# 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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### What is the invariance group of $f(\mathbf{z})=(\mathbf{z}^T\mathbf{z}) (\mathbf{z}^*(\mathbf{z}^*)^T )$?

What is the invariance group of $f(\mathbf{z})=(\mathbf{z}^T\mathbf{z})(\mathbf{z}^*(\mathbf{z}^*)^T)$ ? where $f:\mathbb{C}^n\to \mathbb{R}$. We can think of $\mathbf{z}$ has a complex vector such ...
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### Coprime action and semidirect product

This is a article The Theory of Finite Groups. I'm reading lemma 8.2.1. http://web.math.ku.dk/~olsson/manus/GruFus/Kurzweil-Stellmacher_Theory%20of%20finite%20groups.pdf I don't why "In the ...
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### invariants of a finite group other than the symmetric group

Let $R$ be a Dedekind ring and consider $R[x]$. Let $G$ be a finite group acting on $R[x]$ which acts on $R$ trivially. I want to know if there is a general method to calculate $R[x]^G$. I ...
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### How to use Maple to calculate the degrees by using Molien’s formula?

First, we got the Algorithm3.5.2(Secindary invariants in the non-modular case) by reading the literature： Let G be a Gröbner basis of the ideal $f_{1}, \ldots, f_{n} \subseteq K[V]$ generated by ...
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### Ring of invariants for the action $SL_2(\mathbb{C})$ on binary quadratic forms

In a lecture, I saw the claim that the ring of invariant polynomials for the action of $SL_2(\mathbb{C})$ on binary quadratic forms is $\mathbb{C}[disc]$ - i.e that essentially the only invariant for ...
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### How to find the minimal basis of an invariant ring?

Let $K$ be a field of characteristic $\neq 2$. Let $C_4=\langle\sigma\rangle$ act on $K^2$ by $\sigma (a,b)=(-b,a)$. This translates to $\sigma:s\mapsto t, t \mapsto -s$ on $K(s,t)$, and it is "easily ...
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### Invariant polynomials under dihedral group action

I'm trying to solve the following problem: Find a generating set for the algebra of invariant polynomials $\mathbb C[x_1, x_2]^\Gamma$, where $\Gamma$ is a dihedral group $D_n$, generated by ...
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### The Theory of Finite Groups - An introdution - Hans Kurzweil, Bernd Stellmacher.

Assume that $G$ allows a direct decomposition $$G = E_1 × ··· × E_n$$ that is invariant under A, i.e., $E_i^a \in {E_1,...,E_n}$ for all $a ∈ A$ and $i \in {1,...,n}.$ Under the additional hypothesis ...
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### A-invariant Sylow p-subgroup

This is a article which Antonio Beltran. I'm reading lemma 2.2.c). I see that: "Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every ...
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### Quotient of closed $G$-invariant subset of $G$-variety

Let $X$ be an affine $G$-variety where $G$ is a reductive group. All the varieties are over $k$ , where $k$ is a field (if it necessary we can assume it is algebraically closed). It is a known theorem ...
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### Exercise on Molien theorem

Consider $G_m$ the mutiplicative group of $m$-roots of unity, acting on $\mathbb{C}[x,y]$ via $\epsilon\cdot(x,y)=(\epsilon^a x,\epsilon^{-a} y)$, where $0<a<m$ and $gcd(a,m)=1$. I have to find ...
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### Dependance of an SO(3) invariant function

I am looking to reduce the dependence of a function, knowing that it satisfies some invariance constraints. Let me first formulate my question by explaining the 2-dimensional case. Imagine I have a ...
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### The coprime action and A-invariant Sylow p-subgroup

I read the article (https://www.researchgate.net/publication/291552840_Invariant_Sylow_subgroups_and_solvability_of_finite_groups ) I don't understand how easy this is: "The group $G$ is acted on ...
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### How to find a structure which is invariant under given symmetries?

In brief, the question I have is the following: given a group of symmetry transformations, is there a procedure for defining a structure which is invariant under (only) those symmetries? In more ...
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### Invariant Sylow subgroups and solvability of finite groups

This is a article which Antonio Beltran. I'm reading lemma 2.2.c). I see that: "Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every ...
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### Proof of a formula for Hilbert Series of finitely generated graded rings

Let $S$ be a graded $k$-algebra generated by homogeneous elements $f_1,\dots,f_r$ with degrees $d_1,\dots,d_r$. The following is Proposition 1.9 in "Mukai, An Introduction to Invariants and Moduli" (...
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### How do I find the type-4 tensor product invariants of SL(3) in 3 dimensions?

