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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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Polynomial ring a module over ring of invariants

Let $\mathbf{k}$ be a field of characteristic 0 and $S=\mathbf{k}[X_1,\cdots,X_n]$ be a polynomial algebra over $\mathbf{k}$. Let $G\subset GL_n(\mathbf{k})$ act linearly on $S$ and $R = S^G$ be the ...
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Example of integral extension of degree $p$?

For a prime $p$, does there exist an example of a (pair of) domains (if yes any way to construct?) $R$ and $S$ such that Both $R$ and $S$ are normal complete local domains $S$ contains an alg. ...
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Elementary symmetric polynomials proof

In the book Polynomial Invariants of Finite Groups by Larry Smith, he proved the algebraic independence of elementary symmetric polynomials as follows: Suppose $g(e_1,…,e_n) =0$ where $ g $ is not ...
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Group action in Polynomial invariant

The following is just a basic definition in Invariant Theory, which I copied from wikipedia "Let $G$ be a group, and ${\displaystyle V}$ a finite-dimensional vector space over a field ${\displaystyle ...
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Why do we only consider homogeneous invariants?

Let $G$ be a finite matrix group, $G \subseteq GL_{n}(\mathbb{C})$. Consider the polynomial ring in $n$ variables; $\mathbb{C}[x_1,...,x_n]$. It is known that the ivnariant subring $\mathbb{C}[x_1,.....
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Dimension of the invariant subspace

Let $\Gamma \subseteq GL_{n}(\mathbb{C})$ be a finite matrix group. Let this finite matrix group act on $f(x_1,...,x_n) \in \mathbb{C}[x_1,...,x_n]$ like so: $$\Gamma \cdot f(x_1,...,x_n) = f(\Gamma \...
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Calculating the Hilbert Series for symmetric polynomials

Let $S = \mathbb{C}[x_1,...,x_n]$ be the polynomial ring in $n$ variables, $S_d \subset S$ the subspace of homogeneous polynomials of degree $d$, i.e., the polynomials with the property \begin{align} ...
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Constructing quaternions - proof that square of each imaginary unit is -1

During construction of vector space of quaternions over real numbers I encountered a problem that I can't quite put my finger on. For the context: Hamilton a multiplication in plane that keeps ...
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There are no $\operatorname{Sp}$-invariant linear forms

Let $V, W$ be finite dimensional $\mathbb{Q}_\ell$ vector spaces, with $\dim W = 1 $, $G$ be a profinite group which acts on $V, W$, $\psi : V \times V \to W $ a bilinear slew-symmetric perfect $G$-...
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Basic Question About Integral With Respect to Invariant Measures

Suppose that $(X,\Sigma)$ is a measurable space, $T$ is a measurable map from $(X,\Sigma)$ to itself, and $\mu$ is a $T$-invariant measure. Define the dynamical system $$ f_n\triangleq T\circ f_{n-1} ...
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Prove that every $G$-invariant subspace of $V \otimes U$ has the form $V \otimes U_{0}$.

Knowing the following theorem: Theorem 4. Let $T$ be an irreducible complex representation of the group $G$ in the space $V$, and let $I$ be the trivial representation of $G$ in the space $U$. Then ...
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Prove that the image of an invariant subspace under a morphism of representations is an invariant subspace.

Prove that the image of an invariant subspace under a morphism of representations is an invariant subspace. Could anyone give me a hint on how to solve this Please? Knowing the following: And ...
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Proof for a $n \times m$ checkerboard tiling

I believe it is a basic problem, but I would like some help proving this statement: Prove that a $n\times m$ checkerboard can be filled with $k\times 1$ tiles if and only if k divides m or n.
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Finding all subspaces invariant under F.(2)

I have a question regarding the question in the link here: Finding all subspaces invariant under F. The answer is so convincing in the above link but I just want to know why the matrix corresponding ...
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Symmetric group action on polynomial ring

Let the symmetric group $S_4$ act on $\mathbb R[x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4]$ by permuting the 1st $4$ variables and again permuting the last $4$ variables. We can restrict the action to the ...
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Definition of invariant ideal and invariant set of points?

I am reading a book about Invariant theory. It is said there the ideal $I$ is invariant action of a group of $n\times n$ matrices. Which element of $I$ are polynomials in $n$ variables. and also said ...
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Finding all subspaces invariant under F.

The question and its answer are given below: But I do not know why is this the answer, could anyone explain this for me please?
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When is $S^G \subset \text{Frac}(R)$.

