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2 votes
3 answers
181 views

Stabilizers of the action of Levi on abelianization of nilpotent radical

$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
Zhiyu's user avatar
  • 6,622
1 vote
0 answers
71 views

Component groups of stabilizers for linear representations

Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$. Given a vector $v \in V$, it is natural to consider its stabilizer group $...
Zhiyu's user avatar
  • 6,622
6 votes
2 answers
794 views

Tensor algebra and universal enveloping algebra

Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
jack's user avatar
  • 673
2 votes
0 answers
180 views

Howe duality vs first fundamental theorem in invariant theory

I'm working on Howe duality, and R. Howe proved that the Howe duality of $\mathrm{GL}_n$ is equivalent to the first fundamental theorem (FFT) in invariant theory. So, Howe duality gives a ...
zhichengzhang's user avatar
2 votes
0 answers
48 views

Multiplicative invariants of non-reduced root systems

It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...
G. Gallego's user avatar
8 votes
1 answer
374 views

Invariant ring of $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ under $\textrm{SO}(4)$

Consider the representation of $\textrm{SO}(4)$ on $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ induced by the standard representation of $\textrm{SO}(4)$ on $\mathbb{R}^4$. I am interested in the ring of ...
Hans's user avatar
  • 3,031
5 votes
1 answer
199 views

Coordinate-free description of an alternating trilinear form on pure octonions

Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$. The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$, and ...
Mikhail Borovoi's user avatar
4 votes
1 answer
511 views

Invariants of symmetric forms with respect to the symplectic group

Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...
Giovanni Moreno's user avatar
10 votes
1 answer
375 views

Invariants for $SO_n \backslash \mathfrak{gl}_n / SO_n$

Is there a nice theorem about the algebra of invariants $\mathbb{C}[\mathfrak{gl}_n]^{SO_n \times SO_n}$, where the action is by left and right multiplication? I'm hoping for something along the lines ...
user44191's user avatar
  • 4,991
8 votes
4 answers
3k views

"Why" is every polynomial representation of SL(2) selfdual?

Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional polynomial representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, ...
darij grinberg's user avatar
14 votes
1 answer
681 views

If an equivariant map is smooth on diagonal matrices, is it smooth everywhere?

This is a followup from a question I asked on math.SE, which received a helpful answer but unfortunately not a complete one. $\def\Sym{\mathrm{Sym}_{n\times n}}$ $\def\s{\mathrm{Sym}}\def\sp{\s^+}$Let ...
Anthony Carapetis's user avatar
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 ...
AndreaPaco's user avatar
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 ...
miss-tery's user avatar
  • 755
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 ...
asv's user avatar
  • 21.8k
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}^ * ) $...
Zhaoting Wei's user avatar
  • 9,019
9 votes
2 answers
634 views

Extension of the Weyl dimension formula

Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...
emiliocba's user avatar
  • 2,446
17 votes
0 answers
547 views

Does a symplectic group act on a tensor power of a spin representation?

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$More specifically, let $S_k$ be the spin representation of $\Spin(2k+1)$. Then is there are action of $\Sp(2r-2)$ on $\bigotimes^{2r}S_k$ ...
Bruce Westbury's user avatar
7 votes
1 answer
378 views

Are SL(n) Invariants of this wedge product isomorphic to a symmetric product?

In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special ...
Drew's user avatar
  • 1,509
3 votes
0 answers
274 views

Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$. ...
Feanoris's user avatar
4 votes
1 answer
516 views

Calculation with weights of $E_6$

Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...
Claudio Gorodski's user avatar
4 votes
1 answer
381 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 ...
miss-tery's user avatar
  • 755
0 votes
0 answers
194 views

What are the E7(7) invariants in the adjoint representation?

Take a real vector space $R$ transforming in the adjoint representation of the ${\rm E}_7(7)$ Lie group as $R \rightarrow G R G^{-1}$. One can define invariants using traces of products of $R$ as ${\...
Geoffrey Compere's user avatar
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) ...
Michał Oszmaniec's user avatar
6 votes
1 answer
403 views

Orbits for homogenous complex polynomials under unitary rotation of variables

Let's have two complex homogeneous polynomials of degree $k$: $f(z_1,\cdots,z_n)$ and $g(z_1,\cdots,z_n)$. We consider rotations of variables in the form of $\vec{z}' = U \vec{z}$, where $U\in SU(n)$. ...
Piotr Migdal's user avatar
  • 1,612