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0
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2answers
119 views

Is it possible to find a 2D distribution function such that the higher order moments always exist?

Is it possible to generate a 2D distribution function $f(x,y)$ with function supports specified as $[-a,a]$ and $[-b,b]$ for $x$ and $y$ respectively, such that it always has moments which are NON ...
1
vote
1answer
26 views

Affine transformation invariants and lie groups

Is it possible to generate geometric properties which are invariant under affine transformations? I'm trying to learn about lie groups and lie algebras with the example of the lie group of affine ...
1
vote
1answer
89 views

What is an example of a syzygy in invariant theory or pre-abstract algebra

I need an example involving the old invariant theory that pre-dated abstract algebra.
0
votes
1answer
51 views

What is the general area of mathematics to which this example belongs?

In elementary college-level calculus courses, I've given students a problem which reduces to this: Given $f(p,q)$ and a relation $p=g(q)$ use substitution to derive $\mathfrak{f}(p)$ then proceed ...
6
votes
0answers
205 views

'Galois Resolvent' and elementary symmetric polynomials in a paper by Noether

In Emmy Noether's 1915 paper "Der Endlichkeitssatz der Invarianten endlicher Gruppen", I saw the notion of a 'Galois resolvent', which I don't quite understand. Google didn't really help me with that, ...
4
votes
0answers
99 views

The anti-commutative Molien series

Suppose $V$ is a finite dimensional complex vector space and $f:V\to V$ is an automorphism. There is a natural extension $\Lambda^\bullet(f):\Lambda^\bullet(V)\to\Lambda^\bullet(V)$ to the exterior ...
4
votes
0answers
159 views

Basic semi-invariants

Let $G$ be a (finite) group and $\chi$ be a linear character corresponding to an irreducible representation. A polynomial $f_{\chi}$ is called semi-invariant (of type $\chi$) if $\sigma\circ ...
3
votes
0answers
31 views

Invariants of the symmetric group

Let $V_\lambda$ be an irreducible representation of the symmetric group $S_n$ as usual labeled by parition $\lambda$ of $n.$ Question. Is there any general information about the algebra of ...
3
votes
0answers
38 views

Is invariant theory OK in positive characteristic?

Let G be a connected reductive group over an algebraically closed field of characteristic p. Let X = Spec(A) be an affine variety over the same field, with an action of G. Are the closed points in ...
3
votes
0answers
40 views

notation for invariation

Let $\Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\})$ and $\mathcal{O}$ the maximal order of some quadratic imaginary number field. I write $T[U] := U^* \cdot T \cdot U$ where $U$ is ...
2
votes
0answers
43 views

Multidegree semi-invariants for quiver representations

Given a quiver Q=($Q_0,Q_1$) ($Q_0$ is the set of vertices and $Q_1$ is the set of arrows) and a dimension vector $\alpha$, the coordinate ring may be written as $\bigotimes_{a \in ...
2
votes
0answers
87 views

Invariants of $K\left[\bigoplus_{k=1}^n V^{\bigodot k} \right]$

What are the generators of invariant ring $K\left[\bigoplus_{k=1}^n V^{\bigodot k} \right]^G$, where $G$ is subgroup of $GL(V)$ with natural representation on $\bigoplus_{k=1}^n V^{\bigodot k} ...
2
votes
0answers
59 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
2
votes
0answers
100 views

Invariant polynomials over symmetric matrices under Euclidean transformations

It is a simple question, but I haven't still had a course on this topic and I'm finding it hard to understand some basics. Consider a $2\times2$ symmetric matrix over a field (for example ...
2
votes
0answers
42 views

Condition for a polynomial invariant ring to be a UFD

Let $W$ be a finite-dimensional vector space over $% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $. Let $\rho $ $:G\rightarrow $ GL$(W)$ be a representation of any finite group $G$. ...
2
votes
0answers
45 views

$\mathbb{A}^m/G_{m} = \emptyset$?

Let $\mathbb{A}^m ={\text{Spec}}\mathbb{C}[x_1,\dots,x_m]$, and the multiplicative group $G_m \cong \mathbb{C}^*$ acts on $\mathbb{C}[x_1,\dots,x_m]$ by $$\lambda_t (x_1,\dots,x_m)= ...
2
votes
0answers
44 views

Invariants of parabolic subgroups

Let $G$ be a finite reflection group acting on $R^n$. Then for each point $x\in R^n$ we can look at its stabilizer $G_x$. Since $G$ is a finite reflection group, its ring of invariants is a polynomial ...
2
votes
0answers
123 views

DFT shift theorem generalizations?

The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved. $$ ...
2
votes
0answers
77 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...
2
votes
0answers
35 views

$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...
2
votes
0answers
133 views

GIT quotient for a certain torus action on an affine space

I'm reading various books and some notes and here is my question. Let $(\mathbb{C}^*)^2$ act on $\mathbb{C}^4$ by $$(\lambda_1,\lambda_2).(x_1, x_2, y_1,y_2)=(\lambda_1 x_1, \lambda_2 x_2, ...
2
votes
0answers
131 views

How to compute the character of a matrix group operating on homogeneous polynomials?

I have a little problem in representation and/or invariant theory which I need help with. Let's assume $G \leq \mathbb{C}^{n\times n}$ is a finite complex matrix group which operates linearly via ...
1
vote
0answers
40 views

Show that isotropic function S(A) and A have same eigenvectors

Given $\boldsymbol{A}$ is a positive definite, symmetric second order tensor and $\boldsymbol{Q}\boldsymbol{S}(\boldsymbol{A})\boldsymbol{Q}^T = \boldsymbol{S}(\boldsymbol{QAQ}^T)$ $\forall ...
1
vote
0answers
33 views

Is there any significance to this matrix/operator?

