6
votes
2answers
299 views

Do compact groups acting irreducibly have finite subgroups which do the same?

Let $G$ be a closed subgroup of $U(n,{\bf C})$, not necessarily connected. Regard ${\bf C}^n$ as a complex $G$-module $M$. Q. Suppose $M$ is irreducible as a $G$-module (equivalent, I think, to ...
1
vote
1answer
198 views

Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product?

I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$. What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product ...
3
votes
1answer
154 views

Switching representatives of right coset in a paper on fundamental domain of tree of GL(2)

I have a question concerning the following paper: "The fundamental domain of the tree of GL(2) over the function field of an elliptic curve" by Shuzo Takahashi. (1993, Duke Math. J., Vol. 72, No. 1). ...
9
votes
4answers
776 views

Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation. Since I essentially need $n\le 4$, I think that I can show it ...
23
votes
1answer
620 views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
1
vote
1answer
330 views

Odd subgroup of $\mathrm{GL}(n,\mathbb{Z})$

The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset ...
4
votes
0answers
82 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
1
vote
1answer
208 views

$SO(N^2-1)$ and the adjoint representation of $SU(N)$

It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$. I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...
8
votes
1answer
506 views

$SL_2(\mathbf{Z},8\mathbf{Z})$ differs from $E_2(\mathbf{Z},8\mathbf{Z})$. Has this result appeared in the literature?

I know a proof that the congruence subgroup $SL_2(\mathbf{Z},8\mathbf{Z})$ differs from its subgroup $E_2(\mathbf{Z},8\mathbf{Z})$, but can't find this fact in the literature. Does anyone know a ...
2
votes
0answers
142 views

Compute the discriminant for reductive groups

Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$. The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
3
votes
2answers
590 views

A problem about Determinant of sum of permutation matrices

Let $w_1$ and $w_2$ be two permutations of $\{1, \cdots , k\}$ such that for all $1\leq i \leq k$, $w_1(i)\neq w_2(i)$. Let $m$ and $n$ be two relatively prime integers. Then is there exist two ...
2
votes
1answer
143 views

Conjugacy of torsion subgroups in Gl(n, Z) for small n [duplicate]

Have the conjugacy classes of the torsion subgroups of Gl(n, Z) been determined for small n (say, n<=6)? In general, can much be said about the torsion subgroup?
13
votes
1answer
530 views

Free subgroups of GL(2,Z)

Is there a bound $B$ such that every 2-generator subgroup $G = \langle a, b \rangle < {\rm GL}(2,\mathbb{Z})$ whose generators do not satisfy a relation of length $\leq B$ is free? If it exists, ...
4
votes
2answers
457 views

Matrix groups and presentation

Suppose $K$ is a number field and I have a subgroup of $GL_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group? More precisely, the ...
2
votes
0answers
171 views

Eigenvalues of the products of a fixed unitari matrix with diagonal unitari matrices

How does the spectra of $DU$ change when $D$ runs over all diagonal unitary matrices? Here $U$ is a fixed unitary matrix. Precisely, let spec$(X)$ be a set of eigenvalues of $X$. For a unitary matrix ...
4
votes
3answers
380 views

Finding generators of matrix subgroups

I am particularly interested in Sp$(2n,\mathbb{Z})$, but I think an answer for a more general set of matrices would help. General question: Given a subgroup of a group of matrices, what tools or ...
11
votes
1answer
245 views

A product on the square roots of unit matrix

There is a strange product that takes two square roots of unit matrix, say $A$ and $B$, $A^2=I$, $B^2=I$ to a square root again, $$ A\star B=(A+B)^{-1}(A-B+2I), \qquad (A\star B)^2=I$$ Could anybody ...
5
votes
2answers
415 views

Complexity of Membership-Testing for finite abelian groups

Consider the following abelian-subgroup membership-testing problem. Inputs: A finite abelian group $G=\mathbb{Z}_{d_1}\times\mathbb{Z}_{d_1}\ldots\times\mathbb{Z}_{d_m}$ with ...
8
votes
1answer
545 views

Automorphisms of a matrix in Smith normal form?

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...
10
votes
3answers
757 views

Congruence subgroups as abstract groups

This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup $$ \pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & ...
4
votes
1answer
263 views

When is Out$(SL_n(R))$ a torsion group ?

This question is a follow up question to this question. So my question is: For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL_n(R))$ a torsion group? A consequence of ...
26
votes
1answer
759 views

Understanding “infinite” relations in groups

Consider the matrices $A = \frac{1}{5}\begin{pmatrix}5&0&0\\\ 2&2&1\\\ 2&1&2\end{pmatrix}$, $B = \frac{1}{5}\begin{pmatrix}2&2&1\\\ 0&5&0\\\ ...
4
votes
1answer
576 views

Subgroups of U(M_n)

can any subgroup of the unitary group of full matrix alg $M_d(\mathbb{C})$ be approximated on finite sets by a finite subgroup? i.e. is the following True or false? Let $n, d$ be positive integers ...
4
votes
4answers
981 views

Commuting matrices in GL(n,Z)

Suppose $M$ is a "hyperbolic" matrix in $GL(n,\mathbb Z)$, i.e., that its characteristic polynomial $p$ is irreducible over $\mathbb Z$ and has no roots of modulus 1. Is there a closed description ...
3
votes
4answers
1k views

Parametrization of O(3)

Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?
24
votes
2answers
1k views

Invertible matrices satisfying $[x,y,y]=x$.

I have been thinking about this question for quite some time but now this question by Denis Serre revived some hope. Question. Let $x,y$ be invertible matrices (say, over $\mathbb C$) and ...
21
votes
2answers
2k views

Finite subgroups of unitary groups

Let $n$ be an integer. Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite ...
7
votes
1answer
461 views

What matrix groups can be embedded in $Sp_4$?

In a joint paper with Yifan Yang we constructed an "exotic" embedding of $SL_2(\mathbb R)$ in $Sp_4(\mathbb R)$ (in fact, of $PSL_2(\mathbb R)$ in $PSp_4(\mathbb R)$), namely, $$ ...
1
vote
2answers
151 views

Freeness of the Canonical $SU(n)$ Action

I have another question about $SU(n)$, again I hope it's not too basic. For $n=2$, the action of $SU(2)$ on $C^2$ is free since it's equal to the group of rotations. In general, the group of rotations ...
1
vote
2answers
232 views

Describing $SU(n,C)$

For $A \in SU(2,C)$, it is clear that $A$ is completly determined by its first row (well any row or column, but let's say first column). In the general $SU(n,C)$-case this is no longer true. In fact, ...
8
votes
4answers
760 views

presentation for GL(n,K)

let $K$ be a field, $n \geq 1$. denote $E_{i,j}$ the elementary matrix having $1$ on the diagonale and in the entry $(i,j)$, and $E_i(a)$ the elementary matrix $diag(1,...,a,...,1)$. you know that ...
7
votes
1answer
847 views

Surjective maps given by words, redux

I asked some time ago: Let $w(X,Y)$ be a word in $X$ and $Y$ (i.e., an element in the free group on $X$ and $Y$). Let the variables $x$ and $y$ now range among elements of $SL_n(K)$, $K$ an ...