All Questions
16 questions
11
votes
2
answers
684
views
Invariants of $\mathrm{GL}_n$ representations
$\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the ...
- 673
9
votes
1
answer
248
views
Decomposition of $\bigotimes^{m} \mathbb{C}^{n}$ under the action of $\operatorname{GL}_{n}\times \operatorname{S}_{m}$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\S{S}$I want to know the proof of the following theorem. It is stated somewhere that, a proof can be found in: "Roger Howe, Perspectives on ...
- 179
8
votes
3
answers
803
views
Why can I divide an affine variety by the action of the general linear group?
Let $G\subseteq\mathrm{Gl}_n(\mathbb{C})$ be a subgroup of the general linear group and assume that $\rho:G\to\mathrm{Gl}(V)$ is a representation. Understand the complex vector space $V$ as an affine ...
- 4,902
8
votes
1
answer
402
views
Separating closed $SO(p,q)$ orbits by invariant polynomials
Consider the real Lie group $SO(p,q)$ (I believe that it happens to be a linearly reductive algebraic group over $\mathbb{R}$, if that's relevant). Also, if relevant, I'm mostly interested in the (...
- 21.5k
8
votes
1
answer
549
views
Ring of invariants for the regular representation
The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
- 195
7
votes
1
answer
227
views
Invariants for the isotropy representation of a Riemannian symmetric space
Statement: Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $...
- 32.5k
5
votes
4
answers
623
views
Structure of $[S(\mathfrak{g})\otimes S(\mathfrak{g})]^G$ for semisimple $\mathfrak{g}$
Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. $S(\mathfrak{g})^G$ is a polynomial algebra with rank $\mathfrak{g}$ generators. Call them $c_i(x)$, where $x\in \mathfrak{g}$ and $i=...
- 6,123
5
votes
1
answer
436
views
Is the Veronese variety "enough" to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?
I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...
- 2,527
5
votes
1
answer
1k
views
The Casimir invariant of an irreducible representation of a compact Lie group
Let $G$ be a compact Lie group (not necessarily connected) and $\rho:G\to \mathrm{End}(V)$ an irreducible (hence finite-dimensional) unitary representation of $G$. Let $\mathfrak{g}$ be the Lie ...
- 1,942
4
votes
1
answer
783
views
Three dimensional representations of Alternating group
The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
- 125
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 ...
- 2,527
4
votes
1
answer
185
views
Frobenius series for the $S_n$-module $\mathbb{Q}[X]$
I'm reposting this question, by recommendation of a moderator.
I'm reading Haiman's article titled Conjectures on the quotient ring by diagonal invariants. In what follows, all vector spaces and ...
- 141
3
votes
0
answers
105
views
Invariant theory of the indefinite orthogonal groups
I believe the following statements are true:
Let $V$ be a finite-dimensional real vector space with a positive-definite inner product $g$. Let $g_{\otimes n}$ denote the natural extension of $g$ to $...
- 2,247
2
votes
1
answer
196
views
Dimension of spaces of invariants/tableaux functions
The Hook lenght formula gives the number of standard Young tableaux on a given diagram.
A variant gives the number of semistandard tableuax.
Does there exist a formula for counting "weighted ...
- 3,779
2
votes
0
answers
108
views
Invariants of Lie superalgebras
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
- 673
0
votes
0
answers
189
views
Action of the (special) orthogonal group on differential forms
I was told that the following facts are true. I am looking for a reference to them.
1) The action of $O(n,\mathbb{C})$ on $\wedge^l\mathbb{C}^n$ is irreducible for any $l$.
2) The action of $SO(n,\...
- 21.8k