I am working on a problem involving the the polarized Hessian covariant in Cartesian coordinates on $\mathbb{R}^2$ $[a,b] = \frac{1}{2} \frac{\partial ^2 a}{\partial x ^2} \frac{\partial ^2 ...
1
vote
0answers
30 views

From linear invariants of group to general ones

There is a lot of information about classical/linear invariants of finite groups. But does it lead to general invariants of group (for example, when we consider some action of our group on finite ...
1
vote
0answers
67 views

Finding an invariant polynomial under a matrix action

I asked this question as a mathematica question: http://mathematica.stackexchange.com/questions/41689/finding-a-certain-invariant-polynomial-using-matrix-coordinates but maybe it will get more ...
1
vote
0answers
70 views

Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n} $$ After the linear change of ...
1
vote
0answers
17 views

Looking for a basic reference on propagators (in Topology)

I am looking for a basic (preferably self-contained) reference where I can read about propagators (as they appear in Topology), and in particular Morse propagators. Thanks!
1
vote
0answers
29 views

Description of certain invariant polynomials (not a group action)

Working on a recent question led me to the following invariant-computation problem : let $$ A=\bigg\lbrace P \in {\mathbb Q}[X_1,X_2,X_3,X_4] \ \bigg| \\ \quad\ P(X_1X_3+X_2X_4+X_1X_4,\ X_2X_3,\ ...
1
vote
0answers
242 views

Frobenius normal form

i would like to clarify everything about Frobenius normal form,so please help me to understand it .as i read from wikipedia In linear algebra, the Frobenius normal form, Turner binormal ...
1
vote
0answers
57 views

Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
1
vote
0answers
62 views

Does $k[B]^{B'}=k[\det(b),\det(b)^{-1}]$?

Let $k=\overline{k}$. Suppose $GL_2(k)$ acts on $GL_2(k)$ by left multiplication. Then $k[GL_2]^{SL_2} = k[\det(g),\det(g)^{-1}]$. Now for $B=\left\{ \left( \begin{array}{cc} b_{11} & b_{12} ...
1
vote
0answers
28 views

Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
1
vote
0answers
42 views

Ring of invariants for the action of rotation groups in tensors.

Consider the component-wise action of the group $SO(p)\times SO(q)$ in the tensor product of two real vector spaces $S^2(R^p)\otimes R^q$. How to parametrize orbits of this action ? For $q=1$ we ...
1
vote
0answers
112 views

Ring of invariants for $\Sigma_3$

I've just started reading about classical invariant theory and I'm not seeing how the general pattern should work, maybe it's obvious I don't know... Let $k$ be a field with $\mathrm{char}\ k = 0$, ...
1
vote
0answers
62 views

$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...
1
vote
0answers
61 views

How to characterize the image of $PGL(V)$ in $\mathbb{P}(W)$ for an irreducible $GL(V)$-representation $W$

I'd like to ask two versions of my question, one a very specific case that I suspect may have a fairly easy and classical answer, the other the general case which I would like to find references on. ...
1
vote
0answers
135 views

What was Cayley's formula for the number of invariants? (Lost Formula!?)

I need to find Cayley's formula for the number of linearly independent invariants of homogenous polynomials. This is a combinatorial formula. He is believed to have discovered it in 1854. ...
0
votes
0answers
28 views

Invariant measure for gradient flows

I found here that a stochastic process from the SDE $dX_t=-\nabla V(X_t)dt + \sqrt{2 \beta^{-1}} dB_t $ has a unique invariant measure (if $V$ is suitable smooth), also called Gibbs distribution ...
0
votes
0answers
40 views

Symmetric expressions for homogeneous functions

Suppose $f(x_1, \ldots, x_n)$ is a homogeneous function, i.e. a function such that $$f(\lambda x_1, \ldots, \lambda x_n) = \lambda^d f(x_1, \ldots, x_n)$$ for all $\lambda$ and for some positive ...
0
votes
0answers
24 views

Existence of dense subsets $G$-invariant.

Let $G$ be a group that acts on a manifold $X$. It is well know that the orbir space $X/G$ isn't in general a manifold. But how can I prove that there is always a dense open $U \subset X$ that is ...
0
votes
0answers
109 views

“Invariant integral” for linearly reductive groups, and the Reynolds operator

Let $G$ be an affine algebraic group. Let $$\lambda\colon G\to GL(k[G]),\quad \lambda(g)(f)=(h\mapsto f(g^{-1}h)),$$ be the left-translation, which is a rational representation of $G$ on its ...
0
votes
0answers
59 views

How is the theory of vector invariants related to number theory?

Is there a nice theorem or toy problem that demonstrates the usefulness of invariant theory to tackle a number theoretic problem? I'm looking for some accessible insight into the motivation (for a ...
0
votes
0answers
184 views

Understanding - G-invariants and concomitants

I am trying to understand a set of lecture notes which I have on representation theory, and I have come unstuck on the section about invariant theory. Suppose we have a finite dimensional vector space ...
-1
votes
0answers
21 views

How can I prove that the set of $A$-invariant subspaces forms a lattice?

How can I prove the following proposition: the set of $A$-invariant subspaces forms a lattice.