# Questions tagged [invariant-theory]

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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### Map between rings of invariant differential operators

Suppose we have a reductive group $H$ and a representation $V$. Let $G$ be a group containing $H$ as a closed subgroup and let $W=G\times_H V$. The rings of differential operators $D(V)$ and $D(W)$ ...
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### What are the invariant polynomials of representations of the Lorentz group $SO^+(3,1)$ and $SL(2,\mathbb{C})$?

The Lie Groups $SO^+(3,1)$ and $SL(2,\mathbb{C})$ occupy a particular, unique place in physics. I am interested in the following problem: suppose I have a finite-dimensional representation (not ...
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### Example of a quotient of an elliptic curve by a finite group being rational

I am interested in an example of the following situation, over an algebraically closed field o zero characteristic. Let $E$ be an elliptic curve, and $G$ a finite group of automorphisms of $E$ (as an ...
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### Positive invariance proof for an ODEs system with a constraint

I am new to ODEs. Tring to prove the postivity invariance of the following system $\forall t$: ''' \begin{array}{l} \frac{d X}{d t}=-a X+b, \\ \frac{d Y}{d t}=-Y\left(\frac{p_1 X}{X+a}+p Z\right)+r Y\...
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### Continuous Section of Length Map for Triangles

Interpret $\mathbb{R}^6 = (\mathbb{R}^2)^3$ as the space of ordered triangles in the real plane (degenerate triangles are included). There is a map $L \colon \mathbb{R}^6 \to \mathbb{R}^3$ sending a ...
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### What is a concrete example of a perpetuant (in classical invariant theory)?

I am trying to determine whether an object of my recent research is actually a "perpetuant" in the sense of Sylvester and classical invariant theory. There are a few papers on the topic, ...
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### First fundamental theorem of invariants for the unitary group

Setting Let $V$ transform according to the direct representation of the unitary group $U(d)$. I have a polynomial on $P:V^k \times (V^\star)^l\rightarrow \mathbb R$ where $V^\star$ is the conjugate ...
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### Cooking up invariants of a group action

There are a lot of examples of a group action preserving a specific invariant. For example, $\mathrm{SL}(V)$ preserves volume element on $V$ $\mathrm{SO}(V)$ preserves distance (or more generally, ...
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### Invariants of graded space is the same as the graded space of invariants

I am stuck on Theorem $3.5.1$ of the following book. The author says "In particular, we see that the map $(3.5.1)$ is compatible with the natural filtrations, and the corresponding map of ...
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### is there a relation between the trace of a 3x3 real, symmetric, positive definite matrix and the trace of its inverse

Let $\mathbb{M}^{3\times3}$ denote the set of real, symmetric, and positive definite $3\times3$ matrices. Given $A \in \mathbb{M}^{3\times3}$ with the property $\det A=1$. In my engineering ...
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### Is the Schofield semi-invariant defined at $V/IV$?

Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
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### How to construct translation invariant distribution

There are serveral papers talk about how to construct a translation invariant distribution, however I couldn't get the idea of why those methods work and why we need it. If possible, could please give ...
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### Linear group defined by its invariants

Let $G \subset GL(V)$ be a faithful finite dimensional representation of a Lie group. Let us denote by $I_G\subset \bigoplus_{n+m} \bigotimes_{n} V \otimes \bigotimes_m V^{*}$ the set of all invariant ...
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### Base change and restriction over Galois extenstion

(Sorry for my bad English.) Let $L/K$ be (if necessary finite) Galois extension, and $A$ be $K$-algebra. Then the Galois group $G=\operatorname{Gal}(L/K)$ acts on $K$-algebra $A\otimes_K L$. So by ...
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