Questions tagged [linear-groups]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
7 votes
1 answer
293 views

How can I detect the homology image of a unipotent group in the general linear group?

Suppose $n$ is a positive integer greater than 2, and $F$ is an arbitrary field with at least 4 elements. Denote $\text{GL}_n(F)$ the general linear group in the usual sense and $U_n(F)$ the unipotent ...
user avatar
5 votes
2 answers
220 views

If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?

I've copied over this question from what I asked on StackExchange, in the hope that an expert here can readily answer the question. Is there an example of a group $G=K\rtimes \mathbb{Z}$ satisfying ...
user avatar
0 votes
0 answers
35 views

On lattice generated by integral vectors close to eigenvectors

$M\in\mathbb Z^{n\times n}$ is an unimodular symmetric matrix whose eigenvectors are $v_1$ to $v_n$ and the lattice $\mathcal L$ generated by $M$ have the shortest vectors $u_1$ to $u_n$ where $u_i$ ...
user avatar
  • 13k
0 votes
1 answer
223 views

Generators of $SL(n,\mathbb F_2)$? [closed]

Consider the invertible matrices in $\mathbb F_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z_{\geq1}$ independent of $n$) generators for ...
user avatar
  • 13k
13 votes
1 answer
292 views

Is every finite $d$-dimensional matrix group generated by $d$ elements?

The question is in the title. If $\Gamma\subset\mathrm{GL}(\Bbb R^d)$ is a finite matrix group, can it be generated by (at most) $d$ elements? I suspect that this hope is too naive, but I have no ...
user avatar
  • 9,887
10 votes
2 answers
394 views

Sequence of epimorphisms of residually finite groups stabilizes

Let $G_1 \to G_2 \to \cdots$ be a sequence of epimorphisms of finitely generated residually finite groups. Does it eventually stabilize? That is, are all but finitely many epimorphisms actually ...
user avatar
  • 435
4 votes
0 answers
92 views

Question about generalizing Cauchy identity

One of the Cauchy identities says that $$\prod_{i,j}(1+x_iy_j) = \sum_\lambda s_\lambda (x_1, \cdots,x_m) s_{\lambda'} (y_1, \cdots,y_n) $$ Where $\lambda$ is a Young diagram, $\lambda'$ is the ...
user avatar
  • 121
6 votes
0 answers
310 views

Centralizer of elements in the upper-triangular matrices

Let $p$ be a prime number and $G=\operatorname{GL}_n ( \mathbb{Z} / p \mathbb{Z} )$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. ...
user avatar
1 vote
1 answer
130 views

Upper bounds for difference of entries between matrices and their inverses in $\mathsf{GL}_k(\mathbb Z)$

Let $a(M)$ be the maximum absolute value of entries of matrix $M\in\mathsf{GL}_k(\mathbb Z)$. $M^{-1}\in\mathsf{GL}_k(\mathbb Z)$ holds. What is a good upper bound for $|a(M)-a(M^{-1})|$? I am ...
user avatar
  • 1,728
3 votes
1 answer
111 views

Is the Singer cycle preserved by field automorphisms and graph automorphisms?

Let $T=\operatorname{PSL}_n(q)$ with $n$ a prime number. Then the $\mathscr{C}_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}_{\frac{q^n-1}{(q-1)(n,q-1)}}{:...
user avatar
1 vote
0 answers
62 views

Conjugacy classes and normal form of $O_n$ and $U_n$

I'm interested in characterizing conjugacy classes inside $O_n$ and $U_n$ over local fields of positive characteristic ($\neq 2$). I need this for my research on representation theory of these groups. ...
user avatar
  • 11
2 votes
1 answer
159 views

Relation to the Bruhat cell

Let $g\in\operatorname{SL}_n(\mathbb Z)$ such there exists $v\in\mathbb Q^n$ such that $v, gv, \dotsc, g^{n−1}v$ is a $\mathbb Q$-base of $\mathbb Q^n$ and there exists a $\mathbb Z$-base $w_1, \dotsc,...
user avatar
  • 332
1 vote
1 answer
169 views

If $\Lambda \cap U$ is Zariski-dense in $U$, then $\Lambda$ contains $U(k\mathbb Z)$ for some $k ≥ 1$?

If $G$ is a $\mathbb Q$-defined subgroup of $\operatorname{GL}_n(\mathbb C)$, $\Lambda$ is a subgroup of $G(\mathbb Z)$, and $U$ is a unipotent subgroup of $G(\mathbb C)$ such that $\Lambda \cap U$ is ...
user avatar
  • 332
4 votes
2 answers
435 views

Non-torsion part of the abelianisation of congruence subgroups

I've posted this question on math.stackexchange, but haven't gotten any responses so I'm trying here instead. Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite ...
user avatar
7 votes
1 answer
342 views

$\operatorname{Out}(F_n)$ is not linear for $n > 3$

The paper The Tits alternative for $\operatorname{Out}(F_n)$ I by Bestvina, Feighn and Handel and the paper Automorphisms of free groups and Outer space by Vogtmann both state that $\operatorname{Aut}(...
user avatar
  • 275
2 votes
0 answers
300 views

