Invariant theory deals with an algebraic, geometric or analytic structure X , submited to the action of an (algebraic) group G . It studies G-invariant elements of X as well as the set of G-orbits.

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Stabilizers of pairs of ternary quadratic forms

Let $A,B$ be two ternary quadratic forms with real coefficients, given by symmetric matrices $$\displaystyle 2A = \begin{pmatrix} 2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23}...
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Parametric solutions to a system of equations

Let $s,t$ be two independent real parameters, and let $a_2(s,t), a_1(s,t), a_0(s,t)$ be linear forms in $s, t$ with real coefficients. Put $a_4 = s, a_3 = t$ and consider the quadratic form $$\...
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Tensor and symmetric invariants of Symmetric group

For the action of $S_n$ on $\mathbb C^n$ the elementary symmetric polynomials generate the ring of polynomial invariants. What are the generators for the action of $S_n$ on $\mathbb C^n \otimes \...
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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}, \...
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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 $...
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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 ...
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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 $\mathfrak{g}^*$. ...
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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 $\...
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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 $...
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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 ...
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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 $T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \operatorname{GL}_2(\mathbb{C})$, $T$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $S$...
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$\text{PGL}_2(\mathbb{Q})$-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 ...
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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 ...
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80 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, \...
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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 $...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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,......
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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? ...
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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 = \text{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $n$, ...
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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....
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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 ...
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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?
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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 ${\...
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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 $G$-...
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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 ...
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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 ...
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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}...
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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. ( ...
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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 ...
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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 ...