# 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.

384
questions

3
votes

0
answers

88
views

### Invariants for the Weyl group of $\mathrm{SO}_{2n}$ acting on a certain group scheme

Let $W$ denote the Weyl group of $\mathrm{SO}_{2n}$, so $W = (\mathbf{Z}/2)^{n-1} \rtimes \Sigma_n$. There is a natural action of $W$ on $\mathbf{Z}^n$ given by permutations and even numbers of sign ...

0
votes

1
answer

103
views

### Functions on products of tori

Let $T$ be an algebraic torus over an algebraically closed field $k$.
Let $d\in\mathbb{N}^{*}$. For every $d$-tuple of integers $\underline{n}=(n_1,\dotsc, n_d)$ and a function $f\in k[T]$, we can ...

3
votes

0
answers

217
views

### On Noether's Problem

Noether's problem is a famous problem in invariant theory, introduced in the 1910's by Emmy Noether in relation to the inverse Galois problem. It is as follows:
Noether's Problem: Let $F=k(x_1,\dotsc,...

1
vote

0
answers

94
views

### Software for computing invariant rings

I have an linearly reductive algebraic group $G$ acting regularly on an affine variety $X$(over an algebraically closed field of characteristic 0). I want to compute the invariant ring $\mathbb{K}[X]^{...

1
vote

0
answers

77
views

### Germs of holomorphic functions and invariant functions

Consider a complex vector space $V \cong {\mathbb C}^n$. Consider the ring of germs of holomorphic functions ${\mathcal O}_0 (V)$ at $0\in V$. We know that this ring is Noetherian.
Now consider a ...

6
votes

0
answers

316
views

### $f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured.
$$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$
or variations of it like the weaker
$$f(f(f(f(z)))) = z ,\...

4
votes

1
answer

152
views

### Frobenius series for the $S_n$-module $\mathbb{Q}[X]$

I'm reposting this question, by recommendation of a moderator.
I'm reading Haiman's article titled Conjectures on the quotient ring by diagonal invariants. In what follows, all vector spaces and ...

2
votes

0
answers

40
views

### Modular invariants of special linear groups acting on exterior powers

In general it can be very difficult to compute the invariants of a group acting on a module which is not a vector space over an infinite field. I am interested in the following example, motivated by ...

7
votes

3
answers

1k
views

### Has anyone researched additive analogues of toric geometry in characteristic zero?

One definition of an $ n $-dimensional toric variety is that it is a variety $ Z $ for which there exists an equivariant embedding of
$ \mathbb{G}_{m}^{n} $ as a Zariski dense, open sub-variety of $ Z ...

1
vote

0
answers

75
views

### When is $Y$ not an orbit closure?

Let $G$ be a linearly reductive algebraic group acting regularly on an affine space over $\mathbb{A}^n$ an algebraically closed field $\mathbb{K}$. Let $Y$ be a $G$-invariant (closed) affine ...

2
votes

0
answers

37
views

### Multiplicative invariants of non-reduced root systems

It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...

1
vote

0
answers

202
views

### Is $\langle\chi,\lambda\rangle=0$, whenever the limit exists? Where is the mistake?

Suppose $G$ is a linearly reductive algebraic group acting linearly on a finite dimensional vector space $V$ over $\mathbb{C}$. This induces an action on the coordinate ring $\mathbb{C}[V]$ (see here)....

2
votes

0
answers

101
views

### Getting an equivariant morphism

Let $X\subset\mathbb{A}^n$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristic zero. Suppose we have two linearly reductive algebraic groups $G$, $G'$ ...

1
vote

1
answer

131
views

### Invariant ring of the subvariety

Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $...

4
votes

1
answer

132
views

### Epstein zeta function of Barnes-Wall and related lattices

Sarnak and Strömbergsson studied Epstein zeta function $\zeta(L,s)=\sum\limits_{0\neq v\in L}\langle v,v\rangle^{-s}$ of a number of highly symmetric lattices in their Inventiones Math. paper.
In ...

1
vote

0
answers

49
views

### Is the Schofield semi invariant defined at $V/IV$?

Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...

1
vote

0
answers

118
views

### How to determine if an invariant rational function is defined at the $\theta$-polystable point

Background:
Let $A$ be a finite-dimensional (associative and unital) algebra over $\mathbb{C}$. Assume there is a quiver $Q=(Q_0,Q_1)$, where $Q_0$ is the set of vertices and $Q_1$ is the set of ...