Finite simple subgroups of $\mathrm{GL}_n(\mathbb{C})$

Let $n$ be a fixed positive integer. Is it true that there are only finitely many isomorphism classes of finite nonabelian simple subgroups of $\mathrm{GL}_n(\mathbb{C})$? I'm especially interested ...
user avatar
  • 4,406
4 votes
1 answer
247 views

Automorphisms of products of $GL_n(\mathbb{Z})$ 's

It is a Theorem of Hua y Reiner (1951) that the group or outer automorphisms $Out(GL_n(\mathbb{Z}))$ is either isomorphic to $\mathbb{Z}/2$, if $n$ odd or $n=2$, or to $\mathbb{Z}/2 \times \mathbb{Z}/...
user avatar
12 votes
0 answers
272 views

Bijective proof of an identity involving number of standard Young tableaux and semistandard tableaux

Question. Can you find a bijective proof of the identity $$ \operatorname{dim}(S^{\lambda} \mathbb{C}^m)\ \operatorname{dim}(S^{\lambda'} \mathbb{C}^n) \ f^{n^m} = \dim \Lambda^p (\mathbb{C}^m \...
user avatar
16 votes
1 answer
624 views

Transitive actions of finite subgroups of ${\rm GL}(n,\Bbb Z)$ on projective geometries

For any $n$, the group ${\rm GL}(n,\Bbb Z)$ has a natural action on $\Bbb Z^n$. Modding out a prime $p$ yields an action on the vector space $F_p^n$, where $F_p$ is the finite field with $p$ elements. ...
user avatar
3 votes
1 answer
111 views

Double cosets of $U(n)\times U(n)$ in $U(2n)$

This may be well-known but I couldn't find a way to charcterize the double-cosets of $U(n)\times U(n)$ in $U(2n)$ or couldn't find reference. Is there reference where I can look for?
user avatar
  • 287
3 votes
0 answers
110 views

An $\mathsf{SL(n,F)}$ decomposition problem

Given $n\times n$ square matrix $A\in\Bbb F^{n\times n}_{}$ where $\Bbb F$ is a field is there an easy way to test there is NO decomposition $A=B+C+D$ where $B,C,D\in\mathsf{SL}(n,{\Bbb F})$ are ...
user avatar
  • 13k
2 votes
1 answer
132 views

Intransitive finite irreducible linear groups whose orbits are all large

I am interested in intransitive irreducible linear subgroups $G\subseteq\mathrm{GL}_n(\mathbb{F}_p)$ acting on $V-\{0\}=\mathbb{F}_p^n-\{0\}$ in the natural way, such that all of the orbits are very ...
user avatar
  • 527
1 vote
0 answers
112 views

Profinite rank of Fuchsian groups

Let $G$ be a Fuchsian group whose profinite completion is finitely generated. Must $G$ be finitely generated? A Fuchsian group is a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(...
user avatar
  • 11k
2 votes
1 answer
376 views

When is the image of the adjoint representation of a real algebraic group Zariski closed?

Let $\operatorname{Ad}:\operatorname{SL}_n(\mathbb{R}) \to \operatorname{GL}(\mathfrak{sl}_n(\mathbb{R}))$ be the adjoint representation (i.e. $\operatorname{Ad}(g)X=gXg^{-1}$) of $SL_n(\mathbb{R})$. ...
user avatar
2 votes
1 answer
146 views

The maximal possible rank of a subgroup of a product of special linear groups

In this question I ask for a generalization of What is the maximal possible rank of a subgroup of a special linear group mod a prime? Let $p_1, \dots, p_r$ be $r$ distinct odd primes. Set $$G = \...
user avatar
  • 11k
7 votes
1 answer
158 views

What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$. What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$? Here we denote by $d(G)$ the smallest ...
user avatar
  • 11k
5 votes
1 answer
479 views

Maximal subgroups of special linear groups over finite fields

Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements. Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ? I am interested in ...
user avatar
  • 11k
3 votes
1 answer
227 views

Is there a bound on the rank of finite index subgroup of SL_3(Z)?

Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}_3(\mathbb{Z})$ has a generating set of size $N$?
user avatar
  • 11k
18 votes
0 answers
446 views

Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...
user avatar
  • 15.3k
0 votes
0 answers
240 views

Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
user avatar
  • 11k
19 votes
2 answers
3k views

Minimal number of generators for $GL(n,\mathbb{Z})$

$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = \begin{...
user avatar
  • 1,292
11 votes
1 answer
457 views

The Mordell and Bogomolov problems in linear groups

Many things in the arithmetic of abelian varieties have counterparts not only in linear tori, but also for semisimple linear groups. Two examples are the Tamagawa number and the conjectured finiteness ...
user avatar
2 votes
1 answer
411 views

Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated?

