# 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 ...
1 vote
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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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### 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 $n$...
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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 ...
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### 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 ...
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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 ...
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1 vote
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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 ...
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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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### 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 ...
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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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### 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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### 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.... 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 ...
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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 $E_7$. 521 views

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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 ...
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 ...
### 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. ...