4
votes
0answers
75 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 ...
3
votes
0answers
108 views

Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task: Let $\mathbb{F}$ be a finite field. Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let $G=\langle ...
3
votes
3answers
102 views

For centralizer subgroups, is the endomorphism ring of a restriction generated by endomorphisms and the centralized element?

In some recent doodlings, I got myself to the point where what I was trying to understand would work out if the following claim were true: Let $G$ be a group, $g\in G$, and $\rho:G \to ...
4
votes
1answer
365 views

Checking irreducibility

This is related to this question. Suppose I have an $n$-dimensional representation of a finitely generated group, and I want to know whether it is absolutely irreducible. This can, of course, be done ...
3
votes
1answer
117 views

Is there an algorithm to compute group presentations of or find generators for the centralizer of a matrix in $GL(n, \mathbb{Z})$?

Let $M \in H \leq GL(n, \mathbb{Z})$. Is there an algorithm that computes either matrix generators or even a group presentation for $C_H(M)$ given generators or a presentation of $H$? Also is ...
3
votes
0answers
131 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 ...
0
votes
1answer
160 views

when $g^*$ is invariant under $Ad(G)$?

Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra. Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under ...
1
vote
1answer
200 views

Subgroup of $SL(n,\mathbb{R})$ with positive entries

Let $SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=SL(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in SL(n,\mathbb{R}) \mid M, ...
5
votes
1answer
296 views

A naive question on eigensheaves for group actions on derived categories

In this Mathoverflow question, Examples of Eigensheaves outside of langlands, David Ben-Zvi says " Given a G -space X you can recover quasicoherent sheaves on X from sheaves on X/G (ie equivariant ...
1
vote
1answer
145 views

primitive polynomial in $F_2$

Let $d$ be a prime number. Is the polynomial $x^d+x+1$ a primitive polynomial? In other words I need the minimal polynomial of $\alpha$ in $F_{2^d}=F_{2}(\alpha)$. Thank you.
3
votes
0answers
84 views

“Spectral decomposition” action on the unitary group

Consider a matrix $U$ from the unitary group $U_N(\mathbb{C})$ and consider the map $f:U_N(\mathbb{C})\rightarrow U_N(\mathbb{C})$ where $f(U)$ is the matrix of the eigenvectors of $U$. What is ...
3
votes
1answer
176 views

On the divisibility of the special linear group of degree $n$ over an algebraically closed field

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But ...
0
votes
1answer
235 views

All and the only algebraically closed fields s.t. any regular n-by-n matrix has a k-th root for every k

The title has it all. I'm looking for a proof/disproof of the fact that an algebraically closed field, say $\mathbb K$, has characteristic zero iff the following property (R) holds: For all $n,k \in ...
15
votes
3answers
919 views

Small index subgroups of SL(3,Z)

I would like to know the smallest index subgroups of SL(3,Z). The smallest I could find has even entries $a_{3,1}$ and $a_{3,2}$, along the bottom row. I could not figure out whether there are ...
1
vote
2answers
246 views

When is PSU(2,q^2) = PSL(2,q) ?

The context for this question is from page 284 - 287 of Berger's paper: ...
13
votes
1answer
507 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, ...
2
votes
0answers
161 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 ...
0
votes
1answer
121 views

Name for a particular subgroup of parabolic subgroups of the general linear groups. [duplicate]

Possible Duplicate: Name for a particular subgroup of parabolic subgroups of the general linear groups. Let $V$ be vector space. The subgroup $P$ of $GL(V)$ consisting of all automorphisms ...
2
votes
3answers
141 views

Simultaneous “Monomialization” of a set of operators.

We all know that a set of commuting diagonalizable matrices can be simultaneously put in diagonal form. My general question is: Under what conditions can a set of (diagonalizable) matrices be ...
2
votes
4answers
311 views

A polynomial homomorphism from Gl to the group of units is a power of the determinant

I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism ...
9
votes
1answer
476 views

Computing the Zariski closure of a subgroup of SL(n,Z)

Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is ...
2
votes
1answer
150 views

on the determination of a quadratic form from its isotropy group in char. 2

So this question is a continuation of the following one [1] On the determination of a quadratic form from its isotropy group For some motivations and relevant backgrounds related to this question ...
2
votes
1answer
284 views

equations over (some) lie groups

To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the ...
3
votes
0answers
163 views

normal form of antisymmetric matrices under pseudo-orthogonal transformations

It is well-known that any real anti-symmetric $n \times n$ matrix $A$ can be transformed via $A \to O A O^T$ into block-diagonal form consisting of $2 \times 2$ antisymmetric matrices, where $O \in ...
1
vote
0answers
127 views

Characterizing symplectic matrices relative to a partial Iwasawa decomposition

Fixing notation: for matrices $A,X$ we let $A[X]$ denote ${}^tXAX$. Let $P_n$ denote the collection of real $n\times n$ positive definite symmetric matrices. For $Y\in P_n$ we have the usual ...
6
votes
0answers
272 views

maximal subgroups of $GL_2(Z/p^kZ)$

Hello, is there any classification of proper maximal subroups of $GL_2(\mathbb{Z}/p^k\mathbb{Z})$ for $k>1$ (analogous to the one which exist for $GL_2(\mathbb{Z}/p\mathbb{Z})$)? Could you give ...
0
votes
1answer
235 views

intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$

Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. ...
4
votes
1answer
274 views

Aschbacher classes and $\mathbb{F}_p$-subspace stabilizers in classical linear groups

I am reading the Kleidman-Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost simple ...
3
votes
0answers
155 views

Vector spaces over a field of prime order with certain hyperplanes

Let $V$ be a vector space of finite dimentional $d$ over a field of prime order $p$. For what values of $d$ and $p$, one can find $d+1$ (pairwise distinct) hyperplanes (subspaces of dimension $d-1$) ...
1
vote
4answers
648 views

Presentation of the Clifford group by generators and relations?

The Clifford group $\mathcal{C}_n$ is a matrix group on $\mathbb{C}^{2^n}$ generated by tensor products of the following matrices: $$ P = \begin{pmatrix} 1 & 0 \\\\ 0 & i\end{pmatrix} \quad H ...
1
vote
2answers
913 views

Periodic matrices in SL(3,Z)

Periodic matrices in SL(3,Z) will be conjugated to product of periodic matrices in SL(2,Z) by +- indentity on a third integer direction. Is this true? Sorry, following your comments, maybe ...
5
votes
1answer
302 views

“Orthogonal complement” in $\mathbb{Z}_q^n$

Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the integers mod $q$. Let $V$ be a submodule of $W$. Let $V^{\perp} = \{w ...
13
votes
3answers
916 views

The normalizer of $\mathrm{GL}(n,\mathbf Z)$ in $\mathrm{GL}(n,\mathbf Q)$

It seems that the normalizer of $H=\mathrm{GL}(n,\mathbf Z)$ in $G=\mathrm{GL}(n,\mathbf Q)$ is "almost" equal to itself, that is, $$ N_G(\mathrm{GL}(n,\mathbf Z))=Z(G) \cdot \mathrm{GL}(n,\mathbf Z) ...
1
vote
1answer
567 views

Subgroups of the Euclidean group as semidirect products

Consider the Euclidean group $E(n)$ as the semidirect product for Euclidean vector space $\mathbb{E}^n$ with its orthogonal group $O(\mathbb{E}^n)$: $E(n)=\mathbb{E}^n\rtimes O(\mathbb{E}^n)$ Then ...
4
votes
1answer
155 views

Expressing a element of a Matrix subgroup in terms of subgroup generators

I'm no (computational) algebraist, and my searches have been pretty unyielding (probably due to the vast amounts written on the key words), but perhaps someone may know if this is possible, and if so, ...
4
votes
2answers
462 views

Conjugacy in $GL(n,\mathbb Z)$

How can I determine whether $A_1,A_2\in GL(n,\mathbb Z)$ conjugate in $GL(n,\mathbb Z)$ and if they are, how can I find a $P\in GL(n,\mathbb Z)$ for which $A_2 = P^{-1}.A_1.P$ ? In $GL(n,\mathbb Q)$ ...
3
votes
2answers
196 views

Does the automorphism group of a cone determine the cone?

