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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Generalizing the symplectic group via a trilinear invariant symbol
The symplectic group $\text{Sp}(2n)$ is defined in terms of the set of $2n$ by $2n$ complex matrices that preserve a bilinear form:
$$\text{Sp}(2n) = \{M \in M_{2n \times 2n}(\mathbb{C}) : M^T \Omega ...
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The invariance of a Lagrangian under a Galilean boost
I have the Schrodinger Lagrangian density for a complex scalar field, given by
$L=i\phi^{*}\frac{\partial \phi}{\partial t}-\frac{1}{2m}(\frac{\partial \phi^*}{\partial x}\frac{\partial \phi}{\partial ...
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Is there a non-trivial max invariant family of distributions?
The Problem Traditionally, the value of a position in a chess engine is computed as the maximum of the values of the subsequent positions. These values are typically represented as a single number. ...
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References on invariant subring for $\mathbb{G}_a\curvearrowright\Bbbk[x_0,\cdots,x_n]$.
Consider in characteristic zero. The $\mathbb{G}_a$-action on $\Bbbk[x_0,\cdots,x_n]$ is induced by a derivation $\partial:\Bbbk[x_0,\cdots,x_n]\to\Bbbk[x_0,\cdots,x_n]$, given by
$$\partial(x_i)=x_{i+...
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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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Specific example of fixed field using Artin's Theorem
I'm attempting a question as follows:
Let $K = \mathbb Q(x,y)$, where $x,y$ are independent transcendentals, and consider the group $G$ of automorphisms generated by
$$\sigma: \quad x \mapsto y,\quad ...
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Prove that a rotationally invariant function depends only on the length and relative angles of its arguments.
I have a function $f(v_{1}, v_{2}): (\mathbb{R}^{3} \times \mathbb{R}^{3}) \rightarrow \mathbb{R}$ which maps two vectors in 3D space to a scalar. I know that this function is invariant under ...
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Order $3$ linear transforms invariating a binary cubic form
Consider $P(x,y)$ a homogenous polynomial of degree $3$ in two variables (a binary cubic). To it we associate first the $2\times 2$ matrix
$$\frac{1}{2}\operatorname{Hess}(P) = \frac{1}{2}\cdot\left( ...
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Weight of a binary form
I'm going through P.J.Olver's book on classical invariant theory. He defines the action of $GL(2)$ on binary forms $Q(x, y)$ by a change of variables:
$$
\bar{x} = \alpha x + \beta y, \ \bar{y} = \...
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Involution on $2\times 2$ matrices
Show that the map on $2\times 2$ matrices
\begin{eqnarray}
\left( \begin{matrix} a & b\\ c & d \end{matrix} \right)\overset{\Phi}{\mapsto} \left( \begin{matrix} a & b\\ c & d \end{...
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Invariants principle Sikinia (Engel problem-solving strategies E4)
I'm studying Engel book and the invariant principle, but the solution of problem E4 blocks me.
E4. In the Parliament of Sikinia, each member has at most three enemies. Prove that the house can be ...
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non-linear surface
In the book "Projective Duality and Homogeneous Spaces" by Evgueni A. Tevelev there is an example (Example 7.5) concerning a non-linear surface but I can not find the definition of it ...
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Non vanishing wedge product of differentials of invariant polynomials
Let $V$ be a $n$-dimensional vector space over $\mathbb{C}$ and let $\mathbb{C}[V]$ be the algebra of polynomial functions on $V$. Let $G$ be a finite subgroup of $\operatorname{GL}(V)$. The natural ...
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What is the significance of the quotient of two Hilbert series?
Just a disclaimer, I am not a mathematician so I am sorry for being less than formal.
Let us say we have a Hilbert series $\mathcal{H}(K[V]^G)$. Now I know that by eliminating some parameters in a ...
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Generic $Q_8$-polynomial in characteristic $2$
Context. Apologies for the long post. As the title suggests, I am trying to compute a generic $Q_8$-polynomial in characteristic two.
Let $F$ be a field, and let $G$ be a finite group. A polynomial $f\...
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Hyperplanes in $SL_2(\mathbb{R})$ containing conjugacy classes of matrices
Let $SL_2(K)$ be the special linear group of rank 2 over a field $K$, as an affine group scheme cut out by the equation $ad - bc = 1$ in $\mathbb{A}^4_K$.
Let $H$ be an arbitrary, homogeneous, ...
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Relating the normal sheaf of $S \subset \operatorname{Sym}^2(S)$ to $\Omega_S$.
$\DeclareMathOperator{\Sym}{Sym}\DeclareMathOperator{\Spec}{Spec}$
Let $S$ be a smooth variety over an algebraically closed field $k$, $\operatorname{char}k \neq 2$. Let $\Sym^2(S) = (S \times_k S) / ...
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Orbit-Stabilizer Theorem for Semi-groups?
Given a point $x \in X$ the set of group G elements $$ G^{x} = \{ g \in G: g.x = x\}$$ is called the stabilizer group of x.
Orbit Stabilizer Theorem for Groups: Each left coset of $G^x$ in G is in one-...
