# Questions tagged [invariant-theory]

Invariant theory deals with an algebraic, geometric or analytic structure $X$, submitted 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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### A good stratification of a variety on which an algebraic group acts

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0
(a reduced separated scheme of finite type over $k$).
Let $G$ be a connected linear algebraic group over $k$ (...

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### Projective invariants of the plane and cross ratio

I am looking for a reference for the following admittedly imprecise statement:
Any projective invariant of n points in the projective plane may be
expressed as a function of well-chosen cross-ratios.
...

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### Scalar triple curvature product in $\mathbb R^4$

For a classical 3d conicoid with coefficients $(a,b,c,f,g,h)$ the first two invariants are $I_1= (a+b)/2, I_2= (a b -h^2) $ and the third one is
$$I_3= \begin{pmatrix} a & h & g \\ h & b &...

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### Ring of invariants for $n$-tuples of Lie algebras

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\Tr{Tr}$Consider the diagonal action of $\GL(n,\mathbb{C})$ on the variety of $k$-tuples of matrices, $\M_{n\times n}(\mathbb{C}...

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### Calculating the polynomials which are invariant under the action of a simple finite group

Let $G$ be a simple, finite group. In general, $G$ is not abelian.
Let $\rho$ be a representation of this group, where each $\rho(g)$ for $g\in G$ is a unitary, complex, $d$-dimensional matrix, $\rho(...

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161 views

### Invariant ideal generated by invariant elements

Let $G$ be a complex reductive group acting linearly on $\mathbb{C}^n$ and let $X$ be a $G$-invariant closed subvariety of $\mathbb{C}^n$. Is $X$ the zero-set of finitely many $G$-invariant functions?
...

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### Basis of invariant tensors of rank n in three dimensions

[This is a question motivated by theoretical physics, so apologies if the language is rough...]
In three dimensions the spaces of invariant (or isotropic) tensors of rank $n$ have dimensions 1, 0, 1, ...

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### Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind

Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension,
and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$.
Then $^\...

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329 views

### Do all orbits have the same dimension?

Well, I've already asked this question at math.SE — but no-one's answered or commented. So now I'm posting it here (it's about the research paper — I think that it isn't an off-topic for this forum) — ...

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### Invariant ring of $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ under $\textrm{SO}(4)$

Consider the representation of $\textrm{SO}(4)$ on $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ induced by the standard representation of $\textrm{SO}(4)$ on $\mathbb{R}^4$. I am interested in the ring of ...

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### On four non-cocyclic integral points on ellipse

Let $(x_i,y_i)\in\mathbb Z^2$ at $i\in\{1,2,3,4\}$ be four (not five and I assume an unique curve exists because of Diophantine constraints and not geometric constraints) non-cocyclic integral points ...

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### Decomposition of $\bigotimes^{m} \mathbb{C}^{n}$ under the action of $\operatorname{GL}_{n}\times \operatorname{S}_{m}$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\S{S}$I want to know the proof of the following theorem. It is stated somewhere that, a proof can be found in: "Roger Howe, Perspectives on ...

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### Quotients by algebraic group actions at the level of the Grothendieck ring

$\DeclareMathOperator\SGro{SGro}\DeclareMathOperator\Gro{Gro}\DeclareMathOperator\GL{GL}$For an algebraically closed field $K$, the Grothendieck semiring of $K$ consists of, say, quasi-projective $K$-...

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### Coordinate-free description of an alternating trilinear form on pure octonions

Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$.
The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$,
and ...

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### invariant theory for $G\times \text{O}(n)$ [closed]

We know that the invariants of the orthogonal group $\text {O}(n)$ gives us the Brauer algebra. Is there any known results for the invariants of $G\times \text{O}(n)$ acting on the tensor space where $...

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566 views

### To describe an invariant trivector in dimension 8 geometrically

$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$.
For an integer $p\ge 0$, write $R_p=S^p R$;...

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49 views

### Kernel of the map $\mathbb{C}[G]^U \to \mathbb{C}[U^+]$

$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $G=\SL_k$ be the special linear group, $U$ the unipotent subgroup consisting of all lower unipotent triangular matrices, $U^+$ the unipotent subgroup ...

