Tagged Questions

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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Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
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Diffeomorphism-invariant spaces of smooth functions

Let's start with an interesting story. In his celebrated Partial Differential Relations (p. 146), the great Misha Gromov gives a nice exercise of which the following is a (strict) part. Exercise. ...
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Basis for $\Bbb Z[x_1,\cdots,x_n]$ over $\Bbb Z[e_1,\cdots,e_n]$

I'm reading the introductory bits in Procesi's Lie Groups, and on p. 22 we have (paraphrasing) Theorem 2. $\mathcal{B}=\{x_1^{\large h_1}\cdots x_n^{\large h_n}: 0\le h_k\le n-k\}$ is a basis for ...
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Any affine algebraic group is linear.

It is a well-known result that any affine algebraic group is a closed subgroup of some $\mathrm{Gl}_n(\Bbbk)$. However, I would like to see a proof for that, so I looked it up in various books, more ...
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The unique closed orbit in GIT quotient fibers for polynomial actions of Gl

The following reasoning must contain a flaw somewhere because I end up with something absurd, and I cannot figure out where the mistake is. I hope that someone can point it out to me. Let $M$ be the ...
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'Galois Resolvent' and elementary symmetric polynomials in a paper by Noether

In Emmy Noether's 1915 paper "Der Endlichkeitssatz der Invarianten endlicher Gruppen", I saw the notion of a 'Galois resolvent', which I don't quite understand. Google didn't really help me with that, ...
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Schur-Weyl Duality ( Classical ) and the Double Commutant reference request

I would like to ask for any reference suggestions on the topic of Schur-Weyl Duality for GLn ( directly GLn, not through the lie algebra ) and the double commutant theorem. The section on this ...
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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 ...
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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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Finite presentation of algebra of invariants

(1) Let $R$ be a ring, let $A$ be a finitely presented $R$-algebra, and let $G$ be a finite group of $R$-automorphisms of $A$. Is the algebra of invariant $A^G$ finitely presented over $R$? I can ...
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Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
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Polynomial with given group of symmetries

