**4**

votes

**2**answers

185 views

### Invariant theory for parabolics

Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \...

**11**

votes

**2**answers

367 views

### Affine GIT is an open map?

Let $k$ be a field, $X= \text{Spec}\,A$ be an affine scheme, with $A$ a finitely generated $k$-algebra. $G=\text{Spec}\,R$ is a linearly reductive group acting rationally on A, i.e. every element of $...

**1**

vote

**1**answer

76 views

### slice theorem for proper actions

I'm trying to understand the slice-theorem for proper Lie-group actions.
Having a smooth manifold $M$ and a Liegroup $G$ acting on $M$ in a proper way, we have the slice theorem, saying that at each ...

**3**

votes

**0**answers

129 views

### Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.
...

**2**

votes

**1**answer

93 views

### Irreducible binary quartic form with prescribed $I$ and $J$-invariants

Let $F(x,y) = ax^4 + bx^3y + cx^2y^2 + dxy^3 + ey^4$ be a binary quartic form with integer coefficients. It is well-known that $F$ has two algebraically independent invariants under the action of $\...

**4**

votes

**1**answer

90 views

### Primary invariants

This question is related to the earlier question which is in the given link:
Primary invariants of a finite group
Let $G$ be a finite group and $V$ a complex representation of degree $n$, and let $...

**4**

votes

**2**answers

201 views

### Invariant polynomials under diagonal action of the orthogonal group

Consider the diagonal action of the orthogonal group $O(n)$ on $\mathbb{R}^n\times\mathbb{R}^n$ defined as: $U\cdot (x,y) = (Ux,Uy)$ for $U\in O(n)$ and $x,y\in\mathbb{R}^n$. I am looking for a ...

**3**

votes

**0**answers

60 views

### Can we express the degree 10 and degree 15 Galois resolvents of sextic binary forms in terms of its basic invariants?

Let $V_6$ denote the 7 dimensional $\mathbb{C}$-vector space of binary sextic forms. For $T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \operatorname{GL}_2(\mathbb{C})$, $T$ ...

**1**

vote

**1**answer

86 views

### Primary invariants of a finite group

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 ...

**4**

votes

**0**answers

48 views

### Is the restriction of a graded automorphism linearizable in characteristic zero?

This question follows up a previous one which was answered by Todd Leason. I want to impose two new requirements on the setup.
Let $k$ be a characteristic zero field. Let $A=k[x_1,\dots,x_n]$ be the ...

**2**

votes

**1**answer

67 views

### Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?

Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...

**2**

votes

**0**answers

77 views

### Dimension of curvature invariants

EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group.
Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...

**2**

votes

**0**answers

136 views

### Smoothness of a (given) global section of a vector bundle over G(2,6)

Let $G=Gr(2,6)$ the Grassmannian of two planes in $V=\mathbb C^6$, and let $\mathcal Q(1)$ the rank four quotient bundle on it twisted with $\mathcal O_G(1) \cong $ det$(S^*)$, $S$ being the ...

**4**

votes

**1**answer

285 views

### How to compute the tangent space of a quotient by a finite group

Let $I\subseteq R:=\mathbb C[x_0,\ldots,x_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my previous question, the permutation group $\mathfrak S_{n+1}$ acts on $R$ by ...

**3**

votes

**2**answers

190 views

### Checking smoothness of the components of a highly symmetric scheme via quotient?

Setting
Let $I\subseteq\mathbb C[x_0,\ldots,x_n]=:S$ be a homogeneous ideal and $X\subseteq\mathbb P^n$ the scheme defined by $I$. Consider the action of the symmetric group $\mathfrak S_{n+1}$ on $S$...

**4**

votes

**0**answers

170 views

### $\text{PGL}_2(\mathbb{Q})$-equivalence versus $\text{PGL}_2(\mathbb{Z})$-equivalence

Let $V_{\mathbb{R}}$ be the space of binary quartic forms with real coefficients (so in particular $V_{\mathbb{R}}$ is a 5-dimensional vector space over $\mathbb{R}$), and define the twisted action of ...

**7**

votes

**1**answer

172 views

### How fine an invariant of a representation is its quotient singularity?

This is a refinement of a question asked on MSE.
Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In ...