A cone is a $R_+$-module. That is, a cone is an abelian monoid that is closed under nonnegative real scalar multiplication. An automorphism of a cone is a bijective $R_+$-linear map. That is a map ...
2
votes
1answer
171 views

Computing a generating set of the kernel of a module

Crossposted from math.stackexchange, since I'm not getting any answer and I think the question is suitable here. Given a generating set of a $\mathbb{Z_k}$-module $M \subseteq {\mathbb{Z}_k}^n$, is ...
6
votes
3answers
1k views

Finite subgroup of $Gl(n,\mathbb Z)$ and congruences

Suppose we have an invertible matrix q in a finite subgroup $Q$ of $Gl(n,\mathbb Z)$, the group of all invertible integer matrices. Now I want to find all $x\; mod\; \mathbb Z^n$ for which ...
12
votes
2answers
966 views

Non-degenerate alternating bilinear form on a finite abelian group

I asked this question on math.stackexchange yesterday, but nobody has helped so far, and only 44 people have seen it! So I hope people do not mind me asking it here... Let $A$ be a finite abelian ...
2
votes
1answer
336 views

Heisenberg group over the Gaussian integers

If we take the entries of the (standard $3 \times 3$) Heisenberg group to live in the Gaussian integers $\mathbb{Z}[i]$, what is the structure of this group? Are all of its representations known?
31
votes
7answers
3k views

Does $SL_3(R)$ embed in $SL_2(R)$?

Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$? $SL_{3}(\mathbb{Z})$ is not am amalgam, and has the wrong number of order $2$ elements to be a subgroup ...
3
votes
4answers
984 views

Parametrization of O(3)

Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?
5
votes
2answers
355 views

length of decompositions into elementary matrices

The Gaussian algorithm tells us, that for any field $k$ a $n\times n$-matrix over $k$ can written as a product of at most $C$ elementary matrices ($C\sim n^2$). I am wondering, whether such a ...
1
vote
1answer
317 views

Tori acting on vector spaces

Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now ...
7
votes
2answers
1k views

Centralizers in GL(n,p)

There appear to be a number of rational canonical forms. The best thing about standards is how many there are to choose from. However, the standard I choose seems to have a centralizer that is ...
10
votes
2answers
2k views

AC in group isomorphism between R and R^2

Using the axiom of choice, one can show that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups. In particular, they are both vector spaces over $\mathbb{Q}$ and AC gives bases of ...
4
votes
1answer
1k views

Finite subgroups of GL_n(C)

A finite groups of $\mathrm{GL}_n(\mathbb C)$ of exponent $m$ necessarily have order $C$ verifying $C\leqslant m^n$ and $n! m^n$ divides $C$, but this condition is not sufficient, for instance ...
1
vote
2answers
817 views

Problem with the proof of a corollary of Schur's lemma

Hello, I'm reading the book 'A course in Modern Mathematical Physics' by 'Szekeres' and encountered a problem in interpreting the proof of the following corollary of Schur's lemma. The corollary and ...
6
votes
2answers
653 views

Linear algebra and regular orbits

If $A$ is an $n\times n$ matrix over a field, and $A^{k} = I$, with $k$ the least positive integer such that this occurs, then must there be some vector $v$ such that $\{v,Av,A^{2}v,\dots,A^{k-1}v\}$ ...