20
votes
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 ...
- 156k
16
votes
Accepted
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 $\mathbb{G}_m$. ...
- 148k
14
votes
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 ...
- 31.5k
14
votes
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 ...
- 15.5k
14
votes
Example of non homogenous manifold with a finitely generated algebra of natural functions
No. Here's a counterexample: Let $f(r)$ satisfy the equation $f'' + 2 f^3 = 0$ with the initial conditions $f(0)=1$ and $f'(0)=0$. Then $f$ satisfies $(f')^2+f^4=1$ and is periodic with period
$$
L ...
- 108k
13
votes
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.&...
- 2,851
13
votes
Accepted
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 ...
- 114k
12
votes
Accepted
Standard Monomial basis for other types
Standard monomial theory has been extended to all classical groups by Lakshmibai, Seshadri and others in the series of papers "Geometry of $G/P$ I-IX".
A very concise description of standard tableaux ...
- 15.5k
12
votes
Accepted
To describe an invariant trivector in dimension 8 geometrically
Here's another very nice (but still algebraic) interpretation that explains some of the geometry: Recall that $\operatorname{SL}(2,\mathbb{C})$ has a $2$-to-$1$ representation into $\operatorname{SL}(...
- 108k
11
votes
Accepted
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 ...
11
votes
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 ...
- 8,529
11
votes
Accepted
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 ...
- 15.5k
11
votes
Why is the catalecticant invariant under coordinate changes?
Let $d=2n$ be the degree of your binary form $f$.
Let me introduce $n+1$ pairs of formal variables $\alpha^{(1)}=(\alpha^{(1)}_{1},\alpha^{(1)}_{2}),\ldots, \alpha^{(n+1)}=(\alpha^{(n+1)}_{1},\alpha^{(...
11
votes
Accepted
Explicit formulas for invariants of binary quintic forms
Here's another way to do it that you might find useful:
Recall that $\mathrm{SL}(2,\mathbb{C})$
acts on the polynomial ring $\mathbb{C}[x,y]$
by linear substitution in $x$ and $y$,
making the ...
- 108k
11
votes
Accepted
Moduli of smooth curves in $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)| $ and their invariants
[EDITED to exhibit $j$ as a rational function of $J_2,J_3,J_4$,
and to fix various local errors etc.]
The action of ${\rm SL_2} \times {\rm SL_2}$ on the $9$-dimensional space of
$(2,2)$ forms has a ...
- 79.9k
11
votes
To describe an invariant trivector in dimension 8 geometrically
For a purely geometric construction, see further below, after the following algebraic considerations.
There is a Wronskian isomorphism which as a particular case says that the second exterior power of ...
11
votes
Has anyone researched additive analogues of toric geometry in characteristic zero?
Firstly $\mathbb{G}_a$ and $\mathbb{G}_m$ are definitely not isomorphic as group schemes even in characterstic $0$, as the exponential is not an algebraic map.
But there is a foundational paper on the ...
- 21.1k
10
votes
Chevalley–Shephard–Todd theorem
Torsten's argument is of course beautiful, but it might be worth recording that there is also a slick combinatorial argument, in case you need to teach this to students without algebraic geometry. (...
- 156k
10
votes
Accepted
Ring of invariants of $\operatorname{SL}_6$ acting on $\Lambda^3 \mathbb C^6$
The principal isotropy group is $H=SL(3)\times SL(3)$: it has the right dimension (namely 16) and occurs as an isotropy group (namely of a general element of $W$). Now it is a general result of Luna-...
- 15.5k
10
votes
Accepted
A duality result for Coxeter groups
Yes. By Chevalley-Shepard-Todd, $S(V)^G$ and $S(V)^H$ are polynomial rings. Let $S(V)^G=\mathbb{R}[g_1, \ldots, g_n]$ and $S(V)^H = \mathbb{R}[h_1,\ldots, h_n]$ where the $g_i$ and $h_i$ are ...
- 156k
10
votes
Accepted
Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$
Note added on 26 Nov 2018: I have corrected my answer, which had a serious mistake.
For simplicity of notation, let $(x,y,z) = (x_1,x_2,x_3)$. The Hessian form associated to $f_0 = {x_1}^3+{x_2}^3+{...
- 108k
10
votes
Accepted
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 ...
- 28.1k
10
votes
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Tensor algebra and universal enveloping algebra
The projection from the tensor algebra to the symmetric algebra is a split surjection. Therefore so is the map from $T(\mathfrak g)$ to $U(\mathfrak g)$, by the PBW theorem. Now note that PBW is an ...
- 40.2k
10
votes
Tensor algebra and universal enveloping algebra
To round things up, let's give a counterexample in positive characteristic.
Let $F=\mathbb{F}_2$ the field of two elements and let $L=F X\oplus FY$ be the Lie algebra with $[X,Y]=X$.
We write $\pi$ ...
- 460
10
votes
Accepted
Diagonal analogue of symmetric functions
Yes, these have been studied before. They were studied by MacMahon under the name "symmetric functions of several systems of quantities." Nowadays they are usually called MacMahon symmetric ...
- 17k
9
votes
Accepted
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 ...
- 15.5k
9
votes
Accepted
Invariant theory over $\mathbb R$
The answer depends on what you mean by "one-to-one correspondence". Is it bijective or just injective? Robert Bryant's (standard) argument shows that $\mathbb R^N/\mathrm{SO}(n)\to \mathrm{...
- 15.5k
9
votes
Invariants of $\mathrm{GL}_n$ representations
When $k=1$ the ring of invariants of degree $d$ separately in each of $W = \mathrm{Sym}^2 V$ and $W^\star = \mathrm{Sym^2} V^\star$ has dimension equal to the number of partitions of $d$ with at most $...
- 11.2k
8
votes
How to constructively/combinatorially prove Schur-Weyl duality?
look at
C. De Concini, C. Procesi, A characteristic free approach to invariant theory, Adv. Math. 21 (1976),
330–354.
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8
votes
How to constructively/combinatorially prove Schur-Weyl duality?
This a continuation of my first answer. I was trying to edit the previous one but the MathJax processing was freezing my computer. I suppose that answer was getting too long.
@Darij: There has been ...
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