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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Invariant polynomial function on Lie algebras.

Take $L$ a complex simple Lie algebra and $f \in S(L^*)^L$ where $S(L^*)$ is the symmetric algebra of $L^*$ (that could be seen like algebras of polynomial functions on $L$). For every homogeneous $f \...
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Do the invariants & covariants completely characterize a projective variety up to projective equivalence?

Let $V \subseteq \mathbb{CP}^n$ be a projective variety embedded in complex projective n-space. By the nullstellensatz it is the zero-set of a finite number of homogeneous polynomials, $p_1(x_1,...x_{...
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A question about notation issues with *homogeneous* polynomials

Reading about homogeneous polynomials in an introduction to invariant theory, I'm trying to better understand some notation. In particular, every homogeneous polynomial of degree $d$ can be written as ...
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General formulation of polynomial invariants.

Reading this work https://arxiv.org/pdf/2012.06452.pdf, I'm wondering if it's true that any fundamental invariant polynomial $\{f_i\}_{i=1}^{N_{inv}}$ can be written as $$f_i(x) = \sum_{g \in G}\psi(g ...
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Is the fundamental theorem of symmetric polynomials a particular case of the Hilbert finiteness theorem?

I have found out that the statement of so-called Fundamental theorem of symmetric polynomials, which asserts the following: (Wikipedia) For any commutative ring $A$, we denote the ring of symmetric ...
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A question about approximating a continuous function with a $G$-invariant polynomial.

Given $f:\mathbb{R}^n \rightarrow \mathbb{R}$, a finite group $G$ and a compact set $K \subseteq \mathbb{R}^n$, I want to prove that I can approximate $f$ with a $G$-invariant polynomial on $K$. I ...
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Proof for existence of quotient, finite group acts on affine variety

I am currently studying quotients of varieties. In the book "Algebraic Geometry" by J. Harris (http://userpage.fu-berlin.de/aconstant/Alg2/Bib/Harris_AlgebraicGeometry.pdf), Harris wants to ...
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Invariant ring is a finitely generated $K$-algebra, confusion regarding the proof

I'm currently studying quotients of affine varieties by finite groups. I use the book "Algebraic Geometry" by J. Harris (http://userpage.fu-berlin.de/aconstant/Alg2/Bib/...
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Confusion regarding quotient of varieties

I am currently studying quotients of varieties, I use "Algebraic Geometry" by J. Harris. However, there are a few things I don't quite understand: (p. 123) At first it says that such a map $...
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A question about the definition of equivariant map

If $S$ is a set of functions from $X$ to $Y$ then I can consider the action of a group $G$ on $S$ via its action on $X$ and $Y$ by the formula $$(g \cdot f)(x) = g \cdot f(g^{-1} \cdot x),$$ So we are ...
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The invariant ring of complex polynomials is a subalgebra

I'm trying to see that the invariant ring of polynomials over $\mathbb{C}$ is a subalgebra of $\mathbb{C}[\textbf{x}]$. Given $$\mathbb{C}[\textbf{x}]^G:= \{f \in \mathbb{C}[\textbf{x}] \, | \, g \...
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A question about obtaining an invariant polynomial from any polynomial $f$

Symmetric polynomials in variables $x_1, \dots, x_n$ are invariant under the action of $S_n$ which arbitrary permutes the variables. Now, I'm reading that an invariant polynomial $f_{sym}$ can be ...
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What is the explicit ${\rm GL}_n$-action on the Weyl algebra?

This is a typical classical invariant theory setup. Let $V = \mathbb C^n$, and let $\mathbb C[V] \cong \mathbb C[x_1,\ldots,x_n]$ be the space of polynomial functions on $V$. The group $G={\rm GL}(n,...
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A question about definition of equivariance for a function.

Equivariance in deep learning, it is actually defined as a property of some function $f$ to permute the outputs according to permutation of inputs, i.e. $$f(PAP^T) = Pf(A)P^T$$ for some permutation ...
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Invariant rings of all finite subgroups of GL(1,C)

I am trying to determine the invariant rings of all finite subgroups $\Gamma$ of $\text{GL}(1,\mathbb{C})=\mathbb{C}^\ast$. I know that since $\Gamma$ is finite, the invariant ring is finitely ...
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Show that any permutation invariant function acting on a countable set has a defined form

Reading about invariance and equivariance properties occurring in the field of Machine Learning, I'm trying to understand the proof of the main theorem presented in the paper Deep Sets: https://arxiv....
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Restriction/projection map between invariants of symmetric algebras

$\newcommand{\sym}[1]{\mathfrak{S}_{#1}}$ $\DeclareMathOperator{\Sym}{Sym}$ Given a commutative ring $A$ and an $A$-module $M$, writing $\sym{n}$ for the symmetric group on $n \geq 1$ letters, is the ...
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Combining rational invariants

This turned up with the Euler Brick. It is well known that solving $(p^2-1)(q^2-1)(r^2-1)-8pqr=0$ in rational numbers gives an Euler brick (a quader with rational sides and space diagonals). The ...
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What if the semi-invariant ring is a polynomial ring or hypersurface

I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)? I have been studying about semi-...
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Why do people study semi-invariant ring (in general)?

I'm interested in studying about semi-invariant ring in the context of Quiver representations. I started reading about it in some books and few places on the internet. But, nowhere do they mention why ...
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About Molien's formula and Hilbert series

Given that ${\bf r}_a, {\bf r}_b, \dots$ are irreducible representations of a finite group $G$, the k-fold Kronecker product of $\bf{r}_a$ on a symmetric subspace ${\rm Sym}^k(V)$ is expanded in terms ...
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Invariants of a symmetric bilinear form

The following is a theorem from the Linear Algebra Textbook by Friedberg, Insel, and Spence (5th Edition). Theorem 6.38 (Sylvester's Law of Inertia). Let $H$ be a symmetric bilinear form on a finite-...
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Invariant-coinvariants splitting failure

Consider a finite group $G.$ Over any field where all divisors of $|G|$ are invertible there is a splitting of any $G$-module $M:$ $$ M=M^G\oplus M_G $$ This splitting could be written explicitly ...
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Polynomials invariant under the action of SO(3)

Let $SU(2)$ act on $V_2=\{ax^2+bxy+cy^2:a,b,c\in\mathbb C\}$ by $$\begin{pmatrix}a&b\\c&d\end{pmatrix}f(x,y)=f(ax+cy,bx+dy).$$ Then $SU(2)$ also acts on the $n$-th symmetric power $S^n V_2$. ...
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Invariant functions for irreducible representations of $\mathrm{SU}(2)$.

The orbits of $\mathrm{SU}(2)$ acting irreducibly on $\mathbb{C}^2$ are three-spheres centered around the origin. In other words, an orbit is uniquely specified by the Euclidean norm in $\mathbb{C}^2$....
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Relation between invariant meromorphic and invariant holomorphic functions

Given an affine space $\mathbb{C}^n$ (more generally a Stein space), and an action of a complex Lie group $G$ on it. Is there a relation between (sheaves of) invariant holomorphic and invariant ...
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How can you change the counts on vertices of a tetrahedron if you increase/decrease the counts on all vertices of a face the same amount?

Hard question to ask, but here's the idea: I'm creating a game to teach invariants. I want to make a game where each vertex of a platonic solid (we'll start with a tetrahedron) has a counter - all ...
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$\lim \frac{1}{n} \sum_{i=0}^{n-1} X \circ T^{i}$ T-invariant?

Birkhoff's ergodic theorem goes as follows: Let $T$ be a measure-preserving transformation on $(\Omega,\mathcal{F}, \mathbb{P})$ and $X \in \mathcal{L}^1$. Then $S_n := \frac{1}{n} \sum_{i=0}^{n-1} X \...
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Find invariant ring in modular case

I'm learning Invariant Theory of Finite Groups and saw this problem: "What is $S^G$ where $S := \mathbb{F}_p[x, y]$, and $G$ is the subgroup of $GL_2(\mathbb{F}_p)$ generated by $\begin{pmatrix} ...
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Necessary condition for orbits to be separated by invariant polynomials.

Suppose we have an algebraic subgroup $G \subset GL(V)$, where $V$ is a finite dimensional vector space over the field of complex numbers $\mathbb{C}$. I'm trying to prove the following: If $G$-...
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Quotient field $G$-invariant ring is a ring of rational functions

I am working through Eisenbud's Commutative Algebra and am stuck on a "trivial" part of an exercise. Let $G$ be a finite abelian group that acts by characters on $S = k[x_1,\ldots,x_r]$, ...
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Do all of the orbits have the same dimensions?

Let $G$ be an algebraic group and let $X$ be a $G$-variety. It's stated in the paper (pg. 13) that all orbits are closed and have the same dimensions if the graph $$\Gamma_{X}:=\{(g x, x) \mid g \in ...
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Dimension of the Invariant Subspace $((\mathbb{C}^2)^{\otimes 2n})^{\mathfrak{sl}(2,\mathbb{C})}$

This was posed as a challenging exercise by my professor. The exercise is to compute the dimension of the invariant subspace of $(\mathbb{C}^2)^{\otimes 2n}$ viewed as an $\mathfrak{sl}(2,\mathbb{C})$ ...
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Singular quadrics, covariants of binary quartics and $\operatorname{SL}_2(\Bbbk)$ representation

Consider the space of irreducible $\operatorname{SL}_2(\Bbbk)$ representation: $$R_4 = \langle x^4, x^3y, x^2y^2,xy^3, y^4 \rangle.$$ This is a 5 dimensional vector space, and we have decomposition ...
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Reflection group action on a reflection hyperplane

Let $G$ be a finite reflection group acting on a euclidean vector space $V$ of dimension $n$ and let $H$ be one of the reflection hyperplanes. Then $H$ is stabilized by some (reflection) subgroup of $...
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$\mathbb{C}(V)$ is a finite module over $\mathbb{C}(V)^G$?