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### $\operatorname{SL}_2(k)$ invariant polynomials in $k[x_1,x_2,y_1,y_2]$

Let $k$ be a field and let $\operatorname{SL}_2(k)$ act on $k[x_1,x_2]$ and $k[y_1,y_2]$ in the usual ways. These actions induce an action on the tensor product $k[x_1,x_2,y_1,y_2]$ that preserves ...

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243 views

### Nilpotent orbits in representations of exceptional groups

The first nontrivial irreducible representation of $G_2$ is of 7-dimensional, and the first nontrivial representation of $F_4$ is of 26-dimensional.
My question is: how much is known about the ...

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### invariant theory for non-polynomial functions (eg Hilbert spaces)

I am looking for references regarding the study of group invariant functions that are not polynomials. In particular, I have a (nice) group $G$ and I am interested in what can be said about the $G$-...

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### In char zero $ \operatorname{Cox}(\operatorname{Bl}_{[1:1:1]}(\mathbb{P}(a:b:c))) $ is finitely generated, but not in char p. How?

Let $ X(a_{1}:a_{2}:a_{3}) $ be the blow-up of $ \mathbb{P}(a_{1}:a_{2}:a_{3}) $ at $ [1:1:1] $, the identity of the torus. In Steven Dale Cutkosky's paper Symbolic Algebras of Monomial Primes ...

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### Cover by $K$-invariant affine open sets

Let $X$ be a non-singular complex algebraic variety (quasi-projective if necessary) and $K$ a connected compact Lie group acting on $X$ real algebraically, i.e. the action map $K \times X \to X$ is ...

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### GIT and singularities

Let $G$ be a complex reductive group acting on a complex affine variety $X$ and let $X // G = \operatorname{Spec}\mathbb{C}[X]^G$ be the GIT quotient.
Is there a relationship between the singular ...

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### Invariants of a $kG$-module via its comoposition series, when does $M^P \supsetneq N^P$ hold for a $p$-group for $N\subseteq M$ maximal?

Let $G$ be a finite group, $k$ a field, $M$ a $kG$-module, $M^G$ the invariants of $M$ under $G$, $P$ a Sylow $p$-subgroup of $G$ where $p = \text{char}(k)$, $N$ a maximal submodule of $M$ and $S$ the ...

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### How to find the polynomials that define a compact Matrix Lie group from its Lie algebra?

Consider a compact (connected) Lie group, or more generally, a linear algebraic Lie group. Suppose we are given the Lie algebra corresponding to the Lie group. How can we find a set of polynomial ...

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291 views

### What are all invariant polynomials on the space of algebraic curvature tensors?

Let $V = (\mathbb{R}^n, g)$, where $g$ is the Euclidean inner product on $V$. Denote by $G$ the orthogonal group $O(V) = O(n)$ and by $\mathfrak{g}$ the Lie algebra of $G$.
Let $W \subset \Lambda^2V^* ...

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### Invariant theory in universal algebra

Let $\mathcal{L}$ be a finite first-order language with no relation symbols, and $\mathcal{K}:=\mathcal{V}(\Theta)$ a variety in this language definited by a set of identities $\Theta$.
My questions ...

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224 views

### An explicit negative solution to the Lüroth problem for non-algebraically closed fields

Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$.
If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \...

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### An analogue of Noether's Problem for non-rational varieties

For the sake of simplicity, let $\mathsf{k}$ be algebraically closed and of zero characteristic. Varieties are irreducible.
The (linear) Noether's Problem (which goes back to the early 1910's in ...

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117 views

### Behavior of invariants under reduction mod p

Let $R$ be a finitely generated $\mathbb{Z}$-algebra with an [edit: linear algebraic] action of $G(\mathbb{Z})$ where $G$ is a split simply-connected semisimple group.
Then for any prime $p$ we have a ...

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### Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix

Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$.
My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...

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### Continuous version of the fundamental theorem of invariant theory for the orthogonal group

A standard result in the invariant theory of the orthogonal group states the following.
Theorem
Let $(E, \langle .,. \rangle)$ be an n-dimensional euclidean vector space,
let $f : E^m \rightarrow {\bf ...

