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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36 views

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 $...
12
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1answer
284 views

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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1answer
89 views

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 ...
4
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1answer
89 views

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 ...
4
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1answer
223 views

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, and Rosario ...
4
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1answer
212 views

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 ...
2
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1answer
154 views

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 $U$. ...
3
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2answers
227 views

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 ...
10
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1answer
285 views

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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248 views

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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63 views

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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2answers
236 views

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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1answer
151 views

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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69 views

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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2answers
406 views

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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60 views

Prove that the Mertens function is invariant under matrix inversion for $n>3$

Consider the lower triangular matrix $T$ with the definition: $n \leq3 :$ $$T(n, 1)=1$$ $n>3:$ $$T(n,1)=x_n$$ $n \geq k:$ $n \leq 3:$ $$T(n,k)=1$$ $n \geq k:$ $n>3:$ $k=2 \text{ or } k=3:$ ...
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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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2answers
183 views

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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289 views

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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2answers
279 views

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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196 views

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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147 views

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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105 views

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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2answers
295 views

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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1answer
161 views

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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138 views

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 ...
3
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2answers
377 views

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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89 views

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 ...
6
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1answer
284 views

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 ...
3
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2answers
645 views

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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0answers
155 views

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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183 views

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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1answer
267 views

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 ...
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197 views

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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469 views

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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1answer
121 views

Rotation invariant of surface

Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four ...
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26 views

Does the $G$-norm coincide with the ordinary norm for “quasi-$G$-Galois” extensions

Let $S$ be a commutative ring, let $G$ be a finite group acting on $S$ via automorphisms (not necessarily faithfully), and let $R$ be a subring of $S$ consisting of elements fixed $G$. The extension $...
3
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1answer
123 views

Kernel of restriction for ring of functions on reductive groups

Let $H \subset G$ be an inclusion of reductive groups over an algebraically closed field $k$ of char $0$. For simplicity, let's assume that $G$ is split and $H$ contains a maximal torus for $G$. Then ...
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268 views

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 ...
4
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123 views

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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1answer
112 views

$SU(2)\times SU(2)$ invariant $SU(3)$-structure on $\{t\} \times M^6$

I am reading Jason Lotay and Goncalo Oliveira's paper -$SU(2)^2$ invariant $G_2$-instantons, and have few questions from the same. If we consider the space $M = S^3 \times S^3$. Then the cone metric ...
6
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1answer
112 views

Software for computing equivariants

If $\Gamma$ is a finite group with action on two vector spaces $\mathbb R^n$ and $\mathbb R^m$ denoted by $\gamma_n$ and $\gamma_m$ respectively, the fundamental equivariants are the polynomials $f: \...
8
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0answers
135 views

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 ...
4
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1answer
85 views

Sufficient conditions for secondary invariants

Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension ...
8
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3answers
401 views

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^...
3
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0answers
96 views

Topological criterion for GIT semistability

Let $X$ be a complex algebraic variety and $G$ a complex reductive group acting on $X$. Let $L$ be a linearization of this action, i.e. a line bundle with a linear action of $G$ covering that of $X$. ...

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