Is it true that $\mathbb{C}(V)$ is a finite module over $\mathbb{C}(V)^G$ for any finite subgroup $G \subset GL(V)$ and, moreover, $\dim_{\mathbb{C}(V)^G} \mathbb{C}(V) = |G|$? It possibly follows ...
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Given $f \in V^*$ and $\mathscr A : V \to V$ a linear map, show that $f,f \circ \mathscr A,\dots,f \circ \mathscr A^{n-1}$ is a basis for $V^*$

Let $V$ be a $n$ dimensional vector space, $V^*$ is its dual, $\mathscr{A}$ is a linear transform from $V$ to $V$. Given $g\in V^*$, define $$\mathscr{B}(g)=g\circ\mathscr{A}$$ Then it is easy to ...
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Why is the product map $GL_1(k)\times GL_1(k)\rightarrow GL_1(k)$ not continuous? [duplicate]

I am reading Springer's Invariant Theory. I already have some experience with linear algebraic groups and invariant theory, yet one of the first exercises of the book has already confused me. In the ...
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Show that $ \text{Tr}(XYZ) + \text{Tr}(YXZ)+ \text{Tr}(X)\text{Tr}(Y)\text{Tr}(Z) = ... $

Let $X, Y, Z$ be $2 \times 2$ matrices. Show that these two matrix combinations are equal: $ \text{Tr}(XYZ) + \text{Tr}(YXZ)+ \text{Tr}(X)\text{Tr}(Y)\text{Tr}(Z) $ $ \text{Tr}(X) \, \text{Tr}(YZ)...
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How do we find a minimal form of a Hilbert series?

Before I explain myself better, I would like readers to keep in mind I am a physicist and not a mathematician. Let $G$ be an infinite reductive group such as $\mathrm{SL}(n,\mathbb{C})$ and a k-...
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About a property of Reynolds operator (Invariant theory)

I am learning about classical invariant theory, and I have a question about Reynolds operator. The book I am reading is $\ulcorner$Classical Invariant Theory$\lrcorner$ written by Hansepter Kraft, ...
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Vanishing of certain Ext groups for some two-dimensional singularities

Let $R = \mathbb{C}[x,y]$ (or possibly $R = \mathbb{C}[[x,y]]$, although I think we don't need the local hypothesis, and can work in a graded setting instead) and let $G \leqslant \operatorname{GL}(2,\...
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How to prove a differential entropy is not scale invariant?

For example, $S(X)=-E_X(\log(f_X))=-\int_{-\infty}^{+\infty}f_X(x)\log(f_X(x))dx$ A transformation of X changes the result:$S(aX)=S(X)+\log|a|$ and more in general $S(g(X))=S(X)+\int_{-\infty}^{+\...
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Example of G-invariant ideal

Definition: Let $G$ be a group acting on $R_n:=K[x_1,\dots,x_n]$ with $$\begin{aligned} G\times R_n &\rightarrow R_n\\ (g,f) &\mapsto f^g \end{aligned} $$ where $(f^g)$ acts on a point $p$ in ...
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Help with an algebraic geometry result

I am studying this part of algebraic geometry and I have come to this proposition. I understand the basic idea well but there are two details that escape me. Proposition: Let $G$ be a finite group ...
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Are SU(n) invariants necessarily non-commuting?

SU(3) is a subgroup of O(6). Therefor we can say it can be represented in 6 dimensional space. It has one invariant which is: $$x^2+y^2+z^2+w^2+u^2+v^2$$ So far the group compatible with this is O(6). ...
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231 views

How to prove that the softmax function is not invariant under scalar multiplication.

I want to prove that the softmax function is not invariant under scalar multiplication. How to continue from there to prove that S(x)i is not equal to S(xc)i ?
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A lemma (possibly of Atiyah) in a proof of the fundamental theorems of invariant theory

A classical problem of invariant theory is to describe the space $((V^*)^{\otimes r} \otimes V^{\otimes s})^G$ in terms of generators and relations, or equivalently to determine generators and ...
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86 views

A subalgebra $A(X)^G$ of $G$-invariant functions on $X$ is finitely generated subalgebra where $G$ is a finite group.

Let $X$ be an affine variety, $A(X)$ be a coordinate ring of $X$, and let $G$ be a finite group. Let's assume that $G$ acts on $X$ i.e. we have morphisms $g:X\to X$ for all $g\in G$ s.t. $e:X\to X$ is ...
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112 views

Invariance of $SO(3)$ Lie group when expressed via Euler angles

I am trying to understand the properties of the $SO(3)$ Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions. I am building an Invariant Extended Kalman Filter (IEKF)...

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