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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GIT and singularities
Let $G$ be a complex reductive group acting on a complex affine variety $X$ and let $X // G = \operatorname{Spec}\mathbb{C}[X]^G$ be the GIT quotient.
Is there a relationship between the singular ...
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Invariants of a $kG$-module via its comoposition series, when does $M^P \supsetneq N^P$ hold for a $p$-group for $N\subseteq M$ maximal?
Let $G$ be a finite group, $k$ a field, $M$ a $kG$-module, $M^G$ the invariants of $M$ under $G$, $P$ a Sylow $p$-subgroup of $G$ where $p = \text{char}(k)$, $N$ a maximal submodule of $M$ and $S$ the ...
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How to find the polynomials that define a compact Matrix Lie group from its Lie algebra?
Consider a compact (connected) Lie group, or more generally, a linear algebraic Lie group. Suppose we are given the Lie algebra corresponding to the Lie group. How can we find a set of polynomial ...
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What are all invariant polynomials on the space of algebraic curvature tensors?
Let $V = (\mathbb{R}^n, g)$, where $g$ is the Euclidean inner product on $V$. Denote by $G$ the orthogonal group $O(V) = O(n)$ and by $\mathfrak{g}$ the Lie algebra of $G$.
Let $W \subset \Lambda^2V^* ...
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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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An explicit negative solution to the Lüroth problem for non-algebraically closed fields
Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$.
If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \...
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An analogue of Noether's Problem for non-rational varieties
For the sake of simplicity, let $\mathsf{k}$ be algebraically closed and of zero characteristic. Varieties are irreducible.
The (linear) Noether's Problem (which goes back to the early 1910's in ...
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Behavior of invariants under reduction mod p
Let $R$ be a finitely generated $\mathbb{Z}$-algebra with an [edit: linear algebraic] action of $G(\mathbb{Z})$ where $G$ is a split simply-connected semisimple group.
Then for any prime $p$ we have a ...
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Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix
Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$.
My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
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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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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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Invariant subspace of a nonlinear map
First please see this very simple fact:
Fact: $\ $ Any linear map $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ has a proper invariant linear subspace.
By an invariant subspace we mean a space $M$ ...
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propagation of a invariance along some PDE
Consider the following non linear PDE on $\mathbb{R}^n$
$$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$
with given initial condition $u_0(x)$.
Assume that:
$u_0$ is rotation invariant, ...
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Characterization of all-orthogonal tensors
In the paper [1], it is proven in Theorem 2 that any $n$-tensor $\mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n}$ can be decomposed as
$$
\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n
$$
...
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Invariant theory of the indefinite orthogonal groups
I believe the following statements are true:
Let $V$ be a finite-dimensional real vector space with a positive-definite inner product $g$. Let $g_{\otimes n}$ denote the natural extension of $g$ to $...
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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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Consequences of invariant-subspace problem to Li–Yorke chaos [closed]
The invariant-subspace problem is probably an open problem for reflexive spaces which asks:
Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial ...
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Equivalence classes of a circle of n bits upon flipping 3 consecutive 0s to 1s or vice versa
Consider a circle of n-bits and define the equivalence relation as follow:
Two configurations A and B of the n-bits circle are equivalent if they can be transformed into each other by performing a ...
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Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable?
I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book
Greco, Silvio, ...
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Invariants of symmetric forms with respect to the symplectic group
Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...
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Highest weight vector as a global section of an affine scheme
Let $G$ be a connected, reductive quasi-split group over a field $k$, acting on an afffine $k$-variety $X$. Let $B = TU$ be a Borel subgroup of $G$ with maximal torus $T$ and unipotent radical $U$. ...
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Rosenlicht's theorem and fundamental domain for unipotent group acting on $\mathbb A_k^n$
I have a question about unipotent group actions. I was referred to Rosenlicht's papers, but I had trouble getting much out of these because I don't understand the old algebraic geometry language very ...
