Questions tagged [classical-invariant-theory]

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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 ...
prochet's user avatar
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If $ n \in \mathbb{N} $, then does the Reynolds operator of $ \mathbb{G}_{m}^{n} $ commute with the Frobenius endomorphism?

If $ n \in \mathbb{N} $, then $ \mathbb{G}_{m}^{n} $ is linearly reductive. Let $ \beta: \mathbb{G}_{m}^{n} \to \operatorname{GL}(\mathbf{V}) $ where $ \mathbf{V} $ is a vector space over an ...
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A quadratic $O(N)$ invariant equation for 4-index tensors

Consider an $O(N)$ invariant quadratic equation $$ T_{ijkl}= T_{ijmn}T_{klmn}+ T_{ikmn}T_{jlmn}+ T_{ilmn}T_{jkmn}, $$ where $T_{ijkl}$ is a real, totally symmetric 4-tensor, and the indices run from 1 ...
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3 votes
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Unitary equivalence of symmetric and homogenous polynomials

Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$...
Deepesh Singh's user avatar
3 votes
1 answer
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Generators of polynomial invariant ring of compact Lie groups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$I'm a PhD student in physics working in the broad area of photonic quantum computing. My current project looks at the ...
Deepesh Singh's user avatar
2 votes
1 answer
238 views

What is the Molien series of the SO(2)-invariant ring on the plane (sometimes written C[X]^{SO(2)} )?

Let SO(2) be the group of rotations in the plane. What is the Molien series (sometimes called the Hilbert-Poincare series) of the SO(2)-invariant ring of polynomials? N.B. The main goal being to ...
Victoria's user avatar
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Is the ideal of the Veronese variety $V_{d,n}$ generated by quadrics?

Maybe it sounds like a silly question to the experts but I'm not able to find a proper reference in the web. Anyone knows if the ideal $I_{d,n}$ of the Veronese variety $V_{d,n}$ is generated by ...
gigi's user avatar
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3 votes
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Are there any results on an upper bound for the number of secondary invariants needed to generate the invariant ring of a finite group?

If $ G $ is a finite cyclic group, $ \beta: G \to \operatorname{GL}(\mathbf{V}) $ is a linear $ n $ dimensional representation of $ G $, and $ \{x_{1},\dots,x_{n}\} $ is a basis of $ \mathbf{V}^{\ast} ...
schemer's user avatar
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History of Study's Lemma?

The following theorem is usually attributed to Eduard Study: Let $f(x,y)$ and $g(x,y)$ be polynomials in two variables over a field, with $f$ irreducible. If $f\nmid g$ then the curves $C_f:f=0$ and $...
Drew Armstrong's user avatar
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Parameter of Brauer algebra

Let $O(V)$= the set of orthogonal transformation from the vector space $V$ to $V$ where $\dim V=n$. We know that the centralizer algebra of $O(V)$ on the tensor space $V^{\otimes{f}}$ is Brauer ...
noone 's user avatar
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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 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 $...
Quarto Bendir's user avatar
12 votes
1 answer
486 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. ...
DDT's user avatar
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For what graph does the following algebraic property hold?

Let $G=(V,E)$ be a simple graph. My question: For what graph $G$, does there exist a permutation $\sigma$ on $V$ such that $$\prod_{uv\in E}(x_{\sigma(u)}-x_{\sigma(v)})=-\prod_{uv\in E}(x_u-x_v)?$$ ...
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On a relation between the Hessian and the catalecticant matrix of a binary quartic form

I am currently working on a paper that requires using the theory of invariants of binary quartic forms. Playing around, I have found an interesting identity that gives the Hessian from the minors of ...
JME's user avatar
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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 ...
IMeasy's user avatar
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2 votes
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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 ...
Leox's user avatar
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Can discriminant polynomials become perfect powers on hyperplanes?

Let $$\displaystyle f(x) = a_d x^d + a_{d-1} x^{d-1} + \cdots + a_0.$$ Consider the discriminant of $f$, denoted by $\Delta(f)$, defined as $$\displaystyle \Delta(f) = a_d^{2d-2} \prod_{i < j} (\...
Stanley Yao Xiao's user avatar
4 votes
1 answer
428 views

How to check if a ternary cubic is a product of linear forms?

