All Questions
12 questions
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
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
4
votes
1
answer
317
views
Orbits of tensor product $\operatorname{St}_2\otimes\operatorname{Sym}^2(\mathbb C ^3)$
Let $G_1=\operatorname{GL}_2(\mathbb C)$ act on $V_1=\mathbb C^2$ via the standard multiplication. Denote this representation by $\operatorname{St}_2$. Let $G_2=\operatorname{SL}_3(\mathbb C^3)$ act ...
- 960
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
5
votes
1
answer
158
views
$SL_2$-action on the free lie algebra on a 2-dimensional vector space
Let $V$ be a 2-dimensional vector space (over, say, $\mathbb{Q}$). Let $FL$ be the free lie algebra on $V$, then there is a natural action of the group $SL(V)$ on $FL$, such that the action of $-I$ is ...
- 10.7k
6
votes
1
answer
456
views
How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?
I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...
- 6,206
3
votes
0
answers
236
views
Deligne-Simpson problem for classical groups
Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n =...
- 181
7
votes
2
answers
315
views
Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices
I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...
- 6,206
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
3
answers
482
views
Linear subspaces in cones over orthogonal groups
Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in ...
- 31.2k
3
votes
3
answers
461
views
Multiplicity of eigenvalues in 2-dim families of symmetric matrices
Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
- 1,452
13
votes
2
answers
3k
views
Left and right eigenvalues
A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...
- 12.5k