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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$\mathbb C[X_1, \ldots, X_n]$ is a free module over $\mathbb C[X_1, \ldots, X_n]^G$

Let $G$ be finite subgroup of $GL_n( \mathbb C )$. Let $\mathbb C[X_1, \ldots, X_n]^G$ be the set of all G-invariant polynomials of $\mathbb C[X_1, \ldots, X_n]$. Is there any rule by which we can ...
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What does it take to have a precise definition of volume?

Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of ...
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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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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 ...
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276 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 f=\chi(\...
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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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66 views

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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60 views

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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56 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 ...
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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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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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Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. ...
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134 views

Does Nagata theorem hold in a field that is not algebraically closed?

Let $R$ be a finitely generated $k$ - algebra and $G$ be a reductive group acting rationally on it. Then a theorem of Nagata says that the invariant ring $R^G$ is also finitely generated. Here $k$ ...
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25 views

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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62 views

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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53 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 Q_1}k[Hom(k^{\alpha(...
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106 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} $i.e. ...
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157 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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130 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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57 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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$\mathbb{A}^m/G_{m} = \emptyset$?

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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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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141 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. $$ \mathcal{F}(\{x_{n-m}\})...
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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 $C= \langle A\rangle$ ...
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$k[V]^G = \widetilde{A}$ 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 ...
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157 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, \...
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150 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 $gp\...
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Polar of orthogonal set invariant under group action

I just ask the following question: Set invariant under group action Furthermore, How to prove the green part Original paper: http://arxiv.org/pdf/1403.4914v1.pdf (p.1324) Let $$O(n)=...
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Smoothness of Schubert Variety

Consider the Schubert variety $X(s_3s_2s_1s_4s_3s_2)$ in $SL_5/P_2$, where $P_2$ is the maximal parabolic corresponding to the simple root $\alpha_2$. In one line notation this permutation can be ...
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Every orbit $G\cdot x$ nonmeager is Baire.

For proof of Effros Theorem I have that $G$ is a Polish group and $X$ is a $G-$space Polish, but I need to show that if the orbit $G\cdot x$ is nonmeager then $G\cdot x$ is Baire in its relative ...
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Creating a linear transformation $T$ such that $T$ is invariant in a proper subspace but not normal

Let $V$ be a finite-dimensional complex inner product space and let $W$ be a subspace of $V$ such that $\dim V = 4$, $\dim W = 2$. Construct a linear operator $T$ such that (i) $W$ is $T$-invariant ...
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Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
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49 views

Invariants of a real symmetric matrix

Problem. I have a real symmetric $n \times n$ matrix $A$ and would like to compute a set of real numbers $f(A) = (x_1, \ldots, x_m) \in \mathbb R^m$ which are invariant under multiplication of $A$ ...
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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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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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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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54 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 \...
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37 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 b}{\...
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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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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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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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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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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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282 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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65 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 det=...
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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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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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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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131 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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63 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} = \dfrac{k[...