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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Problem with Molien's formula for covariants

If $E$ and $H$ are finite-dimensional faithful representations (over $\mathbb{C}$) of a finite group $G$, with $H$ irreducible. The Molien formula describer the Poincaré series of the covariants as $$ ...
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Polynomials with $S_n \times \mathbb{Z}_2$ symmetry

Suppose that a polynomial $p(x_1\ldots x_n, y_1\ldots y_n)$ in $2n$ variables is invariant under the following operations: 1) $p(x_1\ldots x_n, y_1\ldots y_n)=p(y_1\ldots y_n, x_1\ldots x_n)$ 2) ...
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Primary and Secondary invariants for finite groups

For a finite group G and complex representation V of degree n, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants ...
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Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite ...
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Almost-invariant polynomials under dihedral group action

Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$. I'd like to ...
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A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n $

(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...
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Does Nagata theorem hold in a field that is not algebraically closed?

Let $R$ be a finitely generated $k$ - algebra and $G$ be a reductive group acting rationally on it. Then a theorem of Nagata says that the invariant ring $R^G$ is also finitely generated. Here $k$ ...
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Creating a linear transformation $T$ such that $T$ is invariant in a proper subspace but not normal

Let $V$ be a finite-dimensional complex inner product space and let $W$ be a subspace of $V$ such that $\dim V = 4$, $\dim W = 2$. Construct a linear operator $T$ such that (i) $W$ is $T$-invariant ...
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Equivalent conditions, reductive groups

I read the book Invariant Theory by T.A. Springer. There is the following definition of a reductive group: Definition. A linear algebraic group $G$ is called reductive if for any rational ...
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How t0 show $U+W$ and $U\cap W$ are invariant under $T$ if $U$ and $W$ are invariant under $T$?

Suppose $T:V\to V$ is a linear transformation, how t0 show $U+W$ and $U\cap W$ are invariant under $T$ if $U$ and $W$ are invariant under $T$ ? My try: $1.$ Let $u\in U,w\in W$. Then ...
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Integer matrix invariant factors

Have question about finding invariant factors of integer matrix: $\begin{pmatrix} 6 & 2 \\ -2 & 6 \end{pmatrix}$ Was sick during lectures, and not completely that what I do is right. What I ...
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Existence of categorical quotient $X/\mathbb{G}_{m,A}$.

Let $A$ be an $\bar{\mathbb{F}}_p$-Algebra of finite type (one might assume $A$ to be reduced). Let $X \subset \mathbb{A}_A^d\backslash \{0\}$ be a closed $A$-subscheme together with a group action of ...
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How to show that $S(U)$ is invariant under $T$ based on the following assumption?

Let $S,T:V\to V$ be a linear transformation such that $ST=TS$. If $U$ is subspace of $V$ invariant under $T$, show that $S(U)$ is invariant under $T.$ If $U$ is subspace of $V$ invariant under $T$, ...
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How to show that if $W\subset V$ is invariant under $T:V\to V$ and $\dim{W}=1$, then $W$ is spanned by an eigenvector for $T$?

$W$ is invariant if $T(W)\subset W$, meaning the result can be expressed in terms of vectors in $W$. But I don't understand how this can be related to eigenvector? Could someone help?
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Can the quotient by a nonabelian group yield an abelian singularity?

Let $V$ be a complex vector space with a faithful linear action of a finite group $G$. Viewing $V$ as affine space (with coordinate ring $\mathbb{C}[V]$), the quotient $V/G$ is the affine variety with ...
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Finding the defining equations for a simple quotient variety

First of all let me note that I have no experience at all with modern algebraic geometry so if at all possible I would appreciate an answer not involving the concept of a scheme. I have however some ...
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Independence of a binary form and its Hessian

Let $f\equiv f(X,Y):=\sum_{i=0}^{n}a_iX^{n-i}Y^i$ be a binary form of degree $n\geq3$ with coefficients over $\mathbb{C}$ and no repeated roots in $\mathbb{C}^2$ (up to scaling). The Hessian of $f$ is ...
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Normalisation of an invariant measure