Specifically I want to find the (irreducible) invariants under SL(3) made from the type-4 symmetric tensor $a_{ijkl}$. Where the indices go from $1..3$ I know that they can be contracted together ...
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### Why are the various invariants of the quartic not symmetric?

To find if a quartic has real roots, one has to look at various quantities made from the coefficients. $$ax^4+bx^3+cx^2+dx+e=0$$ But I would've expected these quantities to be invariant under the ...
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### Stochastic process (Markow)

I have this problem: enter image description here Can someone give some hints for problem 11 and 12: enter image description here? Especially for problem 12 I think I have to use some results that I ...
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### Invariant theory of $SL(n)$ acting on $F^{n\times m}$ by left-multiplication

Consider the linear action of of $SL(n)$ on $n$ by $m$ matrices $F^{n\times m}$ by left-multiplication. Equivalently, $SL(n)$ acts on a set of $m$ vectors from $F^n$ by simultaneous left-...
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### Polynomials invariant up to sign

Let $f(x_1,\ldots,x_n)$ be an unknown homogeneous degree-$d$ polynomial in $\mathbb{Q}[x_1,\ldots,x_n]$. Suppose that a permutation group $G$ acts on $(\mathbb{Q}^n)^*$ so that $f({\bf x}) = 0$ if ...
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### Invariant polynomials of under some linear transformations

I am looking for the polynomials invariant under two linear transformations. That is, if $x\in\mathbb{R}^4$ and given two sets of linear transformations $f$ and $g$, I am looking for the invariant ...
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### Describing invariant subspaces from characteristic polynomial and minimal polynomial

I am working on the following Linear Algebra problem: (a) Suppose $T: \mathbb{R}^4 \longrightarrow \mathbb{R}^4$ is a linear transformation with characteristic polynomial $x^2(x-1)^2$. Describe the ...
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### Existence of a homogeneous system of parameters

In the book on invariant theory by Sturmfels, it is stated that every finitely generated graded $\mathbb{C}$-algebra $R$, of finite Krull dimension $n$, has a homogeneous system of parameters. ...
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### Article: Invariant Sylow subgroups and solvability of finite groups - Antonio Beltran.

This is a article which Antonio Beltran. I'm reading lemma 2.2.b). I see that: "Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every ...
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### Proper invariant subspace given minimal polynomial

I am wondering about the following linear algebra problem: Let the minimal polynomial of $T$ on a finite-dimensional vector space $V$ be $p^2$, where $p$ is irreducible. Is it true that $V$ contains ...
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### Proof of “A geometric quotient is categorical”

I'm reading Geometric Invariant Theory by Mumford-Fogarty, but I can't understand some details in the proof that any geometric quotient is categorical. Let $\sigma$ be an action of $G/S$ on $X/S$ ...
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### Polynomial invariants of finite groups preserved under epimorphism

I am trying to understand Proposition 4.1 of Nakajima's paper 'Invariants of finite groups generated by pseudo-reflections in positive characteristic' (link here: https://pdfs.semanticscholar.org/da5f/...
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### $G$-invariant divisor in affine space

Suppose we have a linearly reductive group $G$ acting on $X=\mathbb{A}^n_{\mathbb{C}}$ and we have a closed subvariety $Z=V(f_1,...,f_k)$, where $f_i$ are $G$-invariant functions. If we further know ...
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### Requirement for the function to be analytic

One-parameter Lie group: When the the infinitesimal group operator $X$ acts upon a function $f=f(x,y)$, it replaces it with $f(x+\xi\delta a,y+\eta\delta a)$. But this can be written as f(x+\xi\...
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### Normal form of the Hironaka decomposition

In "Algorithms in Invariant Theory" (B. Sturmfels) the author gives the following method for testing if a polinomial is part of a Cohen-Macaulay ring $R\in \mathbf{C}[x_1,\dots,x_n]$ that has ...
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### Lie group invariants: are they infinitely differentiable?

In Bluman/Anco's text "Symmetry and Integration Methods for Differential Equations", on p. 46 we find the definition of invariant function F: $\mathbf{F}(\mathbf{x^*})=\mathbf{F}(\mathbf{x})$, where ...
### Find $\mathbb C[x,y,z]^{S_3}$ using Hilbert's Basis Theorem?
Here is the Exercise 2.2.3 from An Invitation to Algebraic Geometry. Let the group $S_{3}$ act on the polynomial $\mathbb C[x,y,z]$. Find the ring of invariant polynomials. Since $S_{3}$ contains ...
### Invariant ring mod $p^n$
Fix a prime number $p\neq 5$. Let $A$ be a finitely generated commutative associative unital $\mathbb{Z}[1/5]$-algebra. Assume the symmetric group $G=S_3$ acts on $A$ by algebra automorphisms. For a ...