Questions tagged [classical-invariant-theory]
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67
questions
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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 ...
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3
answers
2k
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What theorem of Liouville's is Gian-Carlo Rota referring to here?
I am very curious about this remark in Lesson Four of Rota's talk, Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations:
"For second order linear differential ...
- 415
15
votes
3
answers
4k
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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 $l$ ...
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15
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1
answer
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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 $...
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12
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1
answer
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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. ...
- 21.1k
12
votes
1
answer
487
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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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11
votes
5
answers
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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)...
- 13.7k
11
votes
2
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745
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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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11
votes
1
answer
904
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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 ...
- 1,987
11
votes
1
answer
552
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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}...
- 1,764
10
votes
4
answers
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Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...
- 3,453
10
votes
2
answers
1k
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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 ...
- 103
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2
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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
$$\displaystyle H_F(x,y)...
- 25.5k
9
votes
2
answers
741
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Classical invariants involving exterior powers of standard representation
While investigating certain conformal blocks line bundles on $\overline{M}_{0,n}$, I was led to what seems to be an identification between two spaces of invariants, and I am curious if there is a ...
- 1,871
9
votes
3
answers
730
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Invariants of exterior powers
Let $\mathfrak{g}$ be the Lie algebra of $GL(n,\mathbb{R})$. Let $\theta(X) = - X^T$ be the Cartan involution on $\mathfrak{g}$; it induces decomposition as $\mathfrak{g} = \mathfrak{k} \oplus \...
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votes
3
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753
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Explicit formulas for invariants of binary quintic forms
I am looking for explicit formulas for the four basic invariants $I_4, I_8, I_{12}, I_{18}$ of a generic binary quintic form, either given in the shape
$$\displaystyle F(x,y) = ax^5 + 5bx^4y + 10cx^...
- 25.5k
9
votes
2
answers
701
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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 ...
- 4,454
9
votes
4
answers
871
views
A ring of invariants in characteristic 2
Let $K$ be an algebraic closure of $\mathbb{F}_2$. The cyclic group $C_{2^n}$ acts on $K[x_0, \dots, x_{2^n-1}]$ by cyclically permuting the $x_i$: $a : x_i \rightarrow x_{i + a \bmod 2^n}$. Is ...
- 4,626
8
votes
2
answers
316
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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 ...
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votes
1
answer
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The exceptional Lie algebra $\mathfrak{g}_2$ and binary cubics
How is the exceptional 14-dimensional Lie algebra $\mathfrak{g}_2$ related to the covariant algebra for the binary cubic?
Here are some details on this question. This algebra is generated by 4 forms,...
- 1,267
8
votes
1
answer
249
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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} (\...
- 25.5k
8
votes
2
answers
690
views
Enumeration of graphs arising in invariant theory
I've been working on a talk based on some stuff in Olver's "Classical Invariant Theory" book and have been wondering about a related graph enumeration problem.
Start with a triple $(n,v,e)$ of ...
- 15.8k
7
votes
2
answers
394
views
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^* ...
- 5,011
7
votes
2
answers
539
views
Generators for the algebra of GL(n)-equivariant maps from M_n + M_n to M_n
Let $M_n$ be the set of $n$-by-$n$ matrices with complex entries, viewed as a variety over $k=\mathbb{C}$. Equip $M_n$ with the conjugation action of $\mathrm{GL}(n)=\mathrm{GL}(n,\mathbb{C})$. ...
user9154
7
votes
1
answer
727
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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 ...
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6
votes
1
answer
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Is $k[X]^G$ integral closed in $k[X]$.
May assume field $k=\mathbb{C}$.
Let $X$ be an affine variety and $G$ be a reductive group (may assume connected).
Is the ring of invariants $k[X]^G$ integral closed in $k[X]$?
The claim may not ...
- 157
6
votes
1
answer
653
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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 ...
- 17.5k
6
votes
1
answer
228
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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 ...
- 549
6
votes
3
answers
623
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$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) \...
- 1,535
5
votes
2
answers
487
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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{...
- 546
5
votes
2
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372
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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
$$R(x,y) := F\left(A(x,y), B(x,y)\...
- 1,573
5
votes
2
answers
457
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Left U_n-invariants of SL_n - an exercise in Kraft-Procesi
I am sorry for spamming MO with questions I have not thought about for more than 3 hours, but currently I am quite busy with preparing a talk on representations of $S_n$, and I don't want these to get ...
- 33.2k
5
votes
2
answers
1k
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Howe duality for exceptional algebras
There is a nice tool in representation theory, the Howe duality, which as I know works for certain pairs of classical Lie algebras (the reference to the complete list of Howe dual pairs is appreciated ...
- 605
5
votes
1
answer
148
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For what graph does the following algebraic property hold?
Let $G=(V,E)$ be a simple graph.
My question:
For what graph $G$, does there exist a permutation $\sigma$ on $V$ such that
$$\prod_{uv\in E}(x_{\sigma(u)}-x_{\sigma(v)})=-\prod_{uv\in E}(x_u-x_v)?$$
...
- 1,987
5
votes
2
answers
799
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Classical invariant theory: absolute rational invariants and $GL(2)$-orbits
I have a question concerning classical invariant theory. Consider binary $n$-forms (i.e. all homogeneous polynomials of degree $n$ of two variables) over the field of complex numbers. Clearly, the ...
- 103
5
votes
1
answer
492
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The ring of SL_2 invariants in sums of conjugation and tautological modules
Rings of Invariants
Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free ...
- 12.7k
4
votes
1
answer
434
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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 ...
- 25.5k
4
votes
1
answer
568
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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 \...
- 41
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....
user27328
4
votes
3
answers
340
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Generalized symmetric algebras and Dickson algebras over ${\mathbb F}_p$.
Start with the really well-known fact that $R[x_1, \ldots, x_n]^{S_n}$, where $R$ is any commutative ring, is polynomial on elementary symmetric polynomials. Now consider the slight generalization of ...
- 4,970
4
votes
1
answer
270
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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 ...
- 13.7k
4
votes
0
answers
273
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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)$ ...
- 12.1k
3
votes
2
answers
376
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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$.
user21574
3
votes
1
answer
616
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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 \ \...
- 429
3
votes
1
answer
266
views
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 ...
3
votes
1
answer
307
views
When the affine quotient is faithfully flat?
It may be easy for the expert.
Consider the map from $n$ by $m$ matrices (over $\mathbb{C}$ )to the $n$ by $n$ symmetric matrices $\phi\colon A\mapsto A A^T$.
My question is when this map is ...
- 157
3
votes
1
answer
152
views
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} ...
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3
votes
1
answer
289
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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 ...
- 2,982
3
votes
0
answers
134
views
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 $n$...
3
votes
0
answers
95
views
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 $...
- 2,169