ if and only if $K \equiv F$ and $m=n$ (thanks to this question: https://mathoverflow.net/...
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Does there exist an element of order $4$ in $GL_2(\mathbb{Z})/GL_2(\mathbb{Z})'$?
For a group $G$, let $G'$ denote the commutator of the group $G$, and if $H \leq G$ the left cosets will be denoted as $gH$. Now, I understand the fact that $[SL_2(\mathbb{Z}):GL_2(\mathbb{Z})'] = 2.$
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Decomposing (unimodular) matrix over integers into product of matrices mod d
If I have a $n\times n$ unimodular matrix $A \in \text{GL}(n,\mathbb Z)$, i.e., with elements $A_{ij} \in \mathbb Z$, is there some way to decompose the matrix into a product of matrices $A=A^{(1)}A^{(...
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When is the quotient of $\operatorname{GL}_n(\mathbb{R})$ by a discrete subgroup compact?
My question is exactly that on the title.
I'm interested in the action of some (discrete) subgroup $H$ on $\operatorname{GL}_n(\mathbb{R})$ by left multiplication.
For example, $H$ can be
$\...
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Show that as groups, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$, with the isomorphism given by $\theta \mapsto A_\theta$
Show that as groups, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$, with the isomorphism given by $\theta \mapsto A_\theta$. The notation can be found in my attempt.
1st Attempt:
If $\dim_\mathbb{F} ...
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What are the continuous outer automorphisms of the general linear group?
Is the only continuous outer automorphism of $\operatorname{GL}(n, \mathbb{R})$ the transpose inverse map $g \mapsto (g^\intercal)^{-1}$? If not, what other continuous outer automorphisms are there?
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Some property of GL(2,R) and GL(2,Z)
I am trying to show that there exists a family of matrices $(M_n)$ in $GL(2, \mathbb{R}) $ such that $GL(2,\mathbb{Z}) M_n GL(2,\mathbb{Z})$ is the same for every n and $GL(2,\mathbb{Z}) M_n $ is ...
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Showing that the isomorphism of the general linear group of a vector space with the group of invertible matrices is smooth
This is Example 7.3(e) from John Lee's Introduction to Smooth Manifolds.
If $V$ is any real or complex vector space, $GL(V)$ denotes the set of invertible linear maps from $V$ to itself. It is a group ...
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How many elements in $GL_2(\mathbb{F}_p)$ are conjugate to $\begin{pmatrix} \lambda & 0 \\ 0 & \mu\end{pmatrix}$?
How many elements in $GL_2(\mathbb{F}_p)$ are conjugate to $\begin{pmatrix} \lambda & 0 \\ 0 & \mu\end{pmatrix}$ for fixed, distinct $\lambda, \mu \in \mathbb{F}_p$, $p$ prime?
I tried arguing ...
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Proof that $GL(n,\mathbb{C})$ is isomorphic to a properly embedded Lie subgroup of $GL(2n,\mathbb{R})$
Below is an example of embedded Lie Subgroup from John Lee's Introduction to Smooth Manifolds example 7.18 (d).
In this example, why is the image of $\beta$ a properly embedded Lie subgroup of $GL(2n,\...
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$F \subset E$ as a sub-vector bundle of E, Show that it is a $GL_p$-reduction of $Fr(E)$
Suppose $F \subset E$ is a sub-bundle of E,where one uses "frames of $E$ adapted to $F$", meaning that the first $p$ components of the frame are a frame of $F$ (where we denoted by $r$ and $...
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Query Regarding the Proof of Z(SO(2m))={±1} in Section 3.7 of 'Naive Lie Theory' by John Stillwell
I have been reading John Stillwell's 'Naive Lie Theory' and in Section 3.7, the author tries to prove that $Z(SO(2m)) = {\pm 1}$. In the proof, a matrix $I^\star$ is introduced with the form:
$$ I^\...
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Prove that $\operatorname{GL}_n(k)$ and $\operatorname{SL}_n(k)$ cannot be symmetric group?
Problem. Prove that $\operatorname{GL}_n(k)$ and $\operatorname{SL}_n(k)$ cannot be isomorphic to $S_m$, $m\geq 4$ if $k$ is a finite field with at least two elements.
I am trying to argue by looking ...
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is there any unit upper triangular matrix in $GL_n(F_p)$ such that the following holds
Let $U_1$ be the group of $n\times n$ upper triangular matrices with 1's down the main diagonal (called unit upper triangular matrices) over $\mathbb{F}_p$ , which is also a Sylow $p$-subgroup of $G=\...
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Fundamental domains for congruence subgroups $\Gamma_0(N)$ by transformations of the fundamental domain of $SL_2(\mathbb{Z})$
I recently started to study conguence subgroups and quotients of the upper half-plane by their action. I found various proofs of the existence of the fundamental region for congruence subgroups that ...
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Perfect Subgroups of $GL(2,p)$, where $p$ is an odd prime. [closed]
It is known that $SL(2,p)$ is a perfect subgroup of $GL(2,p)$ if $p>3$. My question is:
Are they the only perfect subgroups of $GL(2,p)$?
If not, can $GL(2,p)$ have perfect subgroups whose order is ...
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Number of homomorphisms from $\Bbb Z_p$ to $\operatorname{GL}_2(\Bbb Z_q)$?
I am trying to count the total possible number of group homomorphisms from $\Bbb Z_p$ to $\operatorname{GL}_2(\Bbb Z_q)$, where $p<q$ are primes and $\Bbb Z_p$ denotes the additive group modulo $p$....
