Questions tagged [classical-invariant-theory]
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67
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Is $k[X]^G$ integral closed in $k[X]$.
May assume field $k=\mathbb{C}$.
Let $X$ be an affine variety and $G$ be a reductive group (may assume connected).
Is the ring of invariants $k[X]^G$ integral closed in $k[X]$?
The claim may not ...
- 157
3
votes
1
answer
307
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When the affine quotient is faithfully flat?
It may be easy for the expert.
Consider the map from $n$ by $m$ matrices (over $\mathbb{C}$ )to the $n$ by $n$ symmetric matrices $\phi\colon A\mapsto A A^T$.
My question is when this map is ...
- 157
11
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5
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area of triangle from coefficients of its cubic?
Three points $z_1$, $z_2$, $z_3$ on the complex plane are given by the coefficients $a_k$'s of the cubic polynomial $f(z)=(z-z_1)(z-z_2)(z-z_3)=\sum_{k=0}^3 a_k z^k$. How does one express the (signed)...
- 13.7k
2
votes
1
answer
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Transformation of a cubic form
How can I change an integral binary form
$ax^3+bx^2y+cxy^2+dy^3$ with the usual discriminant $D =b^2c^2-27a^2d^2+18abcd-4ac^3-4b^3d $
into a form $ax^3+dy^3$ which has a simple discriminant $-27a^2d^2$...
- 21
9
votes
2
answers
741
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Classical invariants involving exterior powers of standard representation
While investigating certain conformal blocks line bundles on $\overline{M}_{0,n}$, I was led to what seems to be an identification between two spaces of invariants, and I am curious if there is a ...
- 1,871
5
votes
2
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799
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Classical invariant theory: absolute rational invariants and $GL(2)$-orbits
I have a question concerning classical invariant theory. Consider binary $n$-forms (i.e. all homogeneous polynomials of degree $n$ of two variables) over the field of complex numbers. Clearly, the ...
- 103
21
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3
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What theorem of Liouville's is Gian-Carlo Rota referring to here?
I am very curious about this remark in Lesson Four of Rota's talk, Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations:
"For second order linear differential ...
- 415
5
votes
1
answer
492
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The ring of SL_2 invariants in sums of conjugation and tautological modules
Rings of Invariants
Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free ...
- 12.7k
8
votes
1
answer
513
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The exceptional Lie algebra $\mathfrak{g}_2$ and binary cubics
How is the exceptional 14-dimensional Lie algebra $\mathfrak{g}_2$ related to the covariant algebra for the binary cubic?
Here are some details on this question. This algebra is generated by 4 forms,...
- 1,267
10
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4
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Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...
- 3,453
7
votes
2
answers
539
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Generators for the algebra of GL(n)-equivariant maps from M_n + M_n to M_n
Let $M_n$ be the set of $n$-by-$n$ matrices with complex entries, viewed as a variety over $k=\mathbb{C}$. Equip $M_n$ with the conjugation action of $\mathrm{GL}(n)=\mathrm{GL}(n,\mathbb{C})$. ...
user9154
5
votes
2
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457
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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 ...
- 33.2k
5
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2
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1k
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Howe duality for exceptional algebras
There is a nice tool in representation theory, the Howe duality, which as I know works for certain pairs of classical Lie algebras (the reference to the complete list of Howe dual pairs is appreciated ...
- 605
11
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1
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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 ...
- 1,987
4
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3
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340
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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 ...
- 4,970
9
votes
4
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871
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A ring of invariants in characteristic 2
Let $K$ be an algebraic closure of $\mathbb{F}_2$. The cyclic group $C_{2^n}$ acts on $K[x_0, \dots, x_{2^n-1}]$ by cyclically permuting the $x_i$: $a : x_i \rightarrow x_{i + a \bmod 2^n}$. Is ...
- 4,626
8
votes
2
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690
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Enumeration of graphs arising in invariant theory
I've been working on a talk based on some stuff in Olver's "Classical Invariant Theory" book and have been wondering about a related graph enumeration problem.
Start with a triple $(n,v,e)$ of ...
- 15.8k