Let $S/R$ be an extension of rings where $R$ is a domain, and let $G$ be a finite group acting on $S$, fixing $R$. When do we have $S^G \subset \text{Frac}(R)$? For instance, if $S$ is a domain, $G$ ...
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When is $S^G [G] \cong S$ as a $S^G[G]$-module?

Let $S$ be a ring and take a finite group $G$ acting on $S$. Put $R = S^G$, the fixed subring. When is $S$ a free rank-$1$ $R[G]$-module? That is, when do we have $S \cong R[G]$ as $R[G]$-modules? I ...
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Is this representation completely reducible?

Is these representations completely reducible? Definition: A linear representation is said to be completely reducible if every invariant subspace has an invariant complement. But I have no idea how ...
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A difficulty in understanding an example in Vinberg.

The example is given below: But I have difficulties in understanding the following: 1- why $V_{0}$ is called $(n-1)-$dimensional subspace, I want a concrete example please? 2- Why if the ...
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Any quotient representation of completely reducible is completely reducible. [closed]

Prove that every quotient representation of a completely reducible representation is completely reducible. Could anyone give me a hint for this?
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For two invariant complements in the space of representation $T$. Prove that $T_{W_1}$ isomorphic to$ T_{W_2}$

Let $W_{1}$ and $W_{2}$ be two invariant complements of the invariant space $U$ in the space of the representation $T$ prove that $T_{W_{1}}$ equivalent to $T_{W_{2}}$. Invariant Complement ...
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For $U$ a $G-$invariant subspace of $V$ a $G-$representation, show that $T_g(U)=U$

Prove that if the subspace $U$ of the space of the representation $T:G\to GL(V)$ of $G$ is invariant, then $T(g)U = U$ for all $g \in G.$ Could any one give me a hint how to solve this question ?
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A difficulty in understanding the solution of #2 section 1 Vinberg.

The question and its answer is given below: But I do not know how I should think to find all the invariant subspaces and why the answer is as mentioned above, could anyone explain this for me please? ...
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What is a discrete invariant?

I don´t understand the definition of discrete invariant and I wonder if someone of you would know it. The notion appears in the following sense: Given a set $M$ and equivalence relation $\sim$ on $M$...
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Determining maximal Cohen-Macaulay modules over an invariant ring

Suppose that $G$ is a finite small (i.e. reflection-free) subgroup of $\text{GL}(n,\mathbb{C})$ acting on $S := \mathbb{C}[x_1, \dots, x_n]$. Set $R := S^G$. By 5.20 Corollary of this, the maximal ...
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A new graph invariant? The maximum number of non-equivalent colorings with $n$ colors.

Consider (proper) coloring of a finite graph $G$ with exactly $n$ colors and the following coloring transformation: choose an edge of the graph with the end nodes of colors $a$ and $b$ and swap the ...
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How should one find complete sets of invariants for vectors whose elements can be permuted by a given group?

Disclaimer / Introduction I am a physicist by training who hasn't taken courses in invariant theory. I hope my description of my question that doesn't mis-use the terminology in `invariant theory' --...
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Are there mixed-symmetry, primitive invariant tensors for simple Lie algebras?

I am interested in $\mathfrak{g}$-invariant tensors for a simple Lie algebra $\mathfrak{g}$. That is, in tensors $\kappa_{i_1\dots i_n}$ such that $$ \sum\limits_{s=1}^m f^\rho_{\nu i_s} \kappa_{i_1\...
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How to determinate the convergence the start and the finish points?

I have found the Gridpatterns page. One can apply the next algoritm and obtaine the grid pattern "1-2-3". On square grid paper start in the middle. Draw a line 1-unit long. Turn a right angle ...
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Chevalley-Shephard-Todd theorem

I'm studying Chevalley-Shephard-Todd theorem, in the version that states : let $G \subset GL(V)$ a finite group, where $V$ is a finite dimensional complex space. Let $S=S(V^*)$ indicates the symmetric ...
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Differential invariants of ODEs

I have been reading 'Symmetry and Integration Methods for Differential Equations' by Bluman and Anco. I'm trying to make sense of differential invariants and ODEs... It is confusing. There is an ODE ...
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Rotational invariance of the Lebesgue measure for a unit sphere for any metric.

Let $\rho(\cdot, \cdot)$ be any distance metric on the set $\mathbb{R}^{N+1}$ and let $S^N = \{\mathbf{x} \in \mathbb{R}^{N+1}: \rho(\mathbf{0},\mathbf{x}) = 1\}$ be the unit N-sphere. For a fixed $...
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If $W$ is T-invariant, $W=(W\cap W_{1})\oplus\cdots\oplus (W\cap W_{k})$

Let $T$ an operator over a $\mathbb{F}$ vector space $\mathbb{V}$, with $\dim(\mathbb{V})<\infty$. Let $p=p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}$ the minimal polynomial of $T$, and $\mathbb{V}=W_{1}\...
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Which polynomials in the minors of a matrix are invariant under conjugation?

Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to V$, when $\dim V=n$. Unlike the determinant, the $k$-minors ...
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Does every continuous conjugation-invariant function factor through polynomials in the eigenvalues?

Let $n>2$ and consider the space of $n \times n$ real matrices $M_n(\mathbb{R})$. Let $f:M_n(\mathbb{R}) \to \mathbb{R}$ be a smooth conjugation-invariant function. Let $P_i:M_n(\mathbb{R}) \to \...
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Invariant homogeneous polynomial on quaternions

Let $\mathbb{H}$ denote the quaternions. If $(w_1,\ldots,w_n)\in \mathbb{H}^n$ we can write $w_i=z_i+jz_{n+i}$ with complex numbers $z_1,\ldots,z_{2n}$. Now let $M$ be the group of all matrices of the ...
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Can we characterize the equivalence classes of matrices up to left multiplication by an orthogonal matrix?

Let $M_n$ be the space of $n \times n$ real matrices, and consider the following equivalence relation on $M_n$: $A \sim B$ if there exist $Q \in O(n)$ such that $A=QB$. Can we characterise nicely ...
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About 3-dimensional quadratic space

3.3 Theorem. Assume that every $3$-dimensional quadratic space over $K$ is isotropic. Let $\phi$ be a regular $n$-dimensional quadratic space. Then $$ \phi \cong \langle \delta, 1, \...
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About invariant of quadratic space

The following excerpt is from Scharlau: Quadratic and Hermitian Forms, pages 35-36. One associates "in-variants" with the quadratic space so that the space is determined by its invariants as ...
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Find maximal permutation group $G$ such that a polynomial is $G$-invariant

I don't know if this is a trivial question. But because I lack some background I would need advise or a reference. I have an $n$-variate polynomial over $\mathbb Q$, say $f$, and I am interested in ...
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Does the sum of all tensor invariants have any meaning or application?

I have come across a formula with multiple tensors. After some modification, I ended up with the sum of all the invariants of one of the tensors. E.g., if it was a tensor $A$ with $3\times 3$ matrix ...
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General relativity formulation

I already started a thread about tensors invariance but this question which seems the same is actually about something else. I have read quite a few books where the author (I cannot find what books ...
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Realizing subring as ring of invariants?

Let $R$ be a finite type integral domain over field $k$, $S$ be a subring of $R$, such that $[f.f(R)\colon f.f(S)]=d$, does there exist a group $G$ with $|G|=d$ such that $S=R^G$? (Here $f.f(R)$ means ...
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Shortcut to finding invariant subspaces

I have looked at the following example: $M = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{bmatrix}$ and found $ker(M-1Id)=\langle \begin{bmatrix} 1 \\ 0 \\ 0 ...
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What is the primary decomposition of $Z/70Z \times Z/100Z \times Z/49Z \times Z/40Z$? Its invariant elements?

Is the following the right way to do? Let $G = Z/70Z \times Z/100Z \times Z/49Z \times Z/40Z$ Then its primary decomposition yields to : $$Z/(7 * 2* 5)Z \times Z/(5^2 * 2^2)Z \times Z/(7^2)Z \times ...
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What are the invariant factors of $G = (\Bbb Z/12\Bbb Z) \times (\Bbb Z/25 \Bbb Z) \times (\Bbb Z/45\Bbb Z)$?

What are the invariant factors of $G = (\Bbb Z/12\Bbb Z) \times (\Bbb Z/25 \Bbb Z)\times(\Bbb Z/45\Bbb Z)$ ? My way of solving it: $$G \simeq (\Bbb Z/2^2\Bbb Z \times \Bbb Z/3\Bbb Z)\times(\Bbb Z/5^2\...
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Decomposition of a matrix $C \in M_{q\times p}$

Consider the map $$\Phi:M_{n \times p} \times M_{q \times n} \to M_{q \times p}$$ $$(A,B) \mapsto BA$$ $GL_n(V)$ acts on $M_{n \times p} \times M_{q \times n}$ by $$g(A,B)=(gA,Bg^{-1})$$ Hence $\Phi$ ...
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Identifying group structure of matrix group

I am working on classifying the stabilisers of quadratic binomials in $GL(\mathbb{C}^{n})$ but struggling to identify the groups which are appearing. One example is the binomial $$x_{1}x_{2}-x_{1}x_{...