All Questions
Tagged with rt.representation-theory invariant-theory
128 questions
2
votes
1
answer
42
views
Invariant theory for unitary groups $\mathcal{U}(n)$
I'm trying to understand the invariant theory of the unitary groups $\mathcal{U}(n)$ on tensor powers of their standard representations $V^{\otimes p} \otimes (V^*)^{\otimes q}$. Let $\mathcal{U}(n)$ ...
- 1,104
9
votes
1
answer
158
views
Eigenfunctions of the Laplace–Beltrami operator on the coadjoint orbit of $\mathfrak{su}(n)$
$\DeclareMathOperator\SU{SU}$For $\mathfrak{su}(2,\mathbb{C})$, the generic coadjoint orbit is $\mathbb{S}^2$, and the Laplace–Beltrami operator on it is given by
$$
\Delta \equiv \frac{1}{\sin\theta} ...
- 195
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 ...
- 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 $...
- 6,622
1
vote
0
answers
78
views
Invariant theory (first fundamental theorem) for a direct sum of two fundamental representations
Let $G$ be a simple reductive group over $\mathbb C$, e.g. $G=\mathrm{SO}(V)$ is a special orthogonal group.
Let $W_1$ and $W_2$ be two irreducible representations of $G$. Assume both $W_i$ are ...
- 6,622
1
vote
1
answer
109
views
Orbit spaces of n-tuples of square matrices under simultaneous conjugation
Let $n, p, \geq 1$ be integers. Denote the set of ordered partitions of $p$ by $\Pi$: each $\pi \in \Pi$ is an ordered $k$-tuple $(p_1,p_2, \dotsc, p_k)$ where $p_1+\dotsb+p_k = p$. Write $\pi \leq \...
- 185
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 ...
- 673
5
votes
0
answers
77
views
Tensor product of Brauer algebra modules
Let $V,W$ be two modules for the Brauer algebra $B_n(\delta)$.
Is it known in general how one can regard $V\otimes W$ as a module for the Brauer algebra? That is, is there any analog of a Hopf ...
- 61
2
votes
0
answers
98
views
Invariant subgroups for integer matrix groups
The representation theory of finite groups over the fields $\mathbb{C}$ or $\mathbb{Q}$ is clear. I am wondering what happens if we consider something similar over the group $\mathbb{Z}$. To be more ...
- 151
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 ...
3
votes
0
answers
171
views
Invariants for the Weyl group of $\mathrm{SO}_{2n}$ acting on a certain group scheme
Let $W$ denote the Weyl group of $\mathrm{SO}_{2n}$, so $W = (\mathbf{Z}/2)^{n-1} \rtimes \Sigma_n$. There is a natural action of $W$ on $\mathbf{Z}^n$ given by permutations and even numbers of sign ...
- 5,760
1
vote
0
answers
119
views
Germs of holomorphic functions and invariant functions
Consider a complex vector space $V \cong {\mathbb C}^n$. Consider the ring of germs of holomorphic functions ${\mathcal O}_0 (V)$ at $0\in V$. We know that this ring is Noetherian.
Now consider a ...
- 803
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
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 ...
- 553
4
votes
1
answer
243
views
Epstein zeta function of Barnes-Wall and related lattices
Sarnak and Strömbergsson studied Epstein zeta function $\zeta(L,s)=\sum\limits_{0\neq v\in L}\langle v,v\rangle^{-s}$ of a number of highly symmetric lattices in their Inventiones Math. paper.
In ...
- 14k
1
vote
0
answers
52
views
Is the Schofield semi invariant defined at $V/IV$?
Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
- 839
3
votes
1
answer
260
views
Invariants of general linear groups under torus action
Let $G=GL_n$ be the general linear group (let's say over an algebraically closed field of char $=0$). Let's denote as $T$ the torus of diagonal matrices: is there an explicit description of the ...
- 1,061
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
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
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
10
votes
2
answers
993
views
Character variety of the free group
A classical result of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi_\mathbb{C}$ associated to the free group $F_2$ and the algebraic group $\mathrm{SL}_2(\mathbb{C})$...
- 2,751
5
votes
0
answers
351
views
What representation theoretic properties does the semi-invariant ring tell us?
I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)?
I have been studying about semi-...
- 839
6
votes
0
answers
110
views
subalgebra of invariants for a reductive subgroup
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Spec{Spec}$Trying to understand some tannakian reconstruction, I've stumbled about the following problem in invariant theory. I guess it's something ...
- 887
3
votes
1
answer
152
views
Are there any results on an upper bound for the number of secondary invariants needed to generate the invariant ring of a finite group?
