All Questions
Tagged with lie-groups invariant-theory
10 questions
13
votes
3
answers
693
views
Ring of invariants of $\operatorname{SL}_6$ acting on $\Lambda^3 \mathbb C^6$
Let $G=\operatorname{SL}_6$ act on $V=\Lambda^3 \mathbb C^6$. I would like to find the ring of invariants $\mathbb C[V]^G$. There is an obvious invariant
$$Sq: V \to \mathbb C, \quad \omega \mapsto \...
- 1,980
6
votes
1
answer
1k
views
Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$
Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
- 755
4
votes
1
answer
380
views
The existence of a finite dimensional Lie algebra with a given symmetric invariant metric
The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
- 755
20
votes
6
answers
4k
views
Polynomial invariants of the exceptional Weyl groups
Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
- 2,160
7
votes
2
answers
418
views
About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$
Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) $...
- 9,019
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
3
votes
1
answer
691
views
Invariants of a set of real unit vectors in 3d space, under SO(3)
I have a set of $n$ real unit vectors, in 3-dimensional space.
(It is a follow up of Sets of vectors related by a rotation.)
Is there a construction providing a complete set of independent*) ...
- 1,612
2
votes
1
answer
359
views
Characterization of the weight orbit in the projective space via second order Casimir.
This is the spin-off of the question I previously asked.
First, let me remind you some notation from that question:
$G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
- 965
2
votes
0
answers
808
views
Casimir operators of a given Lie Algebra
I am a Physicist, so let me apologize in advance for some possible imprecisions.
I'm working on a 10-dimensional Lie Algebra. Each element of the algebra represents a quantum mechanical operator, and ...
- 255
1
vote
1
answer
303
views
A representation of Spin(9,1)
Let $Spin(9,1)$ denote the universal (double) cover of $SO(9,1)$. $Spin(9,1)$ acts linearly on $\mathbb{R}^{16}$ (see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ).
Consider the induced ...
- 21.8k