# Tagged Questions

**1**

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**0**answers

79 views

### What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$
$$
F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2)
$$
on the vector space ...

**3**

votes

**1**answer

366 views

### Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...

**3**

votes

**4**answers

349 views

### Reference for an algebraic group preserving a cubic form

Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...

**0**

votes

**1**answer

65 views

### Regular or elliptic elements in the multiplicative group of central division algebra

For an element $g$ of a connected reductive group $G$ over a field $F$,
$g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$,
$g$ is ...

**1**

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**1**answer

142 views

### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

**1**

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**1**answer

132 views

### Form of a block upper triangular matrix of finite order

If I take a block upper triangular matrix and the matrix is known to be diagonalizable and the leading blocks in the diagonal are each of finite order, is it true that away from the leading block ...

**3**

votes

**1**answer

119 views

### On matrices conjugated in a faithful representation

Let $k$ an algebraically closed field.
Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group.
Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular ...

**2**

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**0**answers

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 ...

**0**

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**0**answers

98 views

### a very elementary question on the conjugated matrices

Let $A$ and $B$ two matrices in $GL_{n}(K[[\pi]])$, regular semisimple on $GL_{n}(K((\pi)))$, with $K$ an algebraically closed field of characteristic zero .
We suppose that they have the same ...

**15**

votes

**3**answers

971 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
...

**7**

votes

**4**answers

636 views

### Classification of Tori of GL2, up to conjugation

Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers ...

**3**

votes

**2**answers

313 views

### Explicit equation of Dickson invariant / quasideterminant / special orthogonal group over the integers

Consider $2n$ coordinates $x_1,\ldots,x_n,y_1,\ldots,y_n$ and the quadratic form $q = \sum_{i=1}^n x_i y_i$. Now call $O(q,A)$ (orthogonal group of $q$) the group of $(2n)\times(2n)$ matrices, with ...

**2**

votes

**4**answers

319 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 ...

**16**

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**4**answers

703 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 ...

**6**

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**2**answers

422 views

### Groups of matrices that preserve several quadratic forms

Given two (or more) quadratic forms (on the same vector space) consider the group of matrices that preserve these forms, i.e. $Q_i=U Q_i U^T$, $i=1,2..,k$ What is known about such groups? (at least ...

**3**

votes

**2**answers

293 views

### Infinite products of representations of the additive group

Fix a $\mathbb{Q}$-algebra $R$. Let's call an endomorphism $f : M \to M$ of an $R$-module $M$ locally nilpotent if for every $m \in M$ there is some $n \in \mathbb{N}$ such that $f^n(m)=0$. ...

**0**

votes

**1**answer

237 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. ...

**3**

votes

**2**answers

470 views

### $k$ structures on $K$ vector spaces

The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.
Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is ...

**2**

votes

**2**answers

276 views

### The product of non-commuting semisimple matrices need not be semisimple

In general when one proves that the product of semisimple (i.e. diagonalizable) matrices is semisimple one assumes they commute and are thus simultaneously diagonalizable, and then the result follows. ...

**5**

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**3**answers

461 views

### Zariski-closed subsemigroups of SL_n(C) are groups

I would like to show that any Zariski-closed subsemigroup of $SL_n(\mathbb{C})$ is a group. If I understand correctly, this is consequence 1.2.A of http://www.heldermann-verlag.de/jlt/jlt03/BOSLAT.PDF ...

**1**

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**1**answer

322 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 ...

**21**

votes

**6**answers

3k views

### A slick proof of the Bruhat Decomposition for GL_n(k)?

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the ...

**6**

votes

**3**answers

2k views

### Simultaneous diagonalization

I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ ...

**8**

votes

**5**answers

701 views

### Non-conjugate words with the same trace

Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...