**0**

votes

**1**answer

79 views

### full set of invariant functions on manifold

Let $M$ be a smooth manifold and $G$ a Lie group acting properly on $M$. Let $k$ be the codimension of a maximal dimensional $G$-orbit in $M$.
Is it always possible to construct $k$ functions $f_1, \...

**2**

votes

**0**answers

61 views

### Diagonal invariants of $SO(n)$

Consider a Lie algebra $\mathfrak g$ (I am mostly interested in the case $\mathfrak g=so(n)$), its universal enveloping algebra $U$ and its center $C$. There is an adjoint action of $\mathfrak g$ on $...

**4**

votes

**0**answers

167 views

### The density of quartic polynomials whose Galois group is a subgroup of $D_4$

Let $D_4$ denote the (isomorphism class of) dihedral group of 8 elements. For a given binary quartic form $f$ with integer coefficients, define $\text{Gal}(f)$ to be the Galois group of the Galois ...

**9**

votes

**1**answer

245 views

### Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$

Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$.
Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by ...

**0**

votes

**0**answers

66 views

### Hausdorff limits of fibers of affine maps

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and let
$$
F=(P_1,\ldots, P_m):\mathbb{K}^n\to \mathbb{K}^m
$$
be a polynomial map. I would like to know under what conditions the preimages $F^{-1}(y)$ of ...

**3**

votes

**1**answer

153 views

### Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic

$\newcommand{\al}{\alpha}$
Let $M_n$ be the space of $n \times n$ real matrices.
Question:
For which $n$, is there an inner product on $M_n$ which satisfies:
$$(*) \, \, \langle Q^TXQ,Q^TYQ \...

**10**

votes

**1**answer

254 views

### Invariant ring of $S_5$

The irreducible representations of the Symmetric group $S_5$ are classified by the partitions of $5$. For the standard representation which corresponds to the partition (4,1) the ring of invariants is ...

**-1**

votes

**1**answer

131 views

### Tensor bundles as G structures [closed]

For a smooth, real surface $\Sigma$, its bundle of symmetric, bi-linear forms $S^2T\Sigma$ reduced to a $PGL(2,\mathbb{R})$ structure. A similar reduction(with different structure group) can be done ...

**7**

votes

**1**answer

257 views

### Ring of invariants for the regular representation

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 ...

**5**

votes

**1**answer

152 views

### Common zero of invariants of finite groups

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 $\mathbb{Q}[x_1,......

**2**

votes

**0**answers

78 views

### Irreducible real curves on ${\mathbb C}P^1$ invariant under the finite group action

Let $G$ be a finite subgroup of a Möbius group with a standart action on a real algebraic variety ${\mathbb C}{\mathbf P^1}.$
How one can describe $G$-invariant irreducible real algebraic curves?
...

**5**

votes

**1**answer

188 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{...

**6**

votes

**1**answer

122 views

### Generate harmonic polynomials for a finite group

Let $G$ be a finite group acting on a complex vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A ...

**0**

votes

**1**answer

74 views

### Generalization of a Result about degree bounds of invariant rings

A theorem of Knop states that if $G$ is semisimple and connected acting on a vector space $V$ over a field $K$ of characteristic 0, then the degree of the Hilbert series of $K[V]^G$ is less than or ...

**1**

vote

**1**answer

134 views

### Equivariant polynomial maps

Let $V$ be a complex vector spaces and assume that a compact group G acts linearly on $V$. Then look at the $G$-equivariant polynomial maps from $V$ to $V$. Denote this by $Mor_G(V,V)$. In the case ...

**3**

votes

**0**answers

54 views

### Involutions of binary sextic forms

Let $F(x,y) = a_6 x^6 + a_5 x^5 y + \cdots + a_1 xy^5 + a_0 y^6$ be a binary sextic form with complex coefficients. Let $V_\mathbb{C}$ be the space over $\mathbb{C}$ of binary sextic forms. It is ...

**1**

vote

**2**answers

192 views

### Can a general binary sextic form be put into the following form?

Let $F(x,y) = a_6 x^6 + a_5 x^5 y + \cdots + a_0 y^6$ be a binary sextic form with real coefficients and non-zero discriminant. Can one always find an element $U = \begin{pmatrix} u_1 & u_2 \\ u_3 ...