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Does $S_n$ always embed into $GL_{n-1} (\mathbb{F}_p$)?
$S_n$ is the symmetry group of the standard $n-1$-simplex, which is the convex hull of the standard basis vectors in $\mathbb{R}^n$. One can orthogonally project this shape onto the plane $x_1 +...+ ...
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General linear group inclusion
Do we have $\operatorname{GL}(n,F)\le \operatorname{O}(2n,F)$ where O means general orthogonal group and $F$ is an algebraically closed field?
I checked some finite group cases: $\operatorname{GL}(2,5)...
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Operatornorm of powers of Matricies with integer coefficient
Let $A\in GL_n(\mathbb{Z})$ have infinite order, so $A^k\neq Id_n$ for all $k>0$. The operator norm is defined by $\lVert A \rVert=\max\{\lVert Av\rVert \mid v\in\mathbb{R}^n: \lVert v\rVert=1\}$. ...
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Emil Artin determinant is unique
In book of Emil Artin "Geometric Algebra" i found such definition of determinant Det: function on matrix rows $A_{1}, ..., A_{n}$, that satisfies:
$Det(A_{1}, ..., b*Ai,..., A_{n})=b*Det(A_{...
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Dieudonne determinant is well defined.
Consider $GL_{n}(K)$ over field. By Dieudonne, $\forall A \in GL_{n}(K) \ (A = B*D(x))$ where $B$ is multiplication of some transvections and $D(x)=$
\begin{bmatrix}
1 & 0 &0& \dots \\...
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Is determinant canonical projection $\det:GL(n,\mathbb R) \rightarrow GL(n,\mathbb R)^{ab}$?
Is it possible to define determinant as canonical projection from general linear group to its abelianization? Using determinant we can show, that abelianization of $GL(n,R)$ is isomorphic to $R^{*}$ - ...
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Any homomorphism from $GL(n, F)$ to $F$ is composition of $det$ and $F$-endomorphism. [duplicate]
I found statement of theorem, that for any field $F$, any homomorphism $f:GL(n, F)\rightarrow F^{*}$ is composition $f=g\circ det$ for some $g:F^{*}\rightarrow F^{*}$ - endomorphism, and $det$ - ...
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Linearity of Groups - does it matter which linear groups we consider?
In J. Meier's book "groups, graphs and trees" after remark 3.8 it is stated that
A group that can be faithfully represented as a matrix group is called
a linear group.
Other sources (most ...
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Representations of general linear Lie algebra vs general linear group [duplicate]
I know that $\mathrm{GL}_{n}(\mathbb{C})$ is not simply connected. Therefore I don’t quite understand the correspondence between representations of $\mathrm{GL}_{n}(\mathbb{C})$ and $\mathfrak{gl}_{n}(...
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Why $SL$-invariants and highest weight vectors of rectangular shape coincide?
The groups $\mathrm{SL}(n) := \mathrm{SL}(n,\mathbb{C})$ and $\mathrm{GL}(n) = \mathrm{GL}(n,\mathbb{C})$ acts on $\mathbb{C}^n$ by multiplication from the left. This induces the diagonal action on $\...
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Invertibility of Block Matrix Partial Transpose
Let $$M = \left[\begin{matrix}
M_{1,1} & M_{1,2} & \cdots & M_{1,n}\\
M_{2,1} & M_{2,2} & \cdots & M_{2,n}\\
\vdots & \vdots & \ddots & \vdots\\
M_{n,1} & M_{n,...
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Index of an explicit subgroup of $\mathrm{GL}_4(\mathbb{Z})$
Let $H$ be the subgroup of $\mathrm{GL}_4(\mathbb{Z})$ generated by the $4!$ permutation matrices together with
$$
\begin{pmatrix}
1 & 0 & 0 & 0 \\
-1 & 0 & 1 & 1 \\
0 & 1 &...
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Principal series representation isomorphism
The problem:
Let $G = \mathrm{GL}_2(\mathbb Q_p)$ and $k$ be an algebraically closed field of characteristic $p.$ Denote by $\overline B$ the subgroup of all lower triangular matrices in $G$ and by $...
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Let $R$ be a principal ideal domain with field of fractions $K$, then $N_{GL_n(K)}(GL_n(R)) = K^\times GL_n(R)$
Let $R$ be a principal ideal domain with field of fractions $K$. Let $\mathcal{G}_n(K)$ denote the set of subgroups of $GL_n(K)$, where $GL_n(K)$ acts by conjugation on $\mathcal{G}_n(K)$:
$$GL_n(K) \...
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Is $GL(n)\otimes GL(n)$ is closed in $GL(n^2)$?
Let $GL(n) := \mathrm{GL}(n, \mathbb{C})$ - be space of invertible $n\times n$ matrices over $\mathbb{C}$, i.e. matrix Lie group.
Let $H := GL(n) \otimes GL(n)$ be a group with multiplication given by ...
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Equation in $GL(2,\mathbb{Z}_{2^{n-1}})$
I'm now studying how to embed generalized quaternion group $Q_{2^n}=a^{2^{n-1}}=e$, $u^2=a^{2^{n-2}}$, $ua=a^{-1}u$ in $GL(k,\mathbb{Z}_{2^{n-1}}).$ I got the embedding in case $k=3$, but $k=2$ is ...
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$GL_n(C)$ is isomorphic to a lie subgroup of $GL(2n,R)$
$GL_n(C)$ is isomorphic to a lie subgroup of $GL_{2n}(R)$.
I see some posts concerning this (not the same claim):
$GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
$GL_n\mathbb{C}...
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