Tagged Questions

Invariant theory deals with an algebraic, geometric or analytic structure X , submited 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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Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
10
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2answers
863 views

Chevalley–Shephard–Todd theorem

The wikipedia article claims that the theorem "was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof". I ...
3
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1answer
319 views

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 ...
6
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5answers
814 views

Explicit invariants (under change of basis) of maps $V \to V \otimes V$.

It is standard to construct numbers associated to a linear transformation $f: V \to V$ of a finite-dimensional vector space which are invariant under change of basis. The coefficients of the ...
2
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2answers
227 views

Subrings of rational functions invariant under change of sign

Let $R$ be a ring generated by $k$ rational functions in the variables $x_1,...,x_n$ over the real numbers. Is there an algorithm that computes a set of rational functions $f_1,...,f_l \in R$ which ...
4
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221 views

Invariant forms

Is there a classification of pairs $(\mathfrak g,V)$, with $\mathfrak g$ a Lie algebra and $V$ a $\mathfrak g$-module, such that $(\Lambda^3V)^{\mathfrak g}\neq0$?
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1answer
622 views

Weyl group Invariants

What are the generators of $\mathbb C[V^m]^W$, where $W$ is the Weyl group of type $E_6, E_7, E_8$, V^m denote 'm' (m > 1) copies of the Cartan subalgebra and the action is the diagonal action? Is ...
10
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1answer
505 views

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 ...
4
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3answers
294 views

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 ...
6
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1answer
445 views

Generalizing cosine rule to symmetric spaces

The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both ...
6
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3answers
918 views

Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?

Background/motivation It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
2
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1answer
603 views

What is affine invariant used in computer vision?

Affine invariant for 4 coplanar points ABCD is said to be Area(ACD)/Area(ABC). Can somebody provide the proof of this means why is this invariant under affine ...
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Resources on Invariant Theory

Hi, So my question is pretty much summed up by the summary - basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd ...