# Tagged Questions

**1**

vote

**2**answers

227 views

### A question from the proof of affine algebraic group is a linear

In (some version of) the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", ...

**7**

votes

**2**answers

365 views

### Quotients by the additive group $\mathbb G_a$

Geometric invariant theory doesn't work so well for non-reductive groups, since invariant rings are not generally finitely generated. However, in many cases the action of a non-reductive group has a ...

**5**

votes

**2**answers

240 views

### Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables

Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function
$P_i = px^i + qy^i + rz^i$
where $x,y,z$ are coordinates. I have a few related ...

**5**

votes

**2**answers

190 views

### Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices

I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...

**3**

votes

**4**answers

349 views

### Reference for an algebraic group preserving a cubic form

Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...

**0**

votes

**0**answers

121 views

### Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...

**7**

votes

**2**answers

377 views

### Equivariant normalization?

Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...

**2**

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**0**answers

54 views

### different and discriminant for finite invariants

Let $k$ be an algebraically closed field.
Let $B$ a $k$-algebra of finite type, normal and Cohen-macaulay. Let $G$ a finite group acting on $B$. We assume that the order of $G$ is prime to the ...

**3**

votes

**1**answer

274 views

### quotient by finite group actions that are smooth

Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero.
Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$.
We assume that ...

**3**

votes

**0**answers

139 views

### determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...

**2**

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**0**answers

201 views

### How to determine there exists a unique invariant subspace for a set of matrices

Hi everyone,
Ive been looking at the following problem, but its not entirely in my area and some potential solutions seem to rely on algebraic geometry. Maybe thats just a complicated way to solve ...

**4**

votes

**3**answers

180 views

### Generate a higher degree symmetric polynomial from an existing one

Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as
$q(x_1, x_2, \cdots, x_{n+1}) = ...

**4**

votes

**2**answers

295 views

### Quotient of affine space by cyclic permutation

The quotient of the affine space $\mathbb{A}^n$ by the symmetric group $Sym_n$ is again an affine space of the same dimension, and invariants are given by elementary symmetric polynomials.
What ...

**1**

vote

**1**answer

227 views

### What does this particular geometric quotient locally look like?

Let $k$ be a field and consider the algebraic group $GL_n$ over $Spec(k)$. It has as a closed (but not normal) algebraic subgroup the group $M$ of monomial matrices, i.e. matrices having exactly one ...

**3**

votes

**2**answers

213 views

### Intersection theory for $G$-varieties - an action on the chow ring?

Let $G$ be a reductive algebraic group. Let $X$ be a $G$-variety and consider any closed subvariety $Z$ of $X$. Since any $g\in G$ acts as an automorphism, we know that $g.Z$ is again a closed ...

**6**

votes

**3**answers

308 views

### Invariants of group action: SL_n acts simultaneously on m symmetric matrices

Let $\rm{SL}_n$ be the special linear group and let $\rm{Sym}_n$ be the set of all symmetric matrices of size n. $\rm{SL}_n$ acts on $(\rm{Sym}_n)^m$ by $g(A_1, \ldots , A_m)=(gA_1 g^{\rm T}, \ldots , ...

**5**

votes

**1**answer

621 views

### Is $k[X]^G$ integral closed in $k[X]$.

May assume field $k=\mathbb{C}$.
Let $X$ be an affine variety and $G$ be a reductive group (may assume connected).
Is the ring of invariants $k[X]^G$ integral closed in $k[X]$?
The claim may not ...

**2**

votes

**1**answer

194 views

### When the affine quotient is faithfully flat?

It may be easy for the expert.
Consider the map from $n$ by $m$ matrices (over $\mathbb{C}$ )to the $n$ by $n$ symmetric matrices $\phi\colon A\mapsto A A^T$.
My question is when this map is ...

**4**

votes

**0**answers

235 views

### Group action on Grassmannian: Intersection of two special invariant rings

Let $K$ be a field with characteristic $0$.
Let $G:=G(d,nd)$ the Grassmannian of all $d-$dimensional subspaces of $K^{nd}$ and let $H:=O_d(K)^n$ the n-fold direct product of the orthogonal group. $H$ ...

**7**

votes

**3**answers

480 views

### Why can I divide an affine variety by the action of the general linear group?

Let $G\subseteq\mathrm{Gl}_n(\mathbb{C})$ be a subgroup of the general linear group and assume that $\rho:G\to\mathrm{Gl}(V)$ is a representation. Understand the complex vector space $V$ as an affine ...

**10**

votes

**5**answers

1k views

### area of triangle from coefficients of its cubic?

Three points $z_1$, $z_2$, $z_3$ on the complex plane are given by the coefficients $a_k$'s of the cubic polynomial $f(z)=(z-z_1)(z-z_2)(z-z_3)=\sum_{k=0}^3 a_k z^k$. How does one express the ...

**7**

votes

**1**answer

314 views

### When two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$ are the same?

Let us consider two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$, where $G$ and $G'$ are finite subgroups of $\mathrm{GL}(n,\mathbb{C})$ acting linearly.
It is easy too see, that a different ...

**2**

votes

**1**answer

168 views

### Dimension of spaces of invariants/tableaux functions

The Hook lenght formula gives the number of standard Young tableaux on a given diagram.
A variant gives the number of semistandard tableuax.
Does there exist a formula for counting "weighted ...

**3**

votes

**0**answers

176 views

### Invariant Subvarieties of Variety of Quiver Representations

I'd like to understand a special case of the following rather general algebraic geometry question:
Given an algebraic group $G$ acting on a variety $V$, can we describe the $G$-invariant subvarieties ...

**3**

votes

**2**answers

248 views

### How to compute the ring of invariants of SO_3(k) acting on a polynomial ring

Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.
${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in ...

**10**

votes

**1**answer

504 views

### When Are Quotients Complete Intersections?

Let $S_{n}$ denote the permutation group on $n$ letters and $G\subset S_{n}$ a transitive subgroup. The inclusion of $G$ in $S_{n}$ defines an action of $G$ on $\mathbb{C}^{n}$. By finding a ...

**8**

votes

**8**answers

2k views

### Resources on Invariant Theory

Hi,
So my question is pretty much summed up by the summary - basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd ...