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1answer
75 views

What is an example of a syzygy in invariant theory or pre-abstract algebra

I need an example involving the old invariant theory that pre-dated abstract algebra.
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0answers
103 views

Ring of invariants for $\Sigma_3$

I've just started reading about classical invariant theory and I'm not seeing how the general pattern should work, maybe it's obvious I don't know... Let $k$ be a field with $\mathrm{char}\ k = 0$, ...
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2answers
95 views

$GL_2(\mathbb{C})$-invariant ring for $M_2(\mathbb{C})\times M_2(\mathbb{C})$

For $\mathbb{C}^*$-action on $\mathbb{C}^2$ by $t\circ(x,y)=(t^{-1}x,ty)$, the ring of invariant polynomials is $\mathbb{C}[xy]$. For $\mathbb{C}^*$-action on $\mathbb{C}^4$ by ...
1
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1answer
48 views

Why is it that $\mathbb{C}[\{m_{ij}\}]^{G} \subseteq \mathbb{C}[\textrm{tr}(m),\ldots, \det(m)]$?

Let $G= GL(n,\mathbb{C})$ act on the set of all matrices $M_n$ by conjugation, i.e., for $g\in G$ and $m \in M_n$, $g\circ m = gmg^{-1}$. Then if $m=(m_{ij})$, then the $G$-invariant ring ...
4
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1answer
119 views

How do we know how many branches the inverse function of an elementary function has?

How do we know how many branches the inverse function of an elementary function has ? For instance Lambert W function. How do we know how many branches it has at e.g. $z=-0.5$ , $z=0$ , $z=0.5$ or ...
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0answers
58 views

How is the theory of vector invariants related to number theory?

Is there a nice theorem or toy problem that demonstrates the usefulness of invariant theory to tackle a number theoretic problem? I'm looking for some accessible insight into the motivation (for a ...
4
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2answers
119 views

Finite presentation of algebra of invariants

(1) Let $R$ be a ring, let $A$ be a finitely presented $R$-algebra, and let $G$ be a finite group of $R$-automorphisms of $A$. Is the algebra of invariant $A^G$ finitely presented over $R$? I can ...
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1answer
26 views

Is it possible to have a point $P_1$ not $\chi$-semistable but $P_2$ $\chi$-semistable with these two points in the same orbit?

Let $G$ be a group acting on an affine variety $X\subseteq \mathbb{A}_{\mathbb{C}}^n$. Suppose $P_1$ and $P_2$ are two points in $X$ such that $g\circ P_1=P_2$ for some $g\in G$. This means that ...
2
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0answers
56 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...
8
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1answer
154 views

Any affine algebraic group is linear.

It is a well-known result that any affine algebraic group is a closed subgroup of some $\mathrm{Gl}_n(\Bbbk)$. However, I would like to see a proof for that, so I looked it up in various books, more ...
4
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0answers
89 views

The anti-commutative Molien series

Suppose $V$ is a finite dimensional complex vector space and $f:V\to V$ is an automorphism. There is a natural extension $\Lambda^\bullet(f):\Lambda^\bullet(V)\to\Lambda^\bullet(V)$ to the exterior ...
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2answers
151 views

Diagonalizable linear algebraic group is isomorphic to $(\mathbb{C}^*)^r\times A$, for some finite abelian group $A$

I have three questions about algebraic groups. Let $D$ be a linear algebraic group. Then the following are equivalent: $D$ is diagonalizable. $\mathop{Hom}(D,\mathbb{C}^*)$ is finitely generated ...
2
votes
1answer
44 views

Give a example about invariant ergodic measure and quasi-symmetric mapping

Is there a example $(X,f,\mu)$ such that $X$ is a closed subset of Euclidean space, $f$ be a quasi-symmetric mapping but not a Lipschitz mapping, $f(X)=X$, $\mu$ is a finite measure on $X$ that is ...
2
votes
1answer
151 views