Is there an example of a Markov chain with invariant measure $\pi$ and $\sum_{i \in I}\pi(i) = \infty$ that can be normalised so that we can consider an invariant distribution instead? This is a ...
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31 views

Invariant factors. Prove that $q_i|q_{i+1}$

How could I show that $q_i|q_{i+1}$ without using the theorem of Smith Normal Form ? $q_i$ is the invariant factor. It defines how $q_i = f_i/f_{i-1}$, being $f_i$ the determinantal divisor. I ...
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Lie group action and Lie algebra action.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $r \in g \otimes g$ and $b \in g$. Consider the adjoint action $g \times g \otimes g \to g \otimes g $ given by $(b, x, y) \mapsto b.(x \otimes y) = ...
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Difference between invariant and contractive sets

I came across this particular notion of contractive sets. I know what an invariant set is, but can anyone explain what a contractive set is and the difference between invariant and contractive sets?
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What are the invariants of a number field? [closed]

How is defined an invariant of a field? Given a certain field extension $L/K$, is it related with the Galois group ${\rm Gal}(L/K)$? In the case of number fields, which are the invariants associated ...
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Number of eigenvalues for this operator

Say I have a F - vector space V and a subspace U given by U={va : a is in F}. Now suppose I have an operator defined by $Tv=av$. Clearly, U is invariant under T, since for any element of U, say bv, I ...
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Does Magma let you specify primary invariants?

I am cross-posting this question from scicomp.SE. The computer algebra system Magma can calculate primary invariants (i.e. a homogeneous system of parameters) in an invariant ring of a finite group ...
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Producing a $G$-invariant form from the standard Hermitian product using the averaging process

Problem statement: Let $G$ be a cyclic group of order $3$. The matrix $$A=\begin{bmatrix} -1 && -1 \\ 1 && 0 \end{bmatrix}$$ has order $3$, so it defines a matrix representation of ...
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Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
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45 views

Invariants of a real symmetric matrix

Problem. I have a real symmetric $n \times n$ matrix $A$ and would like to compute a set of real numbers $f(A) = (x_1, \ldots, x_m) \in \mathbb R^m$ which are invariant under multiplication of $A$ ...
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Matrix representation of a 6-dimensional Lie algebra

The question is about the matrix representation of the following 6-dimensional Lie algebra, with 6 generators $t_1,t_2,t_3,t_4,t_5,t_6$. This Lie algebra is nilpotent, non-abelian, non-reductive and ...
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What does it take to have a precise definition of volume?

Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of ...
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If $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, find $t$ operators such that $T=T_1\oplus\dots\oplus T_t$

Be $T\in \mathscr{L}(V)$ a linear operator with characteristic polynomial $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, $n_i\geq 1$ and $\lambda_i\neq\lambda_j$ if $i\neq j$. Show that $T$ ...
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Canonical forms for tensors of type (2,1)

Are there any canonical forms for tensors of type (2,1)? Such a tensor can be defined as a bi-linear map $$ T:V \times V \rightarrow V,$$ for $V$ a finitely dimensional real vector space.
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Is there an invariant similar to the characteristic polynomial for (0,2) and (2,0) tensors?

The characteristic polynomial of a matrix - a (1,1) tensor - is its invariant (independent on basis transformation). Is there a similar invariant for (0,2) and (2,0) tensors? The characteristic ...
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G-invariant polynomials for a concrete G

The collection of matrices$$ \left( \begin{array}{lll} a^2 & 2ab &b^2 \\ ac & bc+ad & bd \\ c^2 & 2cd & d^2 \end{array} \right)$$ indexed by $a,b,c,d \in \mathbb{R}$ is a ...
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4D representation of finite group and projection operator

Suppose I have a finite group, say $C_2$, and its Cartesian representation $ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ ...
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Equivalent definition of almost geometric quotient

I am trying to prove the following lemma - Lemma - Let $X$ be a variety and let $G$ be an algebraic group acting algebraically on $X$. Let $\pi:X\rightarrow X//G$ be a good categorical quotient. Then ...
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Geometric Invariants of a conic section

