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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Group action dimension of orbits

I am working on the following problem right now: Let $V,W,H$ be finite dimensional vector spaces. I have a group action $Gl(V)\times Gl(W) \curvearrowright Hom(V\otimes H,W)$ in the obvious way i.e. ...
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Steenrod Operations an algebraic Approach

Assume that $q=p^{r}$, where $p$ is a prime either 2 or odd and $\mathbb{F}_{q}$ is a Galois field and $V$ a finite dimensional $\mathbb{F}_{q}$-vector space. Then due to Larry Smith in this http://...
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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 ...
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Invariant Problem on colored chips

In a course we were once given the following question There is a finite stack of chips on a table, each chip having one of three different colors $a,b$ and $c$ . At any time, you may choose two ...
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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,\\...
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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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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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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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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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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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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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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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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 ...