# Tag Info

### How to constructively/combinatorially prove Schur-Weyl duality?

This is a quick answer to explain the statement that the hard direction of Schur-Weyl duality is the same thing the First Fundamental Theorem of invariant theory. Let $V$ be a finite dimensional ...
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### Why is Mumford's GIT-quotient so effective?

By my understanding, your question is not "Why is Mumford's construction better than the affine quotient". As you note, Proj is better than Spec of invariants for taking quotients by $\mathbb{G}_m$. ...
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### Why is the catalecticant invariant under coordinate changes?

Dolgachev (2012, p. 57; pdf) observes that your matrix $\left( a_{i+j-2}\right) _{1\leq i\leq n+1,\ 1\leq j\leq n+1}$ (with determinant $\operatorname{Cat} f$) is the matrix of a symmetric bilinear ...
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### Has anyone researched additive analogues of toric geometry in characteristic zero?

The theory of Luna and Vust (Plongements d'espaces homogènes. Comment. Math. Helv. 58 (1983), 186–245.) on equivariant compactifications of homogeneous varieties works actually for any connected ...
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### What generalizes symmetric polynomials to other finite groups?

The name of the body of theory you are asking for is "invariant theory of permutation groups." You will also find relevant papers by searching for "polynomial permutation invariants.&...
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### Continuous version of the fundamental theorem of invariant theory for the orthogonal group

Yes. It suffices to show that if one has a sequence $\vec v^{(n)} = (v^{(n)}_1,\dots,v^{(n)}_m) \in E^m$ whose Gram matrix $(\langle v^{(n)}_i, v^{(n)}_j \rangle)_{i,j=1,\dots,m}$ converges to a Gram ...
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### Are multiplicity-free representations weight multiplicity free?

It is a theorem of Brion and, independently, of Vinberg that varieties with an open $B$-orbit (a.k.a. spherical varieties) have in fact only finitely many orbits. A shorter argument is due to Matsuki (...
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### Three dimensional representations of Alternating group

Let $G_0$ be the image of $A_5$ under one of the $3$-dimensional representations, and $G = \pm G_0$. Then $G$ is the group of symmetries of the icosahedron, which is a Euclidean reflection group (...
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### Geometric interpretation of characteristic polynomial

I am reluctant to answer a question this old that already has a very nice answer, however, looking at the title the first thing that comes to my mind is something quite different from the existing ...
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### Invariant theory for parabolics

The map $\mathbb C[\mathfrak g]^G\to\mathbb C[\mathfrak g]^P$ is an isomorphism for trivial reasons: In any quasi-affine $G$-variety, $P$ and $G$ have the same fixed points. Just look at the orbit map ...
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### Separating closed $SO(p,q)$ orbits by invariant polynomials

The group $SO(p,q)$ is not per se an algebraic group. Rather there is an algebraic group $G$ such that $SO(p,q)=G(\mathbb R)$ is its group of real points. The main point is that also $G(\mathbb C)$ is ...
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### Basis of invariant tensors of rank n in three dimensions

Planar partitions with no singletons works. You need to pick for each $n>1$ some map with certain properties. One way to do this is to just fix a preferred trivalent tree of each size and ...
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### Explicit invariant of tensors nonvanishing on the diagonal

Let me start with some remarks about the classical symbolic method (without which one cannot understand 19th century invariant theory) and multisymmetric functions. I will use an example first. Take ...
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### If an equivariant map is smooth on diagonal matrices, is it smooth everywhere?

I think, one can argue as follows. Let $D\subseteq\text{Sym}$ be the diagonal matrices. Since $\exp:D\to D^+$ and $\exp:\text{Sym}\to\text{Sym}^+$ are compatible diffeomorphisms it suffices to answer ...
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