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.
441 questions
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Injectivity of derivations from the middle transvectant in the free Lie algebra on $\operatorname{Sym}^m$ for $\mathrm{SL}_2$
Let $G=\mathrm{SL}_2(\mathbb C)$, $V$ its standard representation, and $V_m=\operatorname{Sym}^m(V)$ with $m\equiv 2 \pmod 4$. It is classical that
$$
\Lambda^2 V_m \;\cong\; \bigoplus_{\substack{1\le ...
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Rewriting a quaternary cubic as sums of $5$ cubes of linear forms
This question was first asked here but got no answer.
This paper by R. Garver talks about removing 4 terms from the 9th degree equation. Although everything is easy to understand, there was an ...
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Invariance of averages of cubic forms
This is a follow-up discussion to a previous question, posted as a separate question following the suggestion of Stanley Yao Xiao.
The question is about a particular passage in Bhargava-Shankar-...
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Integrals of cubic forms around the unit circle
Given a nondegenerate real cubic form $f(x,y)$ in two variables, consider the integral
$$I(f) = \frac{|\mathrm{Disc}(f)|^{1/6}}{2\pi} \int_0^{2\pi} f(\cos(\theta),\sin(\theta))^{-2/3} \, d\theta$$
The ...
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Exploit realization of Lie super algebra
It is known in the context of Clifford analysis $\mathbb{R}_m$ that the operator $\underline{x}$ and $\overline\partial$ generate the Lie superalgebra $\mathfrak{osc}(1\vert2)$, where $\underline{x}=...
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Invariants of (complex) vectors under simultaneous $\mathrm{SO}_n$ action?
$\DeclareMathOperator\SO{SO}$Consider the natural action of $G=\SO_n$ on $\mathbb{C}^n$. What are the polynomial invariants of vectors under simultaneous rotations by $\SO_n$, i.e. which $P(v_1,v_2,\...
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Sufficient conditions to prove invariance of a state of a $C^*$ algebra under a group $G$ of $*$-automorphisms
This is a follow-up question of a previous question that I asked on Math Stack Exchange: Doubt on invariant states and asymptotic abelianess, where it is observed that having a state $\omega$ over a $...
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Invariants of a maximal unipotent group in GL(n) acting by conjugation on n by n matrices
Let $G = \operatorname{GL}(n, \mathbb{C})$ and let $U \subset G$ be the usual maximal unipotent subgroup (upper triangular matrices with ones on the diagonal). Let $U$ act by conjugation on the space $...
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Fully determined set of equations in coefficients of an unknown polynomial with given discriminant
A polynomial $f \in \mathbb{Z}[x]$ of degree $d$, in general has $d+1$ coefficients. Suppose I know $f$ has discriminant $N \in \mathbb{Z}$. That is one (nonlinear) equation in $d+1$ unknowns using ...
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Doubt on invariant states on C*-algebras
I am reading "Operator algebras and quantum statistical mechanics" by Bratteli and Robinson, and I have one doubt regarding the definition of invariant states under a group $G$ of $*$ - ...
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Commuting variety and invariant polynomials
Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{sl}_n\mathbb{C}$, and let $S_n$ denote the Weyl group of $(\mathfrak{sl}_n\mathbb{C}, \mathfrak h)$. I am interested in understanding the ...
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Is Noether's Problem undecidable?
I begin by recalling Noether's problem over $\mathbb{Q}$:
Let $G$ be a finite group that act faithfully by field automorphisms on $\mathbb{Q}(x_1,\ldots,x_n)$, with the action on $\mathbb{Q}$ trivial. ...
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A noncommutative analogue of GIT quotient by a finite group
Let $\Lambda$ be a commutative Noetherian ring, and an affine algebra over a certain base field $k$. Let $G$ be a finite group of automorphisms of $\Lambda$, and let $X$ be the affine scheme $\...
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Existence of quasi-invariant smooth function with eigencharacter for reductive Lie group
What is known about existence of quasi-invariant smooth function with some eigencharacter on Lie algebra of a reductive Lie group? Consider reductive Lie group $G$ and its Lie algebra $\mathfrak{g}$. ...