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Invariants of rank 2 tensor's and vectors
I am currently trying to obtain the invariants/integrity basis of a rank two tensor $\mathbf{A}\in\mathbb{R}^{3\times 3}$ and a vector $\mathbf{v}\in\mathbb{R}^3$ under the group of proper orthogonal ...
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What does it mean that a quantity is is constant-invariant?
Suppose that $X$ and $B$ are two random variables that are correlated. Then for any $\alpha$ and $\lambda$ real numbers who belong in the $\mathbb{R}-\{0\}$ holds
$$\tag{1}\mathbb{V}ar(\alpha X-\...
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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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Invariant ring of $\mathrm{SO}_3(\mathbb{Z})$
By classical results in invariant theory, we know the invariant ring $\mathbb{C}[X_1, X_2, X_3]^{\mathrm{SO}_3(\mathbb{C})}$ is equal to $\mathbb{C}[X_1^2 + X_2^2 + X_3^2]$. Here
$\mathbb{C}X_1 \oplus ...
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Frobenius series for the $S_n$-module $k[X]$
I'm reading Haiman's article titled Conjectures on the quotient ring by diagonal invariants (which can be found here). In what follows, all vector spaces and algebras are over the field of rational ...
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How to calculate Delta Invariant of of algebraic curve?
I recently asked a question regarding tangent cones here: Tangent cone of an arbitrary algebraic curve
After doing some reading, I have another question on how to calculate the delta invariant of ...
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Is a finitely generated equivariant module always equivariantly finitely generated?
Let $R$ be a commutative algebra over $\mathbb{C}$. Assume $R$ is noetherian. Let $G$ be a finite group acting on $R$ with the action on $\mathbb{C}$ trivial. Let $M$ be a $G$-equivariant $R$ module, ...
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Polynomial ring of invariants and graphs
Let $G$ be a finite simple graph with nodes enumerated as $1, 2, ..., n$. Assign a variable $x_i$ to node number $i$. Consider the action of $\mathrm{Aut}\, G \subset S_n$ on the polynomial ring
$$\...
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Unitary equivalence of symmetric and homogenous polynomials
Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$...
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Precise statement of Weyl's unitary trick?
Let $\rho_1$, $\rho_2$ be representations of $SL(2)$. Given arbitrary $f:V \to \mathbb{R}$, is it possible to construct an "averaged" function such that for all $g \in SL(2, \mathbb{R})$, $\...
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What are the non-identity symmetries of the determinant in $\mathbb{R}^{n \times n}$?
I am curious if the determinant has any symmetries (i.e. transformational invariants). That is mappings $f: \mathbb{R}^{n \times n} \mapsto \mathbb{R}^{n \times n}$ such that
$$\det (f(X)) = \det (X)$$...
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About running variance in Batch Normalization layer
A batch normalization(BN) layer is normally used to reduce the covariance shit problem in neural networks.
Where in a layer, input $x$ will be normalized to something like $x^{\prime}$ = $\frac{x-\mu(...
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Counterexample to Noether's problem not using heavy machinery
Recall first the famous Noether's problem: let $G$ be a finite group acting linearly on a finite dimensional $F$-vector space $V$.
This induces an action on $F[V]$ and $F(V)$.
Nother's problem: is $F(...
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"Twisted" Invariant Theory
Let $ V $ be a finite dimensional representation $ \pi $ of a group $ G $. For any $ k $, there is a natural representation of $ G $ on
$$
V^{\otimes k}
$$
There is also a natural representation of $...
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When is the ring of functions of a quotient variety a smooth algebra?
Let $V$ be a finite-dimensional complex vector space over an algebraically closed field $k$ of characteristic $0$. Let $k[V]$ be the ring of functions on V. Since $V$ is a smooth (non-singular) ...
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What is the meaning of an integral equation for a generator of a finitely generated algebra $R$ over a sub algebra $R^G$?
I'm working in the following book "Computational invariant theory" second edition by Harm Derksen and Gregor Kemper. The book contains the following theorem on page 72
Let $R$ be a finitely ...
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Quotient of affine space under negation
Suppose we are working over a field $k$. Consider the affine space $k^n$ and the $\mathbb{Z}/(2)$-action on it given by $(x_1,x_2,\dots,x_n) \mapsto (-x_1,-x_2,\dots,-x_n)$. I would like to compute ...
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Where can I find information about sets whose subsets produce unique values?
I'm interested in sets whose subsets produce unique values under a given commutative operation. I don't know what this is called. I tried searching for terms like unique commutative invariant, but ...
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Why does wikipedia define $G$ invariant polynomials this way? [closed]
Why does wikipedia(https://en.wikipedia.org/wiki/Invariant_theory) define it as:
$$(g \cdot f)(x) := f(g^{-1}(x))?$$
Instead of,
$$ (g \cdot f)(x) := f(g(x)).$$
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Invariant rational functions under the group action of some semi-direct product
For my research, I need to compute some field of invariants for a specific subgroup of $GL_2(F)$.
My field $F$ has characteristic different from $2$ and contains a primitive $4th$-root of $1$, ...
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