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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3 votes
1 answer
274 views

$\DeclareMathOperator\SU{SU}$$\SU(2)\times \SU(2)$ invariant $\SU(3)$-structure on $\{t\} \times M^6$

$\DeclareMathOperator\SU{SU}$I am reading Jason Lotay and Goncalo Oliveira's paper $\SU(2)^2$ invariant $G_2$-instantons, and have a few questions from the same. If we consider the space $M = S^3 \...
10 votes
1 answer
655 views

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, ...
2 votes
0 answers
72 views

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 $^\...
4 votes
1 answer
485 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) — ...
8 votes
1 answer
354 views

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 ...
0 votes
0 answers
77 views

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 ...
16 votes
0 answers
533 views

Does a symplectic group act on a tensor power of a spin representation?

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$More specifically, let $S_k$ be the spin representation of $\Spin(2k+1)$. Then is there are action of $\Sp(2r-2)$ on $\bigotimes^{2r}S_k$ ...
8 votes
1 answer
233 views

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 ...
10 votes
0 answers
279 views

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 $K$-...
5 votes
1 answer
194 views

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 ...
1 vote
0 answers
83 views

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 $...
11 votes
2 answers
587 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 $R_p=S^p R$;...
31 votes
7 answers
5k views

Invariant polynomials under a group action (hidden GIT)

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$). Now the symmetric group $\mathfrak{S}_n$ ...
3 votes
0 answers
59 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 ...
4 votes
1 answer
263 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 ...
8 votes
1 answer
247 views

$\operatorname{SL}_2(k)$ 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 ...
2 votes
0 answers
111 views

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 $G$-...
2 votes
0 answers
100 views

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 ...
4 votes
0 answers
112 views

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 ...
48 votes
4 answers
5k views

How to constructively/combinatorially prove Schur-Weyl duality?

How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring $\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\...
1 vote
0 answers
56 views

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 ...
0 votes
0 answers
124 views

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 ...
11 votes
2 answers
497 views

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 ...
4 votes
1 answer
454 views

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 ...
7 votes
2 answers
391 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^* ...
14 votes
1 answer
681 views

Donaldson and DT invariants

Let $X$ be a smooth projective surface. Then, using the compactified moduli space of anti self-dual connections or torsion free sheaves we can construct Donaldson invariants of $X$. Similarly, one can ...
7 votes
1 answer
290 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 \...
2 votes
0 answers
144 views

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 ...
3 votes
1 answer
152 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 ...
1 vote
0 answers
182 views

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 ...
6 votes
0 answers
213 views

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,.....
22 votes
2 answers
1k views

Why is the catalecticant invariant under coordinate changes?

Let $\mathbf{k}$ be a commutative $\mathbb{Q}$-algebra. (We could play the same game over any commutative ring $\mathbf{k}$, but this would be a bit more technical, so let me avoid it.) Fix a ...
1 vote
0 answers
123 views

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 $M$ ...
1 vote
0 answers
62 views

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, ...
3 votes
0 answers
94 views

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 $...
12 votes
1 answer
486 views

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. ...
6 votes
1 answer
222 views

Transcendent basis for the field of multisymmetric functions

It is known that the field of multisymmetric rational functions (over a field of characteristic $0$), that is, rational function in variables $x_{11}, \ldots, x_{1m}, \ldots, x_{n1}, \ldots, x_{nm}$ ...
0 votes
1 answer
129 views

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 ...
4 votes
1 answer
105 views

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 ...
17 votes
3 answers
1k views

Noether's bound for anticommutative invariant theory (diff. forms instead of polynomials)?

EDIT: Now with a concrete request to CAS experts (see the end of the post). Let $G$ be a finite group, and $V$ a finite-dimensional representation of $G$. The classical invariant theory of $G$ and $V$...
8 votes
1 answer
233 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 ...
18 votes
1 answer
3k views

Why is Mumford's GIT-quotient so effective?

According to remark 6.14 in Shigeru Mukai's An introduction to invariants and moduli (unfortunately, the page is not available on Google Books, so I explain it below), the GIT-quotient of an affine ...
4 votes
1 answer
334 views

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, ...
2 votes
1 answer
266 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 $U$. ...
5 votes
2 answers
481 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 ...
10 votes
1 answer
375 views

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....
10 votes
1 answer
360 views

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 ...
7 votes
2 answers
657 views

Variants and Generalizations of Arf (-Brown-Kervaire) invariants

(1) I encounter the Arf invariants in Kirby-Taylor, Pin structures on low-dimensional manifolds. The form that I looked at was: $$ S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} ...
2 votes
0 answers
173 views

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 ...
2 votes
0 answers
86 views

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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