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12
votes
1answer
154 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: ...
9
votes
1answer
106 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 ...
8
votes
1answer
162 views

Any affine algebraic group is linear.

It is a well-known result that any affine algebraic group is a closed subgroup of some $\mathrm{Gl}_n(\Bbbk)$. However, I would like to see a proof for that, so I looked it up in various books, more ...
7
votes
1answer
142 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 ...
7
votes
1answer
121 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
222 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 ...
6
votes
0answers
165 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
2answers
325 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 ...
4
votes
1answer
121 views

How do we know how many branches the inverse function of an elementary function has?

How do we know how many branches the inverse function of an elementary function has ? For instance Lambert W function. How do we know how many branches it has at e.g. $z=-0.5$ , $z=0$ , $z=0.5$ or ...
4
votes
2answers
101 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 $$ ...
4
votes
2answers
122 views

Finite presentation of algebra of invariants

(1) Let $R$ be a ring, let $A$ be a finitely presented $R$-algebra, and let $G$ be a finite group of $R$-automorphisms of $A$. Is the algebra of invariant $A^G$ finitely presented over $R$? I can ...
4
votes
1answer
65 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 ...
4
votes
0answers
94 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
135 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
3answers
129 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 ...
3
votes
1answer
110 views

Invariant rings $\mathbb{C}[X,Y]^{GL_2}$ and $\mathbb{C}[X,Y]^{SL_2}$

I feel like I might have made a mistake on this question, and would appreciate some feedback from someone more experienced than me. If $G$ acts on $S$, we write $S^G = \{s \in S: gs = s \, \forall ...
3
votes
2answers
171 views

Quotient of an affine variety by a finite group coincides with topological quotient as a point set?

I have just read the construction of the quotient of a closed subset $X$ of affine space by a finite group $G$ of automorphisms of $X$, in Shafarevich, Basic Algebraic Geometry I. Shafarevich gives ...
3
votes
1answer
48 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 ...
3
votes
1answer
910 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 ...
3
votes
1answer
50 views

Polynomial with given group of symmetries

Let $f$ be a polynomial in $n$ variables and $G$ - its group of symmetries (group of permutations of variables wich left $f$ in place). I'm trying to such $f$ for given group $G$. I have troubles when ...
3
votes
0answers
33 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
39 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
1answer
99 views

Does anyone know of Invariant Theory enough to comment on this question?

I am trying to find a minimal set of invariants for the binary homogenous form $$\displaystyle ax^7 + bx^{6}y + cx^{5}y^{2} + dx^{4}y^{3} + ex^{3}y^{4} + fx^{2}y^{5} + gxy^{6} + hy^{7}$$ What is the ...
2
votes
2answers
73 views

Total dimension of the cohomology of a homogeneous space (or of a graded Tor)

I want to calculate the cohomology ring with rational coefficients of a homogeneous space, but would be happy enough to know its total dimension. Let $G$ be a compact Lie group, $T$ a maximal torus, ...
2
votes
2answers
89 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 ...
2
votes
1answer
44 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 $H_i(X,\mathbb Z)_G$.
2
votes
1answer
96 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 & ...
2
votes
2answers
59 views

Constructing the sequence: $0\rightarrow (x-y)^{S_2} \stackrel{f}{\rightarrow} k[x+y,xy]\stackrel{g}{\rightarrow} k[y]$

Let $S_2$, a group of two elements, act on $k[x,y]$ by permuting $x$ and $y$. It is clear that $$ 0\rightarrow (x-y) \rightarrow k[x,y]\rightarrow \dfrac{k[x,y]}{(x-y)}\cong k[y] \rightarrow 0 $$ ...
2
votes
1answer
85 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 ...
2
votes
1answer
44 views

Give a example about invariant ergodic measure and quasi-symmetric mapping

Is there a example $(X,f,\mu)$ such that $X$ is a closed subset of Euclidean space, $f$ be a quasi-symmetric mapping but not a Lipschitz mapping, $f(X)=X$, $\mu$ is a finite measure on $X$ that is ...
2
votes
1answer
154 views

Identifying $k[x_1,x_2,y_1,y_2]^{\epsilon}$ with $k[x,y]\wedge k[x,y]$

Suppose the symmetric group $S_2$ of order 2 acts on $k^4=Spec \;k[x_1, x_2, y_1, y_2]$ by the following: for $\sigma\not=e$, $$\sigma\circ(x_1, x_2, y_1, y_2)=(x_2,x_1,y_2,y_1).$$ That is, the ...
2
votes
0answers
18 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 ...
2
votes
0answers
77 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
37 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
90 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
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 $G$. ...
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)= ...
2
votes
0answers
39 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
110 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
58 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
124 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
124 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
2answers
95 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 ...
1
vote
1answer
127 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 ...
1
vote
1answer
110 views

A condition in the definition of geometric quotient

I am reading the first several pages of GIT by Mumford, and I need some clarification of one requirement in the definition of geometric quotient (c.f. Definition 0.4, GIT): Suppose a group scheme ...
1
vote
1answer
39 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
2answers
111 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 ...
1
vote
1answer
45 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 ...
1
vote
1answer
49 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 ...