6
votes

0
answers

112
views

### What are Burnside's "fixed systems" in modern language?

I just read Chapter 17, on rational invariants, in Burnside's classic 1911 text Theory of Groups of Finite Order. It was a great read, and mostly it was straightforward to translate the ideas into ...

2
votes

0
answers

43
views

### Different invariants of group actions from isomorphic subgroups

Consider $D_8,$ the dihedral group of order $8$, acting on the unit square $X=[0,1]^2 \subseteq \mathbb{R}^2$ in the natural way– essentially take the unique linear extension of the action on the ...

7
votes

0
answers

142
views

### Ring of invariants for graph automorphism

$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite simple graph with nodes numbered $1$ to $n$. Attach variables $x_1, ..., x_n$ to nodes. The graph automorphism group $\Aut G$ acts on nodes by ...

2
votes

1
answer

190
views

### Invariants of the group algebra of a finite group

Consider a finite group $G$ and its complex group algebra $V_G$, on which $G$ acts. I would like to know: what are the polynomial $G$-invariants of $V_G$ i.e., the polynomial functions $p\in \mathbb{C}...

2
votes

1
answer

200
views

### What is the Molien series of the SO(2)-invariant ring on the plane (sometimes written C[X]^{SO(2)} )?

Let SO(2) be the group of rotations in the plane. What is the Molien series (sometimes called the Hilbert-Poincare series) of the SO(2)-invariant ring of polynomials?
N.B. The main goal being to ...

3
votes

0
answers

34
views

### Different generating sets for conjugation invariants of several matrices

There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...

2
votes

1
answer

556
views

### What goes wrong with this proof of Dirichlet's Theorem?

I am curious what goes wrong with this invariant theoretic argument of Dirichlet's Theorem of primes in arithmetic progression. I know that something goes wrong, but I am really curious about what ...

1
vote

0
answers

72
views

### Codimension of cusp singularities in the space of 2-jets

In trying to prove Cerf's theorem about homotopies between Morse-functions I ended up thinking about the following problem.
For $n>2$, $a= (a_{i,j})\in GL(n-2)$, we define the polynomial map $C_a:\...

3
votes

0
answers

130
views

### Class of finite quotient affine space in Grothendieck ring of varieties

Let $G$ be a finite group acting linearly on affine space $\mathbb{A}^n_k$ over $k = \mathbb{C}$. Since the action is linear, it can be extended to an action of $\operatorname{GL}_n(\mathbb{C})$, ...

0
votes

0
answers

43
views

### Question regarding properties of map which produces measures that are invariant to orthogonal rotation

Let $\mathcal{M}_1$ denote the set of probability measures on the unit ball in $\mathbb{R}^d$ (which comes with its Borel $\sigma$-field). Denote by $\sigma$ the uniform measure on the orthogonal ...

2
votes

1
answer

74
views

### conjugacy in adjoint representation

Let $G$ be an adjoint algebraic group over $\mathbb{C}$, $\mathfrak{g}$ its Lie algebra.
Let $\rho:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let $g,g'\in G$ be two semisimple ...

3
votes

1
answer

212
views

### Invariants of general linear groups under torus action

Let $G=GL_n$ be the general linear group (let's say over an algebraically closed field of char $=0$). Let's denote as $T$ the torus of diagonal matrices: is there an explicit description of the ...

7
votes

1
answer

178
views

### Invariants for the isotropy representation of a Riemannian symmetric space

Statement: Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $...

8
votes

1
answer

199
views

### Coordinates on $N_+ \backslash \overline{B_+ w B_+} / N_+$

Let $G = \text{GL}_n(\mathbb{C})$ and let $N_+$ be the subgroup of upper triangular matrices with $1$'s on the diagonal. Let $w$ be a permutation, let $B_+ w B_+$ be the Bruhat cell and let $\overline{...

2
votes

0
answers

90
views

### Invariants of Lie superalgebras

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...

2
votes

0
answers

161
views

### Understanding the proof of a theorem by Van Den Bergh

I'm trying to understand the proof of a theorem by Van Den Bergh, which is Proposition 6 in the paper Bessenrodt, Christine and Lieven Le Bruyn. “Stable rationality of certain PGLn-quotients.” ...