Let $n \in \mathbb{N}$, $K$ a finite field. Denote by $K[[x]]$ the (profinite) ring of formal power series over $K$. Note that $\text{GL}_n(K[[x]])$ is a profinite group. Is every closed subgroup of $...
user avatar
  • 11k
3 votes
1 answer
616 views

classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [...
user avatar
  • 471
8 votes
2 answers
1k views

Maximal order of finite subgroups of $GL(n,Z)$

I am interested in the finite subgroups of $GL(n,Z)$ of maximal order. Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...
user avatar
  • 1,292
4 votes
2 answers
416 views

differences between character distributions of supercuspidal representations and others

Let $G$ be a $p$-adic linear reductive group. For an irreducible admissible smooth representation $\pi$ of $G$, there is a distribution $\Theta(\pi)$, called the character distribution, attached to $\...
user avatar
  • 2,589
6 votes
3 answers
673 views

Does every linear group admit a subgroup of dimension 1?

Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup? I'm pretty much sure this is true in ...
user avatar
1 vote
1 answer
240 views

centralizer of a n-cyclic permutation matrix over F_2 in GL(n,2)

This is a continuation of this question, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit ...
user avatar
5 votes
1 answer
399 views

centralizer of the order 2^k cyclic permutation matrix over F_2

Let $C$ be the $2^k\times 2^k$-permutation matrix over $\mathbb{F}_2$ of the $2^k$-cycle. We needed to know the structure of its centralizer in $\mathrm{GL}_{2^k}(\mathbb{F}_2)$, and we computed it - ...
user avatar
4 votes
2 answers
350 views

Restricting the composition factors of subgroups of GL_m(Z/nZ)

For a positive integer $m$, let $\mathcal{A}(m)$ be the set of all integers $k \geq 5$ such that: there is a positive integer $n$ and a subgroup $G \subset \operatorname{GL}_m(\mathbb{Z}/n\mathbb{Z})$ ...
user avatar
9 votes
1 answer
347 views

Representation of surface group

Is there a faithful representation of a surface group of genus $>2$ into $GL(n,\mathbb{C})$ for some $n$ for which, for each conjugacy class of each embedded loop in the fundamental group, the ...
user avatar
  • 61.3k
10 votes
2 answers
2k views

When is a Baumslag-Solitar group linear?

The Baumslag-Solitar group $BS(m,n)$ is given by the group presentation $BS(m,n)=(a,b|ba^{m}b^{-1}=a^{n})$. When does it embed into a linear group? Thanks!
user avatar
  • 391
4 votes
0 answers
238 views

A conjecture on Zassenhaus groups

In a 1995 paper by Ali Nesin, "Permutation groups of finite Morley rank", the following conjecture is mentioned: Conjecture 5 An infinite Zassenhaus group $G$ of finite Morley rank is isomorphic ...
user avatar
7 votes
1 answer
357 views

What are the necessary and sufficient conditions for GL(n,Z/p^lZ) to be isomorphic to GL(n,F_p[t]/t^l)?

Let $p$ be a prime number and $n,l$ be natural numbers. I'm interested in the conditions under which the general linear groups of degree $n$ over the following two length $l$ finite discrete valuation ...
user avatar
  • 7,050
47 votes
2 answers
2k views

Isomorphic general linear groups implies isomorphic fields?

Suppose $n > 1$ is a natural number. Suppose $K$ and $L$ are fields such that the general linear groups of degree $n$ over them are isomorphic, i.e., $GL(n,K) \cong GL(n,L)$ as groups. Is it ...
user avatar
  • 7,050
15 votes
1 answer
631 views

distribution of Young diagrams

Consider $\Lambda^p(C^n\otimes C^n)=\oplus_{\pi}S_{\pi}C^n\otimes S_{\pi'}C^n$ as a $GL_n\times GL_n$-module. This space has dimension $\binom {n^2}p$. I would like any information on the shapes of ...
user avatar
11 votes
3 answers
1k views

Sylow subgroups of projective general linear groups

What is known about the Sylow 2-subgroups of $\rm{PGL}_n(\mathbb{F}_q)$, where $q$ is any prime power? For example, according to Theorem 7.9 in Isaacs's Character Theory, these Sylow 2-subgroups ...
user avatar
  • 12.2k
9 votes
3 answers
965 views

Presentations of PSL(2, Z/p^n)

As is well known, the group $PSL(2,\mathbb Z)$ is isomorphic to the free product $C_2 \ast C_3$ of cyclic groups of order $2$ and $3$. Call the generators of the cyclic groups $S$ and $T$. Problem: ...
user avatar
9 votes
1 answer
335 views

On which space does $GL_n(F_p[X])$ act nicely?

The group $GL_n(\mathbb{Z})$ acts properly and isometrically on the space of homothety classes of scalar products on $\mathbb{R}^n$. This is a Riemannian manifold of nonpositive sectional curvature. ...
user avatar
2 votes
2 answers
318 views

Existence of proper invariant subset in an irreducible action

Let $G<\rm{GL}_n(\mathbb{k})$ be a linear group, where $\mathbb{k}$ is an algebraically closed field. Assume that the linear action of $G$ on $\mathbb{k}^n$ is strongly-irreducible (i.e. there are ...
user avatar