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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Geometric interpretation of characteristic polynomial
The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
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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\...
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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 $\mathfrak{S}_n$ ...
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How to make the Capelli's identity less mysterious?
The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity
To ...
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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 ...
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Resources on invariant theory
What are resources on invariant theory? Basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd prefer online / freeish ...
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Invariants for the exceptional complex simple Lie algebra $F_4$
This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise.
Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...
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Polynomial invariants of the exceptional Weyl groups
Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
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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 ...
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Explicit invariant of tensors nonvanishing on the diagonal
The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
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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$...
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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$ ...
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Chevalley–Shephard–Todd theorem
The wikipedia article claims that the theorem "was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof". I ...
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Invariants and orbits of $n$-tensors
My question may be absolutely elementary and is probably answered in 19th century. A reference or a short clear argument would be highly appreciated.
Let $V_1, \ldots V_n$ be finite dimensional ...
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If an equivariant map is smooth on diagonal matrices, is it smooth everywhere?
This is a followup from a question I asked on math.SE, which received a helpful answer but unfortunately not a complete one. $\def\Sym{\mathrm{Sym}_{n\times n}}$
$\def\s{\mathrm{Sym}}\def\sp{\s^+}$Let ...
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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 ...
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What goes wrong with this alternate proof of Dirichlet's Theorem?
I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
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Ring of invariants of $\operatorname{SL}_6$ acting on $\Lambda^3 \mathbb C^6$
Let $G=\operatorname{SL}_6$ act on $V=\Lambda^3 \mathbb C^6$. I would like to find the ring of invariants $\mathbb C[V]^G$. There is an obvious invariant
$$Sq: V \to \mathbb C, \quad \omega \mapsto \...
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Diagonal invariants of the symmetric group on $k[X_1,X_2,...,X_n,Y_1,Y_2,...,Y_n]$
This sounds like something that must have been answered long ago, but for some reason I can find nothing on it in the internet. (There has been lots of recent activity in diagonal covariants, related ...
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Most discriminants are almost squarefree
Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$.
Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...
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A question on invariant theory of $GL_n(\mathbb{C})$.
Let $\rho$ denote the irreducible algebraic representation of $GL_n(\mathbb{C})$ with the highest weight $(2,2,\underset{n-2}{\underbrace{0,\dots,0}})$.
Let $k\leq n/2$ be a non-negative integer. ...
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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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Explicit generators for matrix invariants of the symmetric group
Let $V$ be the space of $n$ by $n$ complex matrices with the conjugate action of the symmetric group $G=S_n$. Is any explicit set of generators for the invariant ring $C[V]^G$ known?
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Earliest use of the term "linearly reductive"?
Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of ...
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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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area of triangle from coefficients of its cubic?
Three points $z_1$, $z_2$, $z_3$ on the complex plane are given by the coefficients $a_k$'s of the cubic polynomial $f(z)=(z-z_1)(z-z_2)(z-z_3)=\sum_{k=0}^3 a_k z^k$. How does one express the (signed)...
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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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When Are Quotients Complete Intersections?
Let $S_{n}$ denote the permutation group on $n$ letters and $G\subset S_{n}$ a transitive subgroup. The inclusion of $G$ in $S_{n}$ defines an action of $G$ on $\mathbb{C}^{n}$. By finding a ...
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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 ...
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Representations of $\mathrm{SL}(2)$ in characteristic 2
$\DeclareMathOperator\SL{SL}$In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of $\SL(2)$-modules. In characteristic $p$, things are more complicated.
I am ...
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Invariant ring of $S_5$
The irreducible representations of the Symmetric group $S_5$ are classified by the partitions of $5$. For the standard representation which corresponds to the partition (4,1) the ring of invariants is ...
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Invariant theory in universal algebra
Let $\mathcal{L}$ be a finite first-order language with no relation symbols and let $\mathcal{K}:=\mathcal{V}(\Theta)$ be a variety in this language defined by a set of identities $\Theta$.
My ...
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Semistability of tensor products under automorphisms of tensored vector spaces
Let $A,B,C,D,E,F$ be vector spaces over a field.
Let $x\in A \otimes B \otimes C$ and $y \in D \otimes E \otimes F$ be tensors that are semistable with respect to the natural actions of $\text{SL}(A) ...
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Equivariant normalization?
Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...
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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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On the field of invariants of a finite group
So let $G$ be a finite group and let $\iota:G\rightarrow S_n$ be an embedding of $G$ in a symmetric group of degree $n$ for some fixed integer $n$. Let $K$ be a fixed field of characteristic $0$. The ...
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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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Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables
Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function
$P_i = px^i + qy^i + rz^i$
where $x,y,z$ are coordinates. I have a few related ...
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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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Progress since Luna's theorem on smooth invariants
In 1976, Luna proved the following important theorem of smooth invariant theory:
Let $G$ be a real reductive Lie group and a representation of $G$ on a real finite dimensional vector space $V$. ...
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Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
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Explicit formulas for invariants of binary quintic forms
I am looking for explicit formulas for the four basic invariants $I_4, I_8, I_{12}, I_{18}$ of a generic binary quintic form, either given in the shape
$$\displaystyle F(x,y) = ax^5 + 5bx^4y + 10cx^...
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Character variety of the free group
A classical result of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi_\mathbb{C}$ associated to the free group $F_2$ and the algebraic group $\mathrm{SL}_2(\mathbb{C})$...
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Standard Monomial basis for other types
For the algebraic group $SL_n$ (type $A_{n-1}$) and for a dominant weight $\lambda$ the standard monomials are indexed by the semi-standard young tableaux of shape $\lambda$ and they form a basis for ...
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A duality result for Coxeter groups
Short version: if $G$ is a Coxeter group and $H \subset G$ is a parabolic subgroup, both acting on a space $V$, is it true that the invariant-coinvariant algebra $(S(V)_G)^H$ has a natural bilinear ...
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Extension of the Weyl dimension formula
Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...
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What is the square of the Weyl denominator?
Let $\Phi$ be a (crystallographic) root system with Weyl group $\mathcal{W}$, and $\Phi^+$ a choice of positive roots, and
$$
q := \prod_{\alpha\in\Phi^+} (\exp(\alpha/2) - \exp(-\alpha/2))
= \sum_{w\...
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answer
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Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$
Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$.
Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by ...
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Invariant polynomials in curvature tensor vs. characteristic classes
Let $M$ be an $4m$-dimensional Riemannian manifold. We can then form the Pontryagin classes $p_k(TM)$ of the tangent bundle using Chern-Weil theory. For any sequence of numbers $k_1, \dots, k_l$ such ...