1
vote

1
answer

161
views

### Is there a permutation invariant for graphs?

Let $V = (v_1,...,v_n)$ be a set of vertices for a directed graph. We denote an edge between the two vertices $v_i$ and $v_j$ by $(i,j)$. The edge set $E$ for a graph $(V,E)$ is then a set of tuples $...

3
votes

0
answers

159
views

### Stability of nodal hypersurfaces

We denote by $\Pi_{n,d}$ the space of homogeneous polynomials of degree $d$ in $n+1$ variables $x_0,\ldots,x_n$, i.e. $\Pi_{n,d}=\Gamma(\mathbb{P}^n(\mathbb{C}),\mathcal{O}(d))$. The group $G=SL(n+1)$ ...

3
votes

1
answer

363
views

### Why is the largest invariant set the following?

Consider this paper:
Hai-Feng Huo and Li-Xiang Feng, "Global stability for an HIV/AIDS epidemic model with different latent stages and treatment", Applied Mathematical Modelling, Volume 37, ...

3
votes

0
answers

151
views

### Nef cone of a GIT quotient

I want to know how to calculate nef cone of a GIT quotient. In particular let $X$ be a projective variety and $L$ be an ample line bundle on $X$ and $G$ be a reductive algebraic group and chosen a $G$ ...

11
votes

2
answers

579
views

### Invariants of $\mathrm{GL}_n$ representations

$\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the ...

4
votes

0
answers

121
views

### Polynomial invariants of infinite reflection groups

It is a famous theorem of Shepard-Todd that the ring of invariant polynomials of a finite complex reflection group $W$ acting on a complex vector space $V$ is actually itself a polynomial ring. In ...

1
vote

1
answer

488
views

### Polynomial invariant — from product formula to monomial expansion

Context
This question deals with the polynomial invariant denoted by $ H_{n} $ in Maksym Fedorchuk and Igor Pak's 2004 paper Rigidity and polynomial invariants of convex polytopes (sections 7.6 and 9)....

1
vote

0
answers

77
views

### Ring structure of coinvariant of $W(U(4))$

I want to know the ring structure of the coinvariant of $W(U(4))$, where $W(G)$ is the Weyl group of G.
I know that the ring structure of the coinvariant of $W(U(3))$ is $\mathbb{Z}[x_1,x_2,x_3]$ with ...

2
votes

0
answers

69
views

### A question about fundamental invariants in the context of neural networks

I'm reading in depth the first part of the following paper: https://arxiv.org/pdf/1804.10306.pdf, paying specific attention to the following result, that I re-write here for the sake of convenience:
[...

1
vote

1
answer

85
views

### A question about finding a system of invariants for a subgroup $H$ of the symmetric group $S_n$

If $H = S_n$ then then the fundamental symmetric polynomials allow to write any $S_n$-invariant polynomial $f$ as a polynomial expression of these elementary symmetric functions. In other words, $\...

4
votes

0
answers

104
views

### Finding special form for integral binary cubic form

Let $f(x,y)=px^3+3qx^2y+3rxy^2+sy^3$, $p,q,r,s\in\mathbb{Z}$, be an integral binary cubic form. Under what conditions is $f$ equivalent to a form $g(x,y)=t x^3+3u x^2y+3v xy^2+w y^3$ with $u=v$?
Here ...

7
votes

1
answer

182
views

### $2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve

Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let ...

6
votes

0
answers

90
views

### $S_n$-invariant polynomials on the dual of reflection representation

Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...

9
votes

2
answers

769
views

### Character variety of the free group

A classical result of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi_\mathbb{C}$ associated to the free group $F_2$ and the algebraic group $\mathrm{SL}_2(\mathbb{C})$...

2
votes

0
answers

66
views

### Periodic functions on groups

I'll give you some context: We can say that a (classic) periodic function $f:\mathbb{R}\rightarrow\mathbb{R}$ with period $\omega$ is an invariant function under the cyclic subgroup $G=\langle\omega\...

2
votes

1
answer

173
views

### Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring

Question 1. Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is ...

5
votes

0
answers

341
views

### What representation theoretic properties does the semi-invariant ring tell us?

I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)?
I have been studying about semi-...