# Tagged Questions

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### 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 ...
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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 ...
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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 ...
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### Algebraic invariants of linear ODE's with constant coefficients

Let consider linear ODE with constant coefficients: $$y^{(n)}(x) + A_{n-1} y^{(n-1)}(x) + ... + A_1 y'(x) + A_0 y(x)= 0.$$ It admits some equivalence point transformations, that preserves its ...
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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 ...
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### 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 $x^2+y^2$ ...
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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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### 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 ...
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### 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....
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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 ...
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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 $E_7$.
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### 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 ...
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### 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 ...
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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. ...
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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 ...
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### 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 ...
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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)...
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### Transformation of a cubic form

How can I change an integral binary form $ax^3+bx^2y+cxy^2+dy^3$ with the usual discriminant $D =b^2c^2-27a^2d^2+18abcd-4ac^3-4b^3d$ into a form $ax^3+dy^3$ which has a simple discriminant $-27a^2d^2$...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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### 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 $\mathrm{GL}(n)=\mathrm{GL}(n,\mathbb{C})$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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 ...
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 ...