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Generators and relations for invariants of the Weyl algebra
Let $k$ be an algebraically closed field of zero characteristic, and $A_n=k\langle x_1, \ldots, x_n, \partial_1, \ldots, \partial_n \rangle$ the rank n Weyl algebra, which can also be described as a ...
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A question about the invariance of maximal commutative Poisson algebra of the standard Poisson algebra
Let $k$ be a base field of characteristic 0, and algebraically closed, for the sake of simplicity. By the standard Poisson algebra, denoted $\mathcal{P}_n(k)$, we mean the polynomial algebra in the ...
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Some clarifications about Noether's Problem
This questions is mainly a question in history of mathematics, and it is a question because I am not skilled enough in german nor have the access to the relevant papers right now.
Question 1: What was ...
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In which paper Noether proved her famous theorem in invariant theory?
I looked at mathscinet, zbmath and many books on invariant theory, but I am having some difficulties to find out precisely in which paper(s) Emmy Noether proved the following theorem: If $A$ is a ...
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Invariants of tuples of matrices under $\mathrm{GL}(p)\otimes \mathrm{GL}(q) \subseteq \mathrm{GL}(n)$?
$\DeclareMathOperator\GL{GL}$Consider $\GL(n)$ over some field of characteristic zero (I'm thinking of either the rationals, reals or complexes) and the subgroup $\GL(p)\otimes \GL(q)$ which embeds ...
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Translation Invariants of Polynomials
The function $f(k)$ is a (numerical) polynomial in $\mathbb{Q}[k]$, and the set
$ S_f = \{ f_d : d \in \mathbb{N} \} $
is a set associated with $ f $, where $f_d(k)=f(k+d)$.
I am interested in finding ...
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Invariant theory for unitary groups $\mathcal{U}(n)$
I'm trying to understand the invariant theory of the unitary groups $\mathcal{U}(n)$ on tensor powers of their standard representations $V^{\otimes p} \otimes (V^*)^{\otimes q}$. Let $\mathcal{U}(n)$ ...
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Orbit space of the action of $\mathrm{GL}(V)$ on the Grassmannian of $V\wedge V$
$
\newcommand{\K}{\mathbb{K}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\Grass}{Grass}
$Consider $\K\in\{\R,...
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Eigenfunctions of the Laplace–Beltrami operator on the coadjoint orbit of $\mathfrak{su}(n)$
$\DeclareMathOperator\SU{SU}$For $\mathfrak{su}(2,\mathbb{C})$, the generic coadjoint orbit is $\mathbb{S}^2$, and the Laplace–Beltrami operator on it is given by
$$
\Delta \equiv \frac{1}{\sin\theta} ...
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Example of non homogenous manifold with a finitely generated algebra of natural functions
Let $(M,g)$ be a Riemannian manifold. Let $C^{\infty}_{Nat}(M,g)$ be the $\mathbb{R}$-algebra of scalar invariants of the curvature tensor and all its higher covariant derivatives. An example of a ...
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Diagonal analogue of symmetric functions
Let $n$ be a positive integer and consider the ring $R$ of power series over $\mathbb{Q}$ in commuting variables $x_1,y_1,x_2,y_2,...$. Let the symmetric group $\mathfrak{S}$ of permutations of the ...
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Stabilizers of the action of Levi on abelianization of nilpotent radical
$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
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Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
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Invariant theory (first fundamental theorem) for a direct sum of two fundamental representations
Let $G$ be a simple reductive group over $\mathbb C$, e.g. $G=\mathrm{SO}(V)$ is a special orthogonal group.
Let $W_1$ and $W_2$ be two irreducible representations of $G$. Assume both $W_i$ are ...
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Orbit spaces of n-tuples of square matrices under simultaneous conjugation
Let $n, p, \geq 1$ be integers. Denote the set of ordered partitions of $p$ by $\Pi$: each $\pi \in \Pi$ is an ordered $k$-tuple $(p_1,p_2, \dotsc, p_k)$ where $p_1+\dotsb+p_k = p$. Write $\pi \leq \...
