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

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 ,\...
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### 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 ...
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### 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 ...
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### 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
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### 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
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\operatorname{GL}_n(\mathbb{C})$, ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### $2$-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 ...
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### 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 ...