Let $F$ be a square-free (as a polynomial) ternary cubic form over $\mathbb{C}$, and let $H_F$ be its Hessian determinant... which is also a ternary cbic form. If $F$ splits over $\mathbb{C}$, so that ...
Stanley Yao Xiao's user avatar
7 votes
1 answer
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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 ...
Vít Tuček's user avatar
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4 votes
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Resultants for compactly represented product form polynomials?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the ...
Turbo's user avatar
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2 votes
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Explicit formulas for polynomial invariants of cubic surfaces in Sylvester standard form

By a cubic surface $X_F$ we mean the zero locus of a homogeneous cubic polynomial $F(x,y,z,w)$. The group $\text{GL}_4$ acts on $X_F$ via substitution. The ring of polynomial invariants induced by ...
Stanley Yao Xiao's user avatar
10 votes
2 answers
495 views

A curious identity involving a covariant of binary cubic forms

Let $F(x,y) = a_3 x^3 + a_2x^2 y + a_1 xy^2 + a_0 y^3$ be a binary cubic form, say with real coefficients. Put $H(x,y) = H_F(x,y)$ for the Hessian covariant of $F$, defined by $$\displaystyle H_F(x,y)...
Stanley Yao Xiao's user avatar
9 votes
3 answers
738 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^...
Stanley Yao Xiao's user avatar
9 votes
3 answers
726 views

Invariants of exterior powers

Let $\mathfrak{g}$ be the Lie algebra of $GL(n,\mathbb{R})$. Let $\theta(X) = - X^T$ be the Cartan involution on $\mathfrak{g}$; it induces decomposition as $\mathfrak{g} = \mathfrak{k} \oplus \...
Vanya's user avatar
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What is currently feasible in invariant theory for binary forms?

When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system ...
Colin McLarty's user avatar
3 votes
1 answer
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Question on the integral $\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx$

From the wikipedia page on Gaussian integral https://en.wikipedia.org/wiki/Gaussian_integral the following formula holds: $$\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx =\frac12 e^f \ \...
sam's user avatar
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2 votes
1 answer
549 views

Homogeneous polynomials and symmetric binary forms

Let $f\in k[x_0,...,x_n]_d$ be a degree $d$ homogeneous polynomial in $n+1$ variables. Is there a way to associate to $f$ a form $g(y_1,...,y_m)$ which is symmetric in the sets of binary variables $...
Puzzled's user avatar
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5 votes
2 answers
371 views

Discriminant of a composition of binary forms

Let $F(x,y), A(x,y), B(x,y)$ be homogeneous polynomials with integer coefficients (a.k.a. binary forms) such that the degrees of $A(x,y)$ and $B(x,y)$ match. Define $$R(x,y) := F\left(A(x,y), B(x,y)\...
Anton's user avatar
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8 votes
2 answers
314 views

Vanishing of Aronhold S-invariant on the cubic forms on $H^2(X, \mathbb Q)$

I am considering several examples of compact complex threefolds $X$ such that $rk H^2(X)=3$. Note that we have a cubic form on $H^2(X, \mathbb Q)$ which comes from the cup product. I calculated the ...
user104109's user avatar
3 votes
0 answers
133 views

Dimension of curvature invariants

EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group. Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...
asv's user avatar
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6 votes
1 answer
647 views

Local differential geometry and invariant theory

Can someone please give me pointers to the literature for local differential differential geometry according to invariant theory in the following sense, provided such a literature exists? Start with ...
David Feldman's user avatar
6 votes
1 answer
227 views

Constructing symmetric invariants of the exceptional simple Lie algebra as restrictions

Let $\mathfrak{g}$ be a complex simple Lie algebra with maximal torus $\mathfrak{h}$, Weyl group $W$. The adjoint representation $\operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)}$ extends ...
Lewis Topley's user avatar
9 votes
2 answers
699 views

Generalization of Pascal's theorem to higher dimensions

Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...
Mostafa's user avatar
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2 votes
0 answers
120 views

