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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Ring of Invariants of symmetric group

The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
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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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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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Producing a $G$-invariant form from the standard Hermitian product using the averaging process

Problem statement: Let $G$ be a cyclic group of order $3$. The matrix $$A=\begin{bmatrix} -1 && -1 \\ 1 && 0 \end{bmatrix}$$ has order $3$, so it defines a matrix representation of ...
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34 views

If $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, find $t$ operators such that $T=T_1\oplus\dots\oplus T_t$

Be $T\in \mathscr{L}(V)$ a linear operator with characteristic polynomial $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, $n_i\geq 1$ and $\lambda_i\neq\lambda_j$ if $i\neq j$. Show that $T$ ...
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Given $\pi:X\rightarrow Y$ how to show $X$ is irreducible (resp. normal) $\Rightarrow$ $Y$ is irreducible(resp. normal)?

Let $G$ act on the affine variety $X=\operatorname{Spec}(R)$ such that $R^G$ is a finitely generated $\mathbb C$ - algebrs and let $\pi:X\rightarrow Y=\operatorname{Spec}(R^G)$ be the morphism of ...
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52 views

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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19 views

What is known about rational points on the ideal of relations / syzygy ideal?

What is known about rational points on the ideal of relations / syzygy ideal? Let $G$ be a finite group, with $|G|=n$. Then $G$ acts on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation ...
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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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109 views

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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What is a trivial linear operator?

The question asks to find a non-trivial linear operator T to make a subspace T-invariant. I'm thinking of $T(x)=2x$, since $2x$ clearly stays in the subspace by scalar multiplication closure, but not ...
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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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4D representation of finite group and projection operator

Suppose I have a finite group, say $C_2$, and its Cartesian representation $ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ ...
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Geometric Invariants of a conic section

There are three independent invariants for every conic section, viz., $$[I_1,I_2,I_3]= [ (a + b + c), (a b -h^2), Det(( a,h,g), (h,b.f), (g,f,c) )] $$ How are they related to the known geometric ...
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equivalent definition of a good categorical quotient?

Working solely with varieties, as you may know, a pair ($Y$, $\pi$) is a "good categorical quotient" for the $G$-variety $X$ if: 1) $\pi$ is surjective and constant along orbits 2) for any open $U ...
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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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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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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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37 views

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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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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about page 358 in burnside in chapter for ring of invariants

context for the problem: I was looking through Burnside's second edition of Theory of Groups of finite order in his chapter "On the invariants of groups of linear substitutions". Let $I_1, I_2, I_3 ...
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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 ...