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Polynomial from degrees of Weyl group
Let $d_1, \dotsc d_n$ be the degrees (of fundamental invariants) of the Weyl group $W$ of a simple Lie group, (in the reflection representation; see table given on the Wikipedia page for their ...
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What is the state of invariant theory?
I have often heard that Hilbert killed invariant theory, but I see that there computational invariant theory seems to be an active field, and I understand that geometric invariant theory arose from ...
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Crepant resolution of cyclic quotient of affine space
Let $ G $ be a cyclic group of order $ n $, acting on $ \mathbb{C}^n $ by the cyclic action $ (z_1, z_2, \ldots, z_n) \rightarrow (z_2, z_3, \ldots, z_1) $. Does the quotient $ \mathbb{C}^n / G $ (...
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Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
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Tensor product of Brauer algebra modules
Let $V,W$ be two modules for the Brauer algebra $B_n(\delta)$.
Is it known in general how one can regard $V\otimes W$ as a module for the Brauer algebra? That is, is there any analog of a Hopf ...
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Invariant subgroups for integer matrix groups
The representation theory of finite groups over the fields $\mathbb{C}$ or $\mathbb{Q}$ is clear. I am wondering what happens if we consider something similar over the group $\mathbb{Z}$. To be more ...
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A variant of Schwarz's theorem on generators of smooth $G$-invariant functions
Let $V$ be a finite dimensional euclidean space and let $G\subset O(V)$ be a finite (or compact) group, let $\mathbb{R}[V]$ be the algebra of polynomial functions on $V$. If $E\subset \mathcal{C}^\...
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Matrix invariants for simultaneous conjugation by a finite subgroup of $\textrm{GL}_n$
Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ ...
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Moduli space of curves away from singular subsets
Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities.
More specifically, I'm interested in ...
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What are the possible symmetry groups of n-point constructions in the projective plane?
Let $k$ be an infinite field, perhaps take $k = \mathbb{C}$ if it simplifies matters.
I will be asking a question about $\mathbb{P}^2$ for definiteness and to simplify definitions/notations, but feel ...
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Noether's Problem and the Inverse Problem on Galois Theory
For the sake of simplicity, assume the base field $k$ as having zero characteristic. I will discuss 4 different formulations of Noether's Problem.
version 1 - original Noether's problem: Let $G<S_n$...
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When is a finitely generated commutative algebra a projective module over its invariant subalgebra?
For the sake of simplicity, I will work over the complex numbers.
Let $A$ be a finitely generated algebra and $G$ any finite group of algebra automorphisms. Then, by Noether's Theorem, $A^G$ is also a ...
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Expositions of symplectic reflection groups
We will work over $\mathbb{C}$.
Remember that a finite subgroup $G$ of $\operatorname{GL}_n(\mathbb{C})$ is called a complex reflection group if it is generated by complex reflections $r$, which are ...
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A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
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Tangent space of a GIT quotient of $GL_{N}$
Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or ...
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Are the two notions of free $\mathbb{G}_a$-actions equivalent?
Consider a finitely generated integral $\mathbb{C}$-domain $B$. An algebraic $\mathbb{G}_a$-action on $X:=\mathrm{Spec}(\mathcal{O}(X))$ is equivalent to a locally nilpotent $\mathbb{C}$-derivation
$$\...
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Can the Weyl algebra be free over its invariant subalgebra?
Let $k$ be an algebraically closed field of zero characteristic, let $P_n$ denote the polynomial algebra in $n$ indeterminates, and let $G$ be a finite group of linear automorphisms. Then, by ...
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List of equivalent conditions for the invariant subalgebra to be polynomial
Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...
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Confusion regarding the invariant rational functions
I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below)
It says that "every invariant rational function can ...
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Does every faithful action on a scheme act freely on a dense open subset?
Disclaimer: I have asked this question on math exchange a week ago (here), but sadly to no avail. So I decided to escalate my question:
Let $G$ be a finite group acting faithfully on a smooth quasi-...
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Automorphism group of the first Weyl field
A related question is this one (Automorphism group of the quantum Weyl field).
Let $A_1$ denote the rank 1 Weyl algebra (over the complex numbers), and $D_1$ its skew field of fractions, called the ...
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