Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. (Def: http://en.m.wikipedia.org/wiki/Invariant_theory)

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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} $i.e. ...
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1answer
144 views

Exercise from Etingof's notes on Representation Theory

I am reading through these notes of Etingof on Representation theory and I am stuck with one exercise (it's problem 4.69 in the notes). The problem is the following. Consider the space $X=Mat_n(\...
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153 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 ...
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51 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 ...
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31 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 set)...
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64 views

Torus orbit closures and rank-1 subtori

Suppose I have a connected complex torus $K$ acting on a quasi-affine complex variety $X$. Suppose also that I have $p,q\in X$ such that the orbit $Kq$ is closed in $X$ and $q\in \overline{Kp}\...
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82 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 ...
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83 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 ...
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21 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!
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50 views

Invariant homology classes

Let $G$ be a finite group acting freely on a manifold $X$. What is the geometrical meaning of invariant homology classes $H_i(X,\mathbb Z)^G$? The same question for coinvariants $H_i(X,\mathbb Z)_G$.
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1answer
528 views

Prove scalar products are invariant under all orthogonal transormation

I wondering how to prove: That scalar products are invariant under all orthogonal transformation: $<\!x, y\!>\; =\;<\!Qx, Qy\!>$ which holds for all vector $x$,$y \in \Re^n$ and all ...
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Is the ring of polynomial invariants of a finite perfect group an UFD?

Let $G$ be a finite group. $G$ acts on $\mathbb K[x_1,...,x_n]$ by automorphisms fixing $K$. $\mathbb K[x_1,...,x_n]^G=\{ T\in \mathbb K[x_1,...,x_n],\forall \sigma \in G, T^{\sigma}=T\}$ is the ring ...
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27 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 $G$-...
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224 views

Using the Invariant Principle to prove a coordinate can't be reached

Problem: A robot wanders around a 2-dimensional grid. Starting at $(0, 0)$, he is allowed 4 different kinds of steps: $(+2, -1)$ $(+1, -2)$ $(+1, +1)$ $(-3, 0)$ He is trying to get to $(0, 2)$. (...
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133 views

What is the field of definition of an invariant ideal?

Let $K/k$ be a finitely generated field extension, such that $k=K^G$ for some (possibly infinite) set $G$ of automorphisms of $K$. Now, consider the extension of polynomial rings $$ k[X_1,\ldots,X_n]\...
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1answer
55 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 ...
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1answer
59 views

Spectrum of $\mathbb{C}[x,y]^{\mathbb{C}^*}$

Let $\mathbb{C}[x,y]$ the ring of polynomials with $\mathbb{C}$-coefficients. We can define an action $\phi: \mathbb{C}^* \times \mathbb{C}[x,y] \rightarrow \mathbb{C}[x,y]$ such that $\phi(\lambda,p(...
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195 views

The unique closed orbit in GIT quotient fibers for polynomial actions of Gl

The following reasoning must contain a flaw somewhere because I end up with something absurd, and I cannot figure out where the mistake is. I hope that someone can point it out to me. Let $M$ be the ...
7
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1answer
365 views

On the automorphisms of the Klein Quartic

I am trying to solve a problem from Miranda's book, Algebraic Curves and Riemann Surfaces. On page 84, problem K gives the Klein curve $X$ as a smooth projective plane curve defined by the equation $...
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1answer
35 views

Image of symmetric group is a reflection group

Let $V$ be $n$-dimensional complex vector space with basis $\{e_1,...,e_n\}$ and let $\phi\colon S_n \to GL(V)$ be homomorphism of groups such that $\phi(\sigma)e_j = e_\sigma(j), j = 1,...,n$. Prove ...
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129 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 $\mathbb{C}$...
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1answer
116 views

Hilbert's finiteness theorem over arbitrary fields; reductive groups

As a generalization of the finiteness result Hilbert proved in his 1890 paper, one usually formulates the following nowadays: Let $G\to\operatorname{GL}(V)$ be a rational representation of a ...
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1answer
403 views

