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 finite groups

Let $G$ be a finite group acting linearly on $\mathbb{C}^n$ and $\mathbb{C}[X]^G$ be the ring of invariant polynomials. If $G$ is a group generated by reflections, this ring is generated by $n$ ...
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Equivariant polynomial maps and gradients of invariants

Let $G$ be a finite group with a linear action on $\mathbb{C}^n$ and $f\in\mathbb{C}[X_1,\ldots,X_n]$ be invariant. Then the gradient of $f$ gives rise to a polynomial map $\phi$ from $\mathbb{C}^n$ ...
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Is there a plane algebraic curve with just 3-fold rotational symmetry, but without reflection symmetry?

I am new to the subject of invariant theory, but the Reynolds operator popped up so I tried to calculated some examples for myself. I computed the invariant polynomials under the cylic group $C_3$ of ...
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Elimination in system of equations

I find in article ,,Model-Based Recognition of 3D Objects from One View,, by I. Weiss M. Ray this system of equations: $$\mu m'_{12} - I_1\mu'm_{12} + \mu\mu'(I_1 - 1)m_{25} = 0 \\ \mu m'_{13} - I_2\...
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For which power of a matrix does it has eigenvectors?

I Have the following question in my exercise: Let $T:\Bbb R^2 \rightarrow \Bbb R^2$ $[T]_E= \left[ \begin{matrix} 2&-4\\5&-2\\\end{matrix} \right]$ The question asks to find all $n \in \Bbb ...
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Why is a graded $\mathbb{C}$-algebra free as a module over the subring generated by a regular sequence?

I am reading Bernd Sturmfels' book Algorithms in Invariant Theory. On p. 38 he makes the following assertion: "If $\theta_1,\dots,\theta_n$ are algebraically independent over $\mathbb{C}$, then the ...
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Is $R$ a finitely generated algebra over $R^G$?

Let $R$ be a noetherian ring, $G$ a finite group acting on $R$, $R^G$ its invariant subring. Is $R$ a finitely generated algebra over $R^G$? Is $R^G$ a noetherian ring?
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Maps from a (f.g.) $k$-algebra (w/action of finite group $G$) to $k$, that agree on the subring of $G$-invariants

Suppose $A$ is a finitely generated $k$-algebra with action of a finite group $G$. Suppose we have two maps $\Phi: A\to k$ and $\Phi': A\to k$ such that $\Phi|_{A^G}=\Phi'|_{A^G}$, i.e. they agree on ...
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47 views

Lyapunov invariant set for affine systems

Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite ...
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Invariant theory and irreducible representation of a group

I am now reading the book 'Invariant theory' by Neusel. It seems that the logic is like this: You have a representation of a group $G$ on some vector space $V$, this might be a reducible ...
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Are similar complex matrices again similar when each is expressed as a real matrix?

We know that, relative to this ordered basis {$(1,0),(i,0),(0,1), (0,i)$}, we can express a 2x2 complex matrix mapping $C^2 -> C^2$ as a $4x4$ real matrix (representing the same transformation of $...
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Automorphism of order 2 of C[x,y] and its ring of invariants

Let $A=\Bbb C[x,y]$. For an automorphism $\sigma$ of order $2$ of the algebra $A$, let $B=\{ f \in A \mid \sigma f = f \}$ be the subalgebra of $A$ consisting of all fixed points of $\sigma$, and ...
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Computing the trace and determinant of a matrix representation relative to a real basis, from the matrix relative to a complex basis,

Let $A$ be the complex matrix representing a transformation of the vector space $C^2$ of 2-tuples over the complex numbers into itself, relative to the natural ordered basis {(1,0),(0,1)}. Let $A_R$ ...
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Local Noetherian ring and its invariants

Let $R$ be a local Noetherian ring and $G$ a finite group. Is $R$ finitely generated as a module over its invariants $R^G$? Thank you.
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Question on an exercise in Atiyah-Macdonald.

I am reading a proof for exercise 5.13 in Atiyah and Macdonald, given below. Exercise 12 is the following I understand everything except one thing, why is $\prod_{\sigma \in G} \sigma^{-1}(x) \in ...
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Proving an eigenspace is a T-Invariant subspace.

I want to know if I'm going about this proof the correct way. Problem Statement: Let $T$ be a linear operator on a vector space $V$, and let $λ$ be a scalar. The eigenspace $V^{(λ)}$ is the set of ...
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Determining ring of invariants of $\pm$ Identity

Determine the ring of invariants $\mathbb C [x,y,z]^\Gamma$ for: $$\Gamma :=\{ \begin{pmatrix} \pm1 & 0 & 0 \\ 0 & \pm1 & 0 \\ 0 & 0 & \...
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finitely generated over invariants

I was wondering if there are some conditions we can add to the following statement to make it true "The ring $R$ is finitely generated as a module over $R^G$, where $G$ is a finite group and $R^G$ ...
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Permutation groups acting on polynomial rings and base change

Let $G\subset S_n$ be a permutation group, and let it act on $R = \mathbb{Z}[x_1,\dots,x_n]$ by permuting the variables, as usual. $G$ acts on $R\otimes_\mathbb{Z} S$ for any unital ring $S$ via its ...
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Algebra of invariants finitely generated

If algebraic torus $T = \mathbb G_m^n$ over an algebraically closed field $k$ acts on finitely generated algebra $A$ over $k$, is the algebra $A^T$ of invariants finitely generated?
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Invariant subspace under representation $\phi$