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### Invariant theory for the orthogonal group and Clifford algebras

The first fundamental theorem of invariant theory for
the orthogonal group $O_n(k)$ asserts that the
ring of invariants is generated by the scalar products:
a polynomial function of $m$ vectors $v_1,.....

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### Bivariate basis functions with span that is closed under rotations in coordinate system

Consider a set of linearly independent functions $\{(x\in\mathbb{R},y\in\mathbb{R})\mapsto f_i(x,y)\in\mathbb{R}\}$ with the property that for any given $\theta\in\mathbb{R}$ and any given $\{a_i\in\...

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### Invariant subspace of a nonlinear map

First please see this very simple fact:
Fact: $\ $ Any linear map $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ has a proper invariant linear subspace.
By an invariant subspace we mean a space $M$ ...

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### propagation of a invariance along some PDE

Consider the following non linear PDE on $\mathbb{R}^n$
$$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$
with given initial condition $u_0(x)$.
Assume that:
$u_0$ is rotation invariant, ...

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### Characterization of all-orthogonal tensors

In the paper [1], it is proven in Theorem 2 that any $n$-tensor $\mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n}$ can be decomposed as
$$
\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n
$$
...

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### Invariant theory of the indefinite orthogonal groups

I believe the following statements are true:
Let $V$ be a finite-dimensional real vector space with a positive-definite inner product $g$. Let $g_{\otimes n}$ denote the natural extension of $g$ to $...

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### Moduli of smooth curves in $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)| $ and their invariants

It is well known that any smooth curve
$C\in |\mathcal{O}_{\mathbf{P}^1\times\mathbf{P}^1}(2,2)| $ has geometric genus equal to 1, so its isomorphism class is determined by its $j$-invariant. ...

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### Consequences of invariant-subspace problem to Li–Yorke chaos [closed]

The invariant-subspace problem is probably an open problem for reflexive spaces which asks:
Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial ...

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### Equivalence classes of a circle of n bits upon flipping 3 consecutive 0s to 1s or vice versa

Consider a circle of n-bits and define the equivalence relation as follow:
Two configurations A and B of the n-bits circle are equivalent if they can be transformed into each other by performing a ...

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### Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable?

I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book
Greco, Silvio, ...

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### Invariants of symmetric forms with respect to the symplectic group

Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...

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194 views

### Highest weight vector as a global section of an affine scheme

Let $G$ be a connected, reductive quasi-split group over a field $k$, acting on an afffine $k$-variety $X$. Let $B = TU$ be a Borel subgroup of $G$ with maximal torus $T$ and unipotent radical $U$. ...

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283 views

### Rosenlicht's theorem and fundamental domain for unipotent group acting on $\mathbb A_k^n$

I have a question about unipotent group actions. I was referred to Rosenlicht's papers, but I had trouble getting much out of these because I don't understand the old algebraic geometry language very ...

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### Lifting $G$-invariants from characteristic $p\gg 0$ to characteristic 0 for a reductive algebraic group $G$

Let $S\subset \mathbb{C}$ be a finitely generated ring, let $R$ be a finitely generated commutative ring over $S$. Let $G$ be a linear algebraic group over $S$, such that $G_{\mathbb{C}}$ is reductive....

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### Invariants for $SO_n \backslash \mathfrak{gl}_n / SO_n$

Is there a nice theorem about the algebra of invariants $\mathbb{C}[\mathfrak{gl}_n]^{SO_n \times SO_n}$, where the action is by left and right multiplication? I'm hoping for something along the lines ...

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### On Shephard-Todd theorem

There is an excellent Torsten Ekedahl's answer to Roman Fedorov's question here: Chevalley–Shephard–Todd theorem. Does anyone know any articles or books where this approach is outlined?
I didn't ...

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### Rational torus invariants

Let $T=(\mathbb{C}^{\times})^n$ be the $n$-dimensional torus acting on the polynomial algebra $\mathbb{C}[x_1,x_2, \ldots,x_n]$ diagonally, i.e.
$$
diag(t^{a_1},t^{a_2},\ldots,t^{a_n})x_i=t^{a_i}x_i, ...

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278 views

### Invariants in the symmetric algebra of a module

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$...