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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“Why” is every polynomial representation of SL(2) selfdual?

Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional polynomial representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, ...
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Is there an invariant theory explanation of the orbit structure of GL₂ acting on second-diagonal symmetric matrices by g∙X = gXJg^tJ ?

Statement of the Specific Result Let $J$ denote the matrices with ones on the "second diagonal", meaning the diagonal between the (1,n) and (n,1) entry, and zeros elsewhere. So in the case $n=2$, ...
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Polynomial invariants of the exceptional Weyl groups

Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let ...
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A relative Noether number for invariants

EDIT: Wrong definition of $\beta\left(G,H\right)$ fixed. One of the results is open (i. e., I cannot prove it). In "Finite Groups and invariant theory" (a paper in Malliavin's LNM #1478 which can ...
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When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?

Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is generated by symplectic reflections, i.e. by elements ...
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How to compute the ring of invariants of SO_3(k) acting on a polynomial ring

Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$. ${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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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633 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 ...
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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 ...
4
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3answers
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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 ...
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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 ...
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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 ...
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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 ...