Invariant space of linear transformation

Let $V$ be a vector space of a finite nonzero dimension $n$ over some field. Let $T$ be a linear transformation of $V$, such that $T$ is nonzero and not one-to one. (a)Give a $T$-invariant linear ...
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37 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,\ ...
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56 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 $G$. ...
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43 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 ...
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1answer
48 views

Verify if the set is invariant with respect to the system

I had three assignments where I had different sets for which I had to verify if they were invariant with respect to the system. I did not really know how to solve them so I hope people are willing to ...
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48 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)= (t^{a_1}x_1,\dots,...
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281 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 ...
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52 views

Verify if set S is invariant for a given system.

Consider the autonomous linear discrete–time system $x(k+1) = Ax(k)$ with $A = \begin{bmatrix} 0.5 & 0 & 0 \\ 2 & -0.6 & 0 \\ -2 & -1 & 1.2 \end{bmatrix}$ and set $S = \{x \in \...
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Invariant Factors vs. Elementary Divisors

I have been studying Cooperstein's Advanced Linear Algebra for about seven months now and I am having problems understanding how to find the elementary divisors of a linear operator and how to find ...
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172 views

Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
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338 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, ...
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137 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 ...
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246 views

Ring of invariants of Klein Four group

Assume $F$ is a field and assume $f\in F[x_1,\ldots,x_4]$ is a polynomial that is invariant under the Klein Four group $V_4$. How can I show that this polynomial can then be rewritten as a polynomial $...
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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 det=...
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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 ...
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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} \\...
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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. $$ \mathcal{F}(\{x_{n-m}\})...
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723 views

Schur-Weyl Duality ( Classical ) and the Double Commutant reference request

I would like to ask for any reference suggestions on the topic of Schur-Weyl Duality for GLn ( directly GLn, not through the lie algebra ) and the double commutant theorem. The section on this ...
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150 views

Invariants of binary forms under a $\begin{pmatrix} 1& 1 \\ 0& 1 \end{pmatrix}$ action

The special linear group $\text{SL}_2(\mathbb{Z})$ of $2\times 2$ invertible matrices in $\mathbb{Z}$ acts on binary cubic forms $\{ax^3 + bx^2y + cxy^2 + dy^3\}$ by acting on the vector $(x,y)^T$. ...
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Two questions on Nagata's counterexample to the Hilbert's fourteenth problem.

Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...
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Geometric interpretation for this invariant in square bracket notation?

Let $X=\{p_1,p_2,p_3,p_4\}$ be four points in $\mathbb{R}^2$, not three of them on a line. We use the square bracket notation $$[i,j,k]=\det \begin{pmatrix} p_i & p_j & p_k \\ 1 & 1 & ...
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33 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 ...
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1answer
101 views

Describe invariant polynomials under action of commutative group of order eight.

I believe the question below should be fairly standard in invariant theory ; I hope someone more familiar with it than me can explain a bit more or point to a reference. Let $F$ be polynomial field ...
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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 ...
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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.
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130 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 $\mathrm{char}\ k = 0$, ...
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101 views

$GL_2(\mathbb{C})$-invariant ring for $M_2(\mathbb{C})\times M_2(\mathbb{C})$

For $\mathbb{C}^*$-action on $\mathbb{C}^2$ by $t\circ(x,y)=(t^{-1}x,ty)$, the ring of invariant polynomials is $\mathbb{C}[xy]$. For $\mathbb{C}^*$-action on $\mathbb{C}^4$ by $t\circ(x,y,z,w)=(t^{-...
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54 views

Why is it that $\mathbb{C}[\{m_{ij}\}]^{G} \subseteq \mathbb{C}[\textrm{tr}(m),\ldots, \det(m)]$?

Let $G= GL(n,\mathbb{C})$ act on the set of all matrices $M_n$ by conjugation, i.e., for $g\in G$ and $m \in M_n$, $g\circ m = gmg^{-1}$. Then if $m=(m_{ij})$, then the $G$-invariant ring $\mathbb{...