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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Meaning of $\Re[\mathbf{z}]\times \Im[\mathbf{z}]$, for $\mathbf{z}\in \mathbb{C}^3$

I came across a function $f(\mathbf{z})$ that maps a vector of $\mathbb{C}^3$, such as $\mathbf{z}=(a+ib,c+id,e+if)$, as follows: $$ f(\mathbf{z})=\mathbf{z}^*\mathbf{z}+(\Re[\mathbf{z}] \times \Im [\...
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Invariant homogeneous polynomials by the action of SL are constant?

It seems to be known that the only invariant homogeneous polynomials in $k[X_0,...,X_n]_d$ by the action of $SL(n+1)$ on $\mathbb{P}^n$, are the constants, ie, $F(gx)=F(x)$ for all $x\in\mathbb{P}^n$...
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Are there any known results on inverse invariant theory?

I understand the basics of invariant theory in commutative algebra. Assume we are working over a polynomial ring in n variables over a field of characteristic 0. What I’m looking for are examples or ...
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Need further explanation on Invariant Subspaces

I'm leaning about invariant subspaces, and the topic seems to be too hard for me to comprehend. Reading the topic off Wikipedia, the following explanation is given about matrix representations of ...
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Invariant theory of the definite and indefinite orthogonal groups

I am vaguely aware of the following facts: Let $V$ be a finite-dimensional real vector space with a positive-definite inner product $g$. Let $g_{\otimes n}$ denote the natural extension of $g$ to $V^{...
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“Series of series of Hyperdeterminants” in Cayley

In reading a paper by Cayley entitled "On Linear Transforms" (listed as number 14 of volume I of Cayley's collected works), I came upon the following, apparently interesting concluding remark: To ...
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On a Hyperdeterminant Computation of Cayley

I recently read a paper by Cayley entitled "On the theory of linear transformations" (it can be found in volume I of his collected works as number 13), wherein Cayley computes a "hyperdeterminant" for ...
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Complete invariants and third-order polynomials

I am interested in understanding some general properties of complete invariant maps. An invariant is a map $$f: X \rightarrow Y$$ defined over objects $X$ and a transformation $\tau$ such that $f(...
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Invariant Sylow subgroups

Today, I'm reading lemma 2.2.c of an article by Antonio Beltran. 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 prime p, ...
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Invariant ring of the alternating group

I suspect that the invariant ring of $A_n$'s action on $K[x_1, ..., x_n]$ are the "alternating polynomials" - ie the symmetric polynomials adjoined with the Vandermonde polynomial $$\prod_{i < j} (...
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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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44 views

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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57 views

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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Dimension problems regarding action of reductive group on variety

Let $k$ a field and $X$ a $k$ algebraic variety. Let $G$ a $k$ reductive group acting on $X$. I denote with $X_d$ the set of points in $X$ which have stabilizer of dimension $d$. It is a fact that $...
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Why the polynomial obtained is invariant under $S_n$ action?

Consider $k$ a field. Identify $\otimes_{i\leq n}k[x]=k[x_1,\dots, x_n]$ where tensor is over $k$ as algebra. I want to see the following holds. Let $z=(v_1,\dots, v_n)\in k^n, z'=(v'_1,\dots, v'_n)\...
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34 views

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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64 views

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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42 views

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 to inversions

We extend the algebraic Lemma used here: http://www.ams.org/amsmtgs/2259_abstracts/1143-53-201.pdf We presented this result here: https://conferinta.ssmr.ro/community For what values of $k$ does$$(...
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68 views

Invariant to inversions 2

Let $a,b,c,d,e,f>0$ satisfying $a+b+c+d+e+f=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}$ . Prove $$ab+bc+cd+da+ac+bd+ae+be+ce+de+af+bf+cf+df+ef+10\sqrt{abcdef}\...
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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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Group action on Artin ring

Let $k$ be a field and $A$ is a finite dimensional commutative $k$-algebra. Suppose a finite group $G$ acts on $A$ by automorphisms. Then $G$ acts on $\operatorname{Spec}(A)$. Suppose $A^G=\{a \in A | ...
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The algebra of invariants of a binary cubic form

How to find the algebra of invariants of a binary cubic form ${a_{\overset{\,}{0}}}x^3+3{a_{\overset{\,}{1}}}x^2y+3{a_{\overset{\,}{2}}}xy^2+{a_{\overset{\,}{3}}}y^3$ The algebra of invariants $I=4\...
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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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1answer
51 views

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 ...
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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 ...
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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 ...

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