**3**

votes

**0**answers

69 views

### Is there a natural covariant of sextic polynomials with the following coefficients?

Let
$$\displaystyle f(x) = a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 $$
be an irreducible sextic polynomial with integer coefficients. Write $\theta_1, \cdots, \theta_6$ for the ...

**4**

votes

**1**answer

259 views

### Is the Veronese variety “enough” to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...

**7**

votes

**0**answers

172 views

### What is the status of this fifty-year-old conjecture of Kostant?

On page 3.27 of his 1963 thesis on the cohomology of homogeneous spaces as approached through the Eilenberg–Moore spectral sequence, Paul Baum states the following conjecture, which he attributes to B....

**2**

votes

**2**answers

121 views

### Invariant polynomials under the action of $H\le\operatorname{GL}_n(\mathbb{F}_p)$

Let $n$ be a positive integer, and $p$ a prime. Any subgroup $H\le \operatorname{GL}_n(\mathbb{F}_p)$ acts on the polynomial ring $\mathbb{F}_p[x_1,\ldots,x_n]$ via $A\cdot x_i=\sum_j a_{ji}x_j$ for ...

**3**

votes

**1**answer

113 views

### Orbits in the adjoint representation of $SU(2,1)$

How can one describe the orbits of the Lie group $G=\mathrm{SU}(2,1)$ in its Lie algebra $\mathfrak{g}=\mathfrak{su}(2,1)$ with respect to the adjoint representation?

**1**

vote

**0**answers

95 views

### What are the E7(7) invariants in the adjoint representation?

Take a real vector space $R$ transforming in the adjoint representation of
the ${\rm E}_7(7)$ Lie group as $R \rightarrow G R G^{-1}$. One can define
invariants using traces of products of $R$ as ${\...

**1**

vote

**0**answers

145 views

### Does the functor of taking invariants commute with tensor products? [closed]

Suppose that $G$ is a group acting on a commutative ring $R$, inducing an action on each $R$-module. For any $R$-module $M$, let $M^G$ denote the collection of elements of $M$ invariant under the $G$-...

**2**

votes

**1**answer

154 views

### Invariant polynomials with respect to group actions on matrices

Let $\mathfrak{gl}_n(\mathbb{R})$ be the Lie algebra of matrices with real entries and $GL_n(\mathbb{R})$ its associated Lie group. Recall that a linear subgroup $G \subseteq GL_n(\mathbb{R})$ acts by ...

**2**

votes

**1**answer

160 views

### Algebraically independent matrix invariants

Let $V$ be the space of pairs of $n \times n$ matrices over $\mathbb{C}$ and let $G$ be the space of $n \times n$ permutation matrices which acts on $(A,B) \in V$ by simultaneous conjugation. It is ...

**5**

votes

**2**answers

242 views

### Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices

Let $G=GL(n,\mathbb{C})$ and let $U\subset G$ be a maximal unipotent subgroup. (For example,assume that U is the set of upper triangular matrices with ones in the diagonal.) Now let $X=M_{n}(\mathbb{C}...

**6**

votes

**2**answers

643 views

### Is there a topological Chevalley-Shephard-Todd Theorem?

Is the following true:
For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections.
( ...

**2**

votes

**0**answers

31 views

### Terminology for research on distributions of inner products

Consider a set of vectors $M$ from an inner product space $V$. The ordered set of inner products of all pairs of elements in $M$ uniquely characterizes $M$ up to isomorphism.
Suppose now that $V$ is ...

**3**

votes

**1**answer

142 views

### Classification of 3-forms in dimension 7

I'm looking for a classification of $3$-forms over a real vector space of dimension $7$ as for the $3$-forms in dimension $6$. References on the latter case are R. Bryant On the geometry of almost ...

**4**

votes

**1**answer

126 views

### A vector version of the Segre embedding: what is the kernel of the ring map?

TL;DR version.
Given a commutative ring $\mathbf{k}$ and $n+m$ "generic" vectors $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n, \mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_m$ in $\mathbf{k}^k$ ...

**13**

votes

**1**answer

419 views

### Most discriminants are almost squarefree

Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$.
Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...

**0**

votes

**0**answers

45 views

### Invariant subalgebra and dual torus for symmetric group

Given permutation module with three generators and corresponding Galois action of symmetric group $\mathfrak S_3$ I am interested in computing corresponding dual torus $T$ (which should be of ...