All Questions
15 questions
13
votes
4
answers
2k
views
Groups of matrices in which all elements have all eigenvalues equal in modulus
I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...
- 6,206
10
votes
1
answer
262
views
What is the hidden symmetry behind four generic planes in $\mathbb{R}^4$?
Consider the action of $\operatorname{GL}(\mathbb{R}^4)$ on the Grassmannian of 2-dimensional subspaces of $\mathbb{R}^4$. In experiments, I observe that four randomly drawn points in this space are ...
- 7,633
8
votes
2
answers
636
views
Bilinear forms in compact/semisimple Lie group theory
If you look up the list of compact or semisimple Lie groups, you will see that three out of four infinite families (B, C and D) are defined in terms of a bilinear form on a vector space, either ...
- 481
8
votes
2
answers
1k
views
Action of symmetric matrices under $\mathrm{O}(n)$
$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\O{O}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\...
- 1,077
8
votes
1
answer
253
views
Simple Lie algebras: making subspaces 'very transversal'
Let $G$ be a Lie group or group of Lie type whose Lie algebra $\mathfrak{g}$ is simple. Because the Lie algebra is simple, for any proper subspace $V\subset \mathfrak{g}$,
there is a $g\in G$ such ...
- 20.2k
6
votes
4
answers
658
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) ...
- 63
6
votes
1
answer
273
views
Simultaneous triangularisation of an exterior power of a set of matrices
I'm working on some research problems relating to random matrix products, and this is taking me into areas of mathematics I've not previously studied: Lie groups, representation theory, and real ...
- 6,206
4
votes
1
answer
333
views
Orbits in the adjoint representation of $SU(2,1)$
How can one describe the orbits of the Lie group $G=\mathrm{SU}(2,1)$ in its Lie algebra $\mathfrak{g}=\mathfrak{su}(2,1)$ with respect to the adjoint representation?
- 14.2k
3
votes
1
answer
138
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 ...
- 3,472
3
votes
0
answers
119
views
Describing the outer automorphism of a special unitary group in terms of the Hermitian form
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}$Let $h$ be a non-degenerate Hermitian form on $\mathbb{C}^n$ with signature $(p,q)$. Let $\U_h$ denote the associated ...
- 7,527
2
votes
1
answer
2k
views
Parametrization of SL(3,R)
Are there any known common parametrizations of SL(3,R)? I know that it is easy to obtain a local parametrization by just exponentiating generators from the Lie algebra, but I do not know if they are ...
- 293
2
votes
0
answers
190
views
Conjugacy classes in centralizers
Let $G$ be a complex reductive group, let $g$ be an element, and let $C$ be the connected component of its centralizer. I'm curious about what is known about the intersection of conjugacy classes in $...
- 491
2
votes
0
answers
212
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 ...
- 3,472
1
vote
1
answer
528
views
Weyl group action on maximal tori
Let $G$ be a semisimple algebraic group over the complex numbers and we fix a maximal torus $T$. Let $w\in W$ be an element in the Weyl group, and let $T^{w}$ be the elements in $T$ that are fixed by $...
- 125
0
votes
0
answers
99
views
Unimodular matrices fixing $(1, 1, \cdots, 1)$
What is known about the subgroup of $GL(n, \mathbb Z)$ fixing (under left multiplication) the vector ${(1, 1, \cdots, 1)}^T$ ('T' denotes transposition). I'm particularly interested in the case $n = 5$...
- 356