If $ G $ is a finite cyclic group, $ \beta: G \to \operatorname{GL}(\mathbf{V}) $ is a linear $ n $ dimensional representation of $ G $, and $ \{x_{1},\dots,x_{n}\} $ is a basis of $ \mathbf{V}^{\ast} ...
- 782
4
votes
2
answers
330
views
Ring of invariants for $n$-tuples of Lie algebras
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\Tr{Tr}$Consider the diagonal action of $\GL(n,\mathbb{C})$ on the variety of $k$-tuples of matrices, $\M_{n\times n}(\mathbb{C}...
- 142
10
votes
1
answer
739
views
Basis of invariant tensors of rank n in three dimensions
[This is a question motivated by theoretical physics, so apologies if the language is rough...]
In three dimensions the spaces of invariant (or isotropic) tensors of rank $n$ have dimensions 1, 0, 1, ...
- 1,038
4
votes
1
answer
553
views
Do all orbits have the same dimension?
Well, I've already asked this question at math.SE — but no-one's answered or commented. So now I'm posting it here (it's about the research paper — I think that it isn't an off-topic for this forum) — ...
- 41
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 ...
- 3,031
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
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 ...
- 14.1k
11
votes
2
answers
589
views
To describe an invariant trivector in dimension 8 geometrically
$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$.
For an integer $p\ge 0$, write $R_p=S^p R$;...
- 14.1k
8
votes
1
answer
256
views
$\operatorname{SL}_2(k)$ invariant polynomials in $k[x_1,x_2,y_1,y_2]$
Let $k$ be a field and let $\operatorname{SL}_2(k)$ act on $k[x_1,x_2]$ and $k[y_1,y_2]$ in the usual ways. These actions induce an action on the tensor product $k[x_1,x_2,y_1,y_2]$ that preserves ...
- 83
4
votes
1
answer
272
views
Nilpotent orbits in representations of exceptional groups
The first nontrivial irreducible representation of $G_2$ is of 7-dimensional, and the first nontrivial representation of $F_4$ is of 26-dimensional.
My question is: how much is known about the ...
- 1,457
1
vote
0
answers
202
views
Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix
Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$.
My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
- 13.6k
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
4
votes
1
answer
343
views
Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable?
I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book
Greco, Silvio, ...
- 2,851
4
votes
1
answer
510
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
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 ...
- 4,991
4
votes
2
answers
398
views
Invariants in the symmetric algebra of a module
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$...
- 1,368
3
votes
0
answers
203
views
Decomposing Schur functor applied to a tensor product
I want to compute
$$
S^{2,2,\dots,2,1}(\mathbb C^{2m-1} \otimes W)^{SL(2m-1)}
$$
Here $m$ numbers should appear in the superscipt of the Schur functor, and the last superscript means to take $SL(2m-1)$...
- 1,509
7
votes
2
answers
450
views
Ideals invariant under ring automorphisms
I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...
- 11.1k
3
votes
2
answers
2k
views
Casimir operator of a given Lie algebra and relation with its matrix representation
I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...
- 255
9
votes
0
answers
361
views
Which polynomials in the minors of a matrix are invariant under conjugation?
$\newcommand{\Cof}{\operatorname{cof}}$
This is a cross-post.
Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...
- 6,741
8
votes
1
answer
799
views
higher Casimirs for $\mathfrak{sl}$
The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
- 8,597
3
votes
1
answer
180
views
Kernel of restriction for ring of functions on reductive groups
Let $H \subset G$ be an inclusion of reductive groups over an algebraically closed field $k$ of char $0$. For simplicity, let's assume that $G$ is split and $H$ contains a maximal torus for $G$. Then ...
- 689
4
votes
1
answer
108
views
Sufficient conditions for secondary invariants
Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension ...
- 768
9
votes
1
answer
294
views
A duality result for Coxeter groups
Short version: if $G$ is a Coxeter group and $H \subset G$ is a parabolic subgroup, both acting on a space $V$, is it true that the invariant-coinvariant algebra $(S(V)_G)^H$ has a natural bilinear ...
- 651
10
votes
0
answers
238
views
Progress since Luna's theorem on smooth invariants
In 1976, Luna proved the following important theorem of smooth invariant theory:
Let $G$ be a real reductive Lie group and a representation of $G$ on a real finite dimensional vector space $V$. ...
- 21.5k
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
9
votes
0
answers
268
views
Clebsch-Gordan coefficients of $SO(5)$
The reduced CG coefficients for $SO(d):SO(d-1)$ are in principle known in full generality for $d\leq 4$: they are trivial for $d=2$, equivalent to $3j$ symbols of $SU(2)$ for $d=3$, and to $9j$ ...
- 1,488