The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
0answers
65 views
+100

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} ...
4
votes
1answer
54 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 ...
2
votes
0answers
26 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 ...
3
votes
0answers
28 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 ...
0
votes
0answers
13 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 ...
1
vote
0answers
29 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
1answer
32 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 ...
1
vote
0answers
57 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
51 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
13 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!
0
votes
0answers
29 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 ...
2
votes
1answer
42 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 $H_i(X,\mathbb Z)_G$.
1
vote
1answer
91 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 ...
2
votes
2answers
86 views

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 ...
0
votes
0answers
18 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 ...
1
vote
2answers
104 views

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

The problem is as follows: A robot wanders around a 2-dimensional grid. Starting at (0, 0) he is allowed 4 different kinds of step: 1. (+2, -1) 2. (+1, -2) 3. (+1, +1) 4. (-3, 0) He is trying to get ...
4
votes
2answers
99 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 $$ ...
0
votes
1answer
46 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 ...
1
vote
1answer
39 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 ...
9
votes
1answer
93 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
votes
1answer
117 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 ...
0
votes
1answer
31 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 ...
0
votes
0answers
79 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 ...
2
votes
0answers
82 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 ...
3
votes
1answer
44 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 ...
0
votes
1answer
131 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 ...
1
vote
0answers
27 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,\ ...
2
votes
0answers
39 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 $G$. ...
3
votes
0answers
38 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 ...
0
votes
1answer
32 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 ...
2
votes
0answers
43 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)= ...
1
vote
0answers
154 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 ...
0
votes
1answer
47 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 ...
3
votes
1answer
730 views

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 ...
12
votes
1answer
147 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: ...
6
votes
0answers
145 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, ...
0
votes
2answers
87 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 ...
3
votes
3answers
125 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 ...
1
vote
0answers
43 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
36 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 ...
1
vote
0answers
60 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} ...
2
votes
0answers
105 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. $$ ...
4
votes
2answers
288 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 ...
7
votes
1answer
116 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 $(x,y)^T$. ...
7
votes
1answer
201 views

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 ...
2
votes
1answer
92 views

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 & ...
1
vote
0answers
26 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
104 views

Two definitions of “semistable locus” for GIT quotient

I am reading Nick Proudfoot's notes on geometric invariant theory and projective toric varieties in which he gives a definition of "semistable" which is unfamiliar to me but which is apparently ...
2
votes
1answer
82 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 ...
1
vote
0answers
39 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 ...