Let $f$ be a polynomial in $n$ variables and $G$ - its group of symmetries (group of permutations of variables wich left $f$ in place). I'm trying to such $f$ for given group $G$. I have troubles when ...
Let $K/k$ be a finitely generated field extension, such that $k=K^G$ for some (possibly infinite) set $G$ of automorphisms of $K$. Now, consider the extension of polynomial rings k[X_1,\ldots,X_n]\... 1answer 38 views 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 = \text{... 1answer 145 views Exercise from Etingof's notes on Representation Theory I am reading through these notes of Etingof on Representation theory and I am stuck with one exercise (it's problem 4.69 in the notes). The problem is the following. Consider the space X=Mat_n(\... 0answers 36 views 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 ... 0answers 42 views 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 ... 0answers 108 views The anti-commutative Molien series Suppose V is a finite dimensional complex vector space and f:V\to V is an automorphism. There is a natural extension \Lambda^\bullet(f):\Lambda^\bullet(V)\to\Lambda^\bullet(V) to the exterior ... 0answers 277 views Basic semi-invariants Let G be a (finite) group and \chi be a linear character corresponding to an irreducible representation. A polynomial f_{\chi} is called semi-invariant (of type \chi) if \sigma\circ f=\chi(\... 2answers 3k views Invariant Factors vs. Elementary Divisors I have been studying Cooperstein's Advanced Linear Algebra for about seven months now and I am having problems understanding how to find the elementary divisors of a linear operator and how to find ... 1answer 129 views Invariant rings \mathbb{C}[X,Y]^{GL_2} and \mathbb{C}[X,Y]^{SL_2} I feel like I might have made a mistake on this question, and would appreciate some feedback from someone more experienced than me. If G acts on S, we write S^G = \{s \in S: gs = s \, \forall \,... 2answers 150 views Is the ring of polynomial invariants of a finite perfect group an UFD? Let G be a finite group. G acts on \mathbb K[x_1,...,x_n] by automorphisms fixing K. \mathbb K[x_1,...,x_n]^G=\{ T\in \mathbb K[x_1,...,x_n],\forall \sigma \in G, T^{\sigma}=T\} is the ring ... 2answers 359 views Quotient of an affine variety by a finite group coincides with topological quotient as a point set? I have just read the construction of the quotient of a closed subset X of affine space by a finite group G of automorphisms of X, in Shafarevich, Basic Algebraic Geometry I. Shafarevich gives ... 1answer 66 views 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 ... 1answer 62 views Permutation groups acting on polynomial rings and base change Let G\subset S_n be a permutation group, and let it act on R = \mathbb{Z}[x_1,\dots,x_n] by permuting the variables, as usual. G acts on R\otimes_\mathbb{Z} S for any unital ring S via its ... 1answer 100 views How does Molien series describe polynomial invariants? As I understood from wiki page, Given a finite group acting on a vector space, Molien series gives a generating function, although I am not sure what this means. And how is this related to the ... 1answer 118 views Hilbert's finiteness theorem over arbitrary fields; reductive groups As a generalization of the finiteness result Hilbert proved in his 1890 paper, one usually formulates the following nowadays: Let G\to\operatorname{GL}(V) be a rational representation of a ... 1answer 103 views Fixed points of polynomial ring homomorphism S=\mathbb R[x+y+z, xy+yz+zx, xyz] is the ring of the symmetric polynomials in \mathbb R[x,y,z]. Let \psi\colon S \to R[x,y,z] be a ring homomorphism such that \begin{align} x &\mapsto -x,\\... 1answer 110 views The disease problem Students are sitting in a n * n grid. There's a disease spreading among them in a particular fashion. At start, there a 'k' students infected(At random). After every time step(equal intervals), the ... 2answers 82 views 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 ... 0answers 22 views 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 ... 0answers 66 views Local Noetherian ring and its invariants Let R be a local Noetherian ring and G a finite group. Is R finitely generated as a module over its invariants R^G? Thank you. 0answers 60 views finitely generated over invariants I was wondering if there are some conditions we can add to the following statement to make it true "The ring R is finitely generated as a module over R^G, where G is a finite group and R^G ... 0answers 56 views Invariants of the symmetric group Let V_\lambda be an irreducible representation of the symmetric group S_n as usual labeled by parition \lambda of n. Question. Is there any general information about the algebra of ... 0answers 51 views Is invariant theory OK in positive characteristic? Let G be a connected reductive group over an algebraically closed field of characteristic p. Let X = Spec(A) be an affine variety over the same field, with an action of G. Are the closed points in ... 0answers 43 views notation for invariation Let \Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\}) and \mathcal{O} the maximal order of some quadratic imaginary number field. I write T[U] := U^* \cdot T \cdot U where U is ... 2answers 110 views 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 ... 1answer 169 views Does anyone know of Invariant Theory enough to comment on this question? I am trying to find a minimal set of invariants for the binary homogenous form\displaystyle ax^7 + bx^{6}y + cx^{5}y^{2} + dx^{4}y^{3} + ex^{3}y^{4} + fx^{2}y^{5} + gxy^{6} + hy^{7}$$What is the ... 2answers 108 views Total dimension of the cohomology of a homogeneous space (or of a graded Tor) I want to calculate the cohomology ring with rational coefficients of a homogeneous space, but would be happy enough to know its total dimension. Let G be a compact Lie group, T a maximal torus, ... 1answer 15 views The true definition of invariant functions on Matrix algebra According to terminologies in "Invariant theory" a true definition for an invariant function f:M_{n}(\mathbb{R})\to \mathbb{R} is the following: Definition 1: A continuous function f is ... 1answer 43 views Calculating the ring of invariants for the action of \mathbb C^* on \mathbb C^2\setminus \{0\} Let \mathbb{C}^*=\mathbb C\setminus\{0\} act on \mathbb C^2\setminus \{0\} by scalar multiplication, where \mathbb C^2=\operatorname{Spec}(\mathbb C[x_0,x_1]). Then \mathbb C^2\setminus \{0\}=... 2answers 66 views Determining ring of invariants of \pm Identity Determine the ring of invariants \mathbb C [x,y,z]^\Gamma for:$$\Gamma :=\{ \begin{pmatrix} \pm1 & 0 & 0 \\ 0 & \pm1 & 0 \\ 0 & 0 & \...
Let $X=\{p_1,p_2,p_3,p_4\}$ be four points in $\mathbb{R}^2$, not three of them on a line. We use the square bracket notation [i,j,k]=\det \begin{pmatrix} p_i & p_j & p_k \\ 1 & 1 & ...