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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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On Shephard-Todd theorem
There is an excellent Torsten Ekedahl's answer to Roman Fedorov's question here: Chevalley–Shephard–Todd theorem. Does anyone know any articles or books where this approach is outlined?
I didn't ...
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Rational torus invariants
Let $T=(\mathbb{C}^{\times})^n$ be the $n$-dimensional torus acting on the polynomial algebra $\mathbb{C}[x_1,x_2, \ldots,x_n]$ diagonally, i.e.
$$
diag(t^{a_1},t^{a_2},\ldots,t^{a_n})x_i=t^{a_i}x_i, ...
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Invariants in the symmetric algebra of a module
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$...
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Transcendent basis for the field of multisymmetric functions
It is known that the field of multisymmetric rational functions (over a field of characteristic $0$), that is,
rational function in variables $x_{11}, \ldots, x_{1m}, \ldots, x_{n1}, \ldots, x_{nm}$ ...
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Decomposing Schur functor applied to a tensor product
I want to compute
$$
S^{2,2,\dots,2,1}(\mathbb C^{2m-1} \otimes W)^{SL(2m-1)}
$$
Here $m$ numbers should appear in the superscipt of the Schur functor, and the last superscript means to take $SL(2m-1)$...
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What generalizes symmetric polynomials to other finite groups?
Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...
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Algorithmic invariant theory of finite groups acting on finitely generated $\mathbb{Z}$-algebras: reference request
Does there exist a book discussing algorithmic invariant theory for finite groups that does not assume that the algebras involved are defined over a field (e.g. base rings $\mathbb{Z}$ and $\mathbb{Z}/...
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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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Chirality and Anti-Chirality of links in 3 and in 5 dimensions
We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory:
https://en.wikipedia.org/wiki/Chiral_knot
My ...
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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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Variety of conjugacy classes
Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some ...
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Dimension of orbit versus invariant functions
$\def\CC{\mathbb{C}}$Let $K = \CC(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$; in the cases I care about, $G \cong \mathbb{Z}$. Then the field of $G$-invariants, $K^G$, ...
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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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Reference request: invariants/tableaux functions for 4 lines in $P^3$
Does anybody have a reference for invariants of configurations of linear subspaces in the projective space?
In particular I would be curious to see an explicit expression of the invariant functions ...
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Orbits of unipotent groups
Let $V$ be a real vector space (of finite dimension) and let $G$ be a unipotent Lie subgroup of $\mathrm{GL}(V)$. The orbits of points under the action of $G$ (that is, the sets $Gx = \{T(x) \ : \ T \...
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Ideals invariant under ring automorphisms
I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...
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On a special type of subring of $\mathbb C[x_0,...,x_{q-1}]$
Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let
$$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
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Invariants of the group $SO(2)$
Let $V_d$ be the complex vector space of binary forms of degree $d$ endowed with the natural
action of the special orthogonal group $SO(2).$ Consider the corresponding action of the
group $SO(2)$ on ...
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Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$
For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$.
For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$...
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Alternatives to the ring of invariants depicting the orbit closures?
Let $G$ be an affine algebraic group over $\mathbb{C}$ and $V=\textrm{Spec}(A)$ an affine $G$-variety. Assume that $G$ is reductive. My understanding is that a central object in studying the action of ...
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The outer automorphism of the dihedral group $D_4$ and quartic polynomials
Let $L$ be a dihedral quartic field. That is, we have $[L : \mathbb{Q}] = 4$ and the Galois closure $M$ of $L$ is a degree 8 extension of $\mathbb{Q}$ with Galois group isomorphic to the dihedral ...
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Casimir operator of a given Lie algebra and relation with its matrix representation
I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...
2
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Space of change of basis matrices between two similar matrices - how to reduce it with additional tests?
Assume we have two real symmetric $n\times n$ matrices: $A, B$. We can easily test their similarity: $\textrm{Tr}(A^k)=\textrm{Tr}(B^k)$ for $k=1..n$. In this case both can be rotated to the same ...
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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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higher Casimirs for $\mathfrak{sl}$
The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
9
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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 ...