# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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 ...