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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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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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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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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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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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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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 ...
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Which polynomials in the minors of a matrix are invariant under conjugation?
$\newcommand{\Cof}{\operatorname{cof}}$
This is a cross-post.
Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...
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On a paper of Formanek about $PGL_4$
In his paper "The center of the ring of 4 × 4 generic matrices" (Journal of Algebra, 1980) Formanek shows that, if $V$ is the representation of $PGL_4$ given by simultaneous conjugation of pairs of ...
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Clebsch-Gordan coefficients of $SO(5)$
The reduced CG coefficients for $SO(d):SO(d-1)$ are in principle known in full generality for $d\leq 4$: they are trivial for $d=2$, equivalent to $3j$ symbols of $SU(2)$ for $d=3$, and to $9j$ ...
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When do almost all these invariants of tensors vanish?
Let $A,B,C,D$ be $n$-dimensional vector spaces over a field $k$.
There is a natural homomorphism from the $mn^m$th tensor power $A^{\otimes (m n^m)} $ of $A$ to $k$ given by the determinant map $A^{\...
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A question related to Conways 99 graph problem
I have observed that the number of triangles $\frac{vk}{6}$ of a strongly regular graph with parameters $(v,k,1,2)$ is given by the coefficient $2(k-1)$ in the molien series of the "4-D extraspecial ...
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Semisimple Lie groups admitting a free algebra of invariants
Assume we work over an algebraically closed field of characteristic zero.
I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of ...
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Generators for invariant tensors of lie algebras
EDITED FOR (hopeful) CLARITY:
For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products ...
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Can one give a "nice" expression for this determinant?
I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague.
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Characteristic classes from moduli of alternating forms
Suppose, just for example, that you have a smooth manifold $M$ of dimension greater than $8$,and a cohomology class $q$ in $H^3(M,{\Bbb R})$. Suppose further that one can represent $q$ with a ...
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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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Chern-Weil theory on some noncompact groups, and characteristic classes in differential cohomology
$\newcommand{\Z}{\mathbb Z}\newcommand{\HdR}{H_{\mathrm{dR}}} \newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\g}{\mathfrak g}$I have a specific question about invariant polynomials for some Lie groups,
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Invariant theory over rings
Apologies if this is a silly question, but I have had cause to briefly introduce myself to invariant theory. I have noticed that authors primarily work over (algebraically closed) fields. I was ...
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Relation between Donaldson invariants and GW invariants
What is known about the relation of Donaldson invariants on a complex surface $\Sigma$ and GW invariants (or equivalent) of local Calabi-Yau 3folds such as the canonical bundle of $\Sigma$? (if any of ...
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The $GL(N)$ chicken versus the $SL(N)$ egg, the Erlangen Program and relations between FFTs
Here FFT is the standard abbreviation for First Fundamental Theorem of classical invariant theory.
If one has a group inclusion $H\subset G$ (and I suspect also more generally a morphism $H\rightarrow ...
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What is the status of this fifty-year-old conjecture of Kostant?
On page 3.27 of his 1963 thesis on the cohomology of homogeneous spaces as approached through the Eilenberg–Moore spectral sequence, Paul Baum states the following conjecture, which he attributes to B....
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Invariant theory of $SL_2$ over a field of positive characteristic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...
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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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$S_n$-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=\{ (...
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subalgebra of invariants for a reductive subgroup
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Spec{Spec}$Trying to understand some tannakian reconstruction, I've stumbled about the following problem in invariant theory. I guess it's something ...
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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,.....
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Translates of a fixed point set (of stable points) by a reductive group
I have the following set up:
Suppose that I am working over $\mathbb{C}$. Suppose that I have a reductive group $G$ acting linearly on $V$, and that I have a $G$-invariant smooth affine variety $X$ ...
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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 $G=SL(n+1)$ ...
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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-...
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Finite generation of kernel of derivations
Let $A$ be a finitely generated regular $k$-algebra, $k$ algebraically closed of characteristic zero, elements $x_1,\dots,x_n\in A$, such that $dx_1,\dots,dx_n$ give rise to a trivialization of the ...