Identifying $k[x_1,x_2,y_1,y_2]^{\epsilon}$ with $k[x,y]\wedge k[x,y]$

Suppose the symmetric group $S_2$ of order 2 acts on $k^4=Spec \;k[x_1, x_2, y_1, y_2]$ by the following: for $\sigma\not=e$, $$\sigma\circ(x_1, x_2, y_1, y_2)=(x_2,x_1,y_2,y_1).$$ That is, the ...
4
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0answers
116 views

Basic semi-invariants

Let $G$ be a (finite) group and $\chi$ be a linear character corresponding to an irreducible representation. A polynomial $f_{\chi}$ is called semi-invariant (of type $\chi$) if $\sigma\circ ...
2
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2answers
59 views

Constructing the sequence: $0\rightarrow (x-y)^{S_2} \stackrel{f}{\rightarrow} k[x+y,xy]\stackrel{g}{\rightarrow} k[y]$

Let $S_2$, a group of two elements, act on $k[x,y]$ by permuting $x$ and $y$. It is clear that $$ 0\rightarrow (x-y) \rightarrow k[x,y]\rightarrow \dfrac{k[x,y]}{(x-y)}\cong k[y] \rightarrow 0 $$ ...
1
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0answers
61 views

$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...
2
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0answers
35 views

$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...
2
votes
0answers
118 views

GIT quotient for a certain torus action on an affine space

I'm reading various books and some notes and here is my question. Let $(\mathbb{C}^*)^2$ act on $\mathbb{C}^4$ by $$(\lambda_1,\lambda_2).(x_1, x_2, y_1,y_2)=(\lambda_1 x_1, \lambda_2 x_2, ...
1
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0answers
51 views

How to characterize the image of $PGL(V)$ in $\mathbb{P}(W)$ for an irreducible $GL(V)$-representation $W$

I'd like to ask two versions of my question, one a very specific case that I suspect may have a fairly easy and classical answer, the other the general case which I would like to find references on. ...
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0answers
124 views

What was Cayley's formula for the number of invariants? (Lost Formula!?)

I need to find Cayley's formula for the number of linearly independent invariants of homogenous polynomials. This is a combinatorial formula. He is believed to have discovered it in 1854. ...
3
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2answers
157 views

Quotient of an affine variety by a finite group coincides with topological quotient as a point set?

I have just read the construction of the quotient of a closed subset $X$ of affine space by a finite group $G$ of automorphisms of $X$, in Shafarevich, Basic Algebraic Geometry I. Shafarevich gives ...
3
votes
1answer
110 views

Invariant rings $\mathbb{C}[X,Y]^{GL_2}$ and $\mathbb{C}[X,Y]^{SL_2}$

I feel like I might have made a mistake on this question, and would appreciate some feedback from someone more experienced than me. If $G$ acts on $S$, we write $S^G = \{s \in S: gs = s \, \forall ...
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0answers
144 views

Understanding - G-invariants and concomitants

I am trying to understand a set of lecture notes which I have on representation theory, and I have come unstuck on the section about invariant theory. Suppose we have a finite dimensional vector space ...
2
votes
1answer
94 views

Does anyone know of Invariant Theory enough to comment on this question?

I am trying to find a minimal set of invariants for the binary homogenous form $$\displaystyle ax^7 + bx^{6}y + cx^{5}y^{2} + dx^{4}y^{3} + ex^{3}y^{4} + fx^{2}y^{5} + gxy^{6} + hy^{7}$$ What is the ...
1
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1answer
108 views

A condition in the definition of geometric quotient

I am reading the first several pages of GIT by Mumford, and I need some clarification of one requirement in the definition of geometric quotient (c.f. Definition 0.4, GIT): Suppose a group scheme ...
2
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0answers
121 views

How to compute the character of a matrix group operating on homogeneous polynomials?

I have a little problem in representation and/or invariant theory which I need help with. Let's assume $G \leq \mathbb{C}^{n\times n}$ is a finite complex matrix group which operates linearly via ...