There are three independent invariants for every conic section, viz., $$[I_1,I_2,I_3]= [ (a + b + c), (a b -h^2), Det(( a,h,g), (h,b.f), (g,f,c) )] $$ How are they related to the known geometric ...
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Calculating the ring of invariants for the action of $\mathbb C^*$ on $\mathbb C^2\setminus \{0\}$

Let $\mathbb{C}^*=\mathbb C\setminus\{0\}$ act on $\mathbb C^2\setminus \{0\}$ by scalar multiplication, where $\mathbb C^2=\operatorname{Spec}(\mathbb C[x_0,x_1])$. Then $\mathbb C^2\setminus ...
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Given $\pi:X\rightarrow Y$ how to show $X$ is irreducible (resp. normal) $\Rightarrow$ $Y$ is irreducible(resp. normal)?

Let $G$ act on the affine variety $X=\operatorname{Spec}(R)$ such that $R^G$ is a finitely generated $\mathbb C$ - algebrs and let $\pi:X\rightarrow Y=\operatorname{Spec}(R^G)$ be the morphism of ...
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Invariant ring for $S_5$ [closed]

For the standard representation of $S_5$, the ring of invariants is generated by the elementary symmetric polynomials and hence it is a polynomial ring. Now if we take the tensor product of standard ...
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Ring of Invariants of symmetric group

The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
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What is known about rational points on the ideal of relations / syzygy ideal?

What is known about rational points on the ideal of relations / syzygy ideal? Let $G$ be a finite group, with $|G|=n$. Then $G$ acts on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation ...
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equivalent definition of a good categorical quotient?

Working solely with varieties, as you may know, a pair ($Y$, $\pi$) is a "good categorical quotient" for the $G$-variety $X$ if: 1) $\pi$ is surjective and constant along orbits 2) for any open $U ...
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Is a polynomial which is invariant in the roots of some separable polynomial also invariant in the usual sense?

Let $\alpha_1,\cdots,\alpha_n \in \mathbb{C}$ be the roots of a separable polynomial with rational coefficients. Let $K := \mathbb{Q}(\alpha_1,\cdots,\alpha_n)$. Then the field extension ...
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Does the invariant ring determine the group?

Let $G$ be a finite group $n = |G|$. Let $\sigma : G \rightarrow GL(n,\mathbb{C})$ be the regular representation. Hence every element of $G$ can be seen as a permutation matrix. Let $I_G := ...
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What is a trivial linear operator?

The question asks to find a non-trivial linear operator T to make a subspace T-invariant. I'm thinking of $T(x)=2x$, since $2x$ clearly stays in the subspace by scalar multiplication closure, but not ...
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$\mathbb C[X_1, \ldots, X_n]$ is a free module over $\mathbb C[X_1, \ldots, X_n]^G$

Let $G$ be finite subgroup of $GL_n( \mathbb C )$. Let $\mathbb C[X_1, \ldots, X_n]^G$ be the set of all G-invariant polynomials of $\mathbb C[X_1, \ldots, X_n]$. Is there any rule by which we can ...
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Description of the algebra of $G$-invariant polynomials by generators and relations

Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = ...
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Invariants of finite groups

Let $G$ be a finite group acting linearly on $\mathbb{C}^n$ and $\mathbb{C}[X]^G$ be the ring of invariant polynomials. If $G$ is a group generated by reflections, this ring is generated by $n$ ...
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Equivariant polynomial maps and gradients of invariants

Let $G$ be a finite group with a linear action on $\mathbb{C}^n$ and $f\in\mathbb{C}[X_1,\ldots,X_n]$ be invariant. Then the gradient of $f$ gives rise to a polynomial map $\phi$ from $\mathbb{C}^n$ ...
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Is there a plane algebraic curve with just 3-fold rotational symmetry, but without reflection symmetry?

I am new to the subject of invariant theory, but the Reynolds operator popped up so I tried to calculated some examples for myself. I computed the invariant polynomials under the cylic group $C_3$ of ...