Anti-Invariant Polynomials of the Dihedral group

I'm interested in the one-dimensional irreducible representations of $D_{2n}$ acting on $\mathbb{R}[x,y]$. I have found that the trivial representations for an algebra freely generated by $x^2+y^2$ ...
Nate S's user avatar
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15 votes
3 answers
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Generators of invariant polynomials of semisimple Lie algebra

Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ ...
Qijun Tan's user avatar
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4 answers
543 views

Equivalent binary forms

Two binary forms $f, g \in k[x, y]$ are equivalent when there exists an $M \in GL_2 (k)$ such that $f^M = g$. For simplicity we take $k$ such that $char (k) =0$ and $k=\bar k$. The equivalence ...
user avatar
11 votes
1 answer
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Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry

Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a $\mathrm{O}...
Xin Nie's user avatar
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6 votes
3 answers
619 views

$SL(n) \times SL(n)$-invariants of $m$-tuples of matrices

I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows $$ (A,B) \cdot (M_1, \ldots, M_m) \...
Sasha Pavlov's user avatar
  • 1,535
1 vote
1 answer
455 views

Schur-Weyl duality for arbitrary tensor products of simple finite-dimensional $GL_n$-modules

Let $M$ and $N$ be two simple finite dimensional $GL_n$ modules. Is there a way of expressing the heighest weight vectors of the simple submodules of $M\otimes N$ in terms of the heighest weight ...
Alex's user avatar
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4 votes
0 answers
273 views

First Fundamental Theorem for Alternating Group

I know it fails but is there an answer? More precisely, let $V$ be the standard complex $n$-dimensional representation of the alternating group $A_n$, $kV$ the direct sum of its $k$ copies, $S(kV)$ ...
Bugs Bunny's user avatar
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5 votes
2 answers
485 views

Linearisation of a group

If $G$ is a connected Lie group acting on a vector $\mathbb{C}$-space $V$ then it is well known that the algebra of invariants $\mathbb{C}[V]^G$ coincides with the algebra of invariants $\mathbb{...
Leox's user avatar
  • 546
4 votes
3 answers
271 views

Invariants in $S^n(S^k(\mathbb{C}^w)$

Are the formulas for the multiplicity of $SL(w)$ invariants in $S^n(S^k(\mathbb{C}^w)$ known? This is a very classical topic. If no, in what ranges one can compute it (for certain paramaters fixed - e....
user avatar
10 votes
2 answers
1k views

Invariant polynomials for a product of algebraic groups

Let $G_1$ and $G_2$ be connected reductive algebraic groups defined over $\mathbb{C}$ and let $V_1$ and $V_2$ be irreducible representations of $G_1$ and $G_2$ respectively. I'm interested in general ...
Neil's user avatar
  • 103
3 votes
2 answers
376 views

degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$

How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.
user avatar
4 votes
1 answer
563 views

Igusa invariants of genus 2 curves as Siegel modular functions?

Hi, Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that $\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I_2I_4-3I_6), \quad \...
Joshua's user avatar
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0 votes
0 answers
175 views

The transcendence degree of the algebras of invariants

Let $V_n,V_m$ be the vector $\mathbb{C}$-spaces of the binary forms of degrees $n,m$ considered as usual $SL_2$-modules. Let $I_{n,m}=\mathbb{C}[V_n \oplus V_m]^{SL_2}$ and $C_{n,m}=\mathbb{C}[...
Melania's user avatar
  • 301
2 votes
1 answer
331 views

Modules of invariants?

Let $G \subset SL(V, \mathbb{C})$ be a finite group and $R=(\operatorname{Sym}\[V\])^G$ is the ring of polynomial invariants, $W$ some irreducible complex representation of $G$. I want to know is ...
Sasha Pavlov's user avatar
  • 1,535
2 votes
2 answers
215 views

Computing the relations in invariant algebra

Suppose we have a ring $R$ and a finite group $G$ acting on it, Is there a way to compute the invariant ring $R^G$ explicitly? Infact I am more interested in the case of affine ring and the symmetric ...
George's user avatar
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12 votes
1 answer
654 views

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. ...
asv's user avatar
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