Let $\phi: \Bbb{Z}/n\Bbb{Z}\to GL_{2}(\Bbb{C})$ be the representation which takes $\bar{m} \to \begin{bmatrix} \cos (\theta) & -\sin (\theta) \\ \sin (\theta) & \cos (\theta) \end{bmatrix}$ ...
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Regular sequence in rings of invariants

When I was reading a book in invariant theory, I've come across this assumption: "Let $f_1, f_2 \in \mathbb{F}[V]^G$ be a regular sequence in $\mathbb{F}[V]$". I know that we cannot assume anything ...
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Local Lie derivative on $G$-space at the zero of a vector field

Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian). $L_X$ is a vector field on $M$, at a ...
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Find the invariant polynomial space of a finite matrix group.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Denote the finite matrix group $\{\pm I_2\}\subset GL(2,k)$ by $C_2$. It is known that the invariant polynomials under $C_2$ ...
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Do we have $\mathbb{C}[SL_n] = \oplus_{\lambda, \text{ht}(\lambda)\leq n} V_{\lambda} $?

The coordinate algebra $\mathbb{C}[SL_n]=\mathbb{C}[x_{ij}: i, j \in \{1, \ldots, n\}]/(\det(x_{ij}) - 1)$ is a representation of $SL_n$: $(g'.f)(g)=f(g'^T g)$. Let $V_{\lambda} = \langle e_T : T \...
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153 views

A simple question on forward and backward invariant sets

A subset $A\subset X$ is forward invariant if $f^{t}(A)\subset A$ for all $t\ge 0$ and backward invariant if $f^{-t}(A)\subset A$ for all $t\ge 0$. I want to show that the complement of a forward ...
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$SL(2)$ invariant polynomials are generated by determinant?

Suppose $SL_2(\mathbf{C})$ acts on the space of quadratics $aX^2+2bXY+cY^2$ by $X\to \alpha X+\beta Y, Y\to\gamma X+\delta Y$, where $\alpha,\beta,\gamma,\delta$ consists a matrix in $SL_2(\mathbf{C})...
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Diffeomorphism-invariant spaces of smooth functions

Let's start with an interesting story. In his celebrated Partial Differential Relations (p. 146), the great Misha Gromov gives a nice exercise of which the following is a (strict) part. Exercise. ...
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Quotients of curves

Magma (link) has a lot of functionality for computing quotients of curves by group actions. I am interested to know how one does this in general and I am finding it oddly difficult to find literature ...
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Partial derivatives with respect to algebraically independent polynomials

Suppose that $\{f_1, \ldots, f_n\}, \{g_1, \ldots, g_n\}$ and $\{h_1, \ldots, h_n\}$ are algebraically independent polynomials that generates the same algebra of $\mathbb{R}[x_1, \ldots, x_n]$. Then I ...
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The elementary symmetric functions are a homogeneous system of parameters for the invariant ring of a permutation representation

For a permutation representation of order $n$ over a field $F$, I need to show that the elementary symmetric functions $s_1,s_2,\ldots,s_n$ form a homogeneous system of parameters for the ring of ...
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Show that $\mathbb{F}[x^2,y^2,xy]$ is not polynomial

$\mathbb{F}[x^2,y^2,xy]$ is the polynomials in two variables whose terms all have even degrees. Of course, this generating set $x^2,y^2,xy$ is not algebraically independent, but I need to show that no ...
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Characters of Linear Algebraic Groups

Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering ...
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Write $V=P_2(\mathbb{R})$ as a direct sum of $V=W_1\oplus W_2 \oplus W_3$

So, if I let $T:P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R})$ and is a linear endomorphism given by $T(f(x))=f(x)-f(2x-1)$. Then I have to write$V=P_2(\mathbb{R})$ as a direct sum of $V=W_1\oplus W_2 \...
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Discriminant is the unique invariant of $\text{SL}_2\mathbb{Z}$ acting on polynomials.

The following is a really wonderful theorem that I really have no idea how to prove. Consider $p=ax^2+bxy+cy^2$, and let $\text{SL}_2\mathbb{Z}$ act on all such $p$ by $\begin{pmatrix} a&b \\ ...
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What does invariant exactly mean and how does it get the invariant?

I have read many journal about simulation of regularized long wave. In numerical test section,many researcher use invariant of mass,momentum and energy to check accuracy of their method but i found ...
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Representations of symmetric groups of order $2n$ and $n$

Background: Denote by $S_n$ the symmetric group of order $n$. There are many ways to embed $S_n$ as a subgroup into $S_{2n}$. Given a symmetric group, we can use Young diagrams to classify all ...
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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 ...
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How does Molien series describe polynomial invariants?

As I understood from wiki page, Given a finite group acting on a vector space, Molien series gives a generating function, although I am not sure what this means. And how is this related to the ...
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Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
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Algebraic invariants for first order equivalence between fields

I know that every two models of the theory $ACF$ (namely two algebraic closed fields) with the same characteristic are elementary equivalent. But what about generic fields? Are there any algebraic ...
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The disease problem

Students are sitting in a n * n grid. There's a disease spreading among them in a particular fashion. At start, there a 'k' students infected(At random). After every time step(equal intervals), the ...
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Generating set of the algebra invariants of finite group.

Let a finite group $G$ acts on a complex vector space $V$ and let $\mathbb{C}[V]^G$ be corresponding algebra of polynomial invariants. Let $f_1,f_2,\ldots,f_m$ be a generating set of this algebra of ...
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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 Q_1}k[Hom(k^{\alpha(...
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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 \...
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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 ...
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Name of quantity that is not invariant, but only changes in one direction

How do you call a quantity that is not an invariant, but only changes in one direction during the process? Example: The degree of the polynomials go down when Euclidean division is applied, so the ...
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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 b}{\...
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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, ...
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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 ...