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What is the Jarlskog invariant, conceptually?
Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity:
$$J_{ij,k\ell} := \operatorname{...
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Can we express the degree 10 and degree 15 Galois resolvents of sextic binary forms in terms of its basic invariants?
Let $V_6$ denote the 7 dimensional $\mathbb{C}$-vector space of binary sextic forms. For $T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \operatorname{GL}_2(\mathbb{C})$, $T$ ...
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The density of quartic polynomials whose Galois group is a subgroup of $D_4$
Let $D_4$ denote the (isomorphism class of) dihedral group of 8 elements. For a given binary quartic form $f$ with integer coefficients, define $\text{Gal}(f)$ to be the Galois group of the Galois ...
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The action of Steenrod squares on the indeterminate in Bullett-Macdonald identities
The Bullet-Macdonald identity (c.f. On the Adem relations)is the following:
$$P(s^2+st)P(t^2)=P(t^2+st)P(s^2).$$
where we have $P(s)=\Sigma _s s^iSq^i$. This is known to be equivalent to the
Adem ...
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algebra of endomorphisms over the diagonal invariants
Let $k$ be a field of characteristic 0 (say $\mathbb{C}$).
Consider the ring of polynomials $R = k[X_1,...,X_n]$ and its subring of invariant polynomials $S = R^{S_n}$.
It is known that the ...
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FFT like theorems for tensor product
The fundamental theorem of symmetric functions states that $\mathbb C[V]^{S_n}$ is generated by the elementary symmetric polynomials, where $V$ is the natural representation of $S_n$. Then a theorem ...
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Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
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Ring of invariants of a finite subgroup of $GL_2(\mathbb{C})$
In the paper:
Kac, Victor; Watanabe, Keiichi, Finite linear groups whose ring of invariants is a complete intersection. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 221–223
it is said in Remark 2 ...
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Is the field of invariants $k(V)^G$ purely transcendental over $k$?
Reference: http://www.math.u-psud.fr/~colliot/mumbai04.pdf
Proposition 4.3. on page 18 in the above reference reads as follows:
Assume $k = \overline{k}$. If $V$ is a finite dimensional vector space ...
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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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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 ...
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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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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 ...
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Invariants of linear endomorphisms of tensor products
Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero.
Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space ...
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Length of fibers of $(\mathbb{A}^n)^d\to\mathrm{Sym}^d(\mathbb{A}^n)$
Let $k$ be a field, consider the canonical morphism $f\colon (\mathbb{A}_k^n)^d\to\mathrm{Sym}^d\mathbb{A}_k^n$.
Is there an explicit bound on the length of fibers of $f$ in terms of $n,d,\mathrm{...
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Invariants and subgroups
Let $G$ be an affine algebraic group over some algebraically closed field $K$, and let $H$ be a closed subgroup.
Assume that $G$ acts algebraically on an affine variety $X$.
Assume that $X'\subseteq ...
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Categorifying the Cauchy kernel as a filtration of $\operatorname*{Sym}\left( F\otimes G\right) $ over any commutative ring
Question 1 (short version). Let $R$ be a commutative ring with unity. Let
$F$ and $G$ be two $R$-modules. Let $n\in\mathbb{N}$. Is it true that the
$n$-th symmetric power $\operatorname*{Sym}\...
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Infinite-dimensional Chevalley–Shephard–Todd theorem
The Chevalley–Shephard–Todd theorem states that given a finite-dimensional faithful representation of a finite group $G$ on a vector space $V$ over a field $k$ whose characteristic does not divide the ...
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Global sections of the tangent bundle on Grassmannian and derivations
Let G=Gr(2,n) the grassmannian of 2-planes in $\mathbb{C}^n$, $R= \bigoplus_{k \geq 0} H^0(G, \mathcal{O}_G(k))$ its coordinate ring under the Plücker embedding, and consider the space of global ...