Articles of invariant theory

Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. So according to the article $T$ should be $(U,V)\cdot X$ $UXV^T\in \mathbf{R}^{n\times n}$; therefore, $\mathbf{R}^{n\times n}$ is invariant under the group action. Why does the […]

Describe invariant polynomials under action of commutative group of order eight.

I believe the question below should be fairly standard in invariant theory ; I hope someone more familiar with it than me can explain a bit more or point to a reference. Let $F$ be polynomial field $F={\mathbb R}(X_1,X_2,X_3, \ldots ,X_8)$. Denote by $S$ the group of permutations on $\lbrace X_1,X_2,X_3, \ldots ,X_8\rbrace$, which acts […]

finitely generated over invariants

I was wondering if there are some conditions we can add to the following statement to make it true “The ring $R$ is finitely generated as a module over $R^G$, where $G$ is a finite group and $R^G$ the ring of invariants”. Many thanks

Invariant Factors vs. Elementary Divisors

I have been studying Cooperstein’s Advanced Linear Algebra for about seven months now and I am having problems understanding how to find the elementary divisors of a linear operator and how to find the invariant factors of a linear operator as well. I feel as though if I’m having trouble with these then I won’t […]

Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite dimensional $G$-invariant linear subspace of $A$.) By Nagata’s theorem, $A^G$ is a f.g. $k$-algebra. We have the affine GIT quotiont […]

Schur-Weyl Duality ( Classical ) and the Double Commutant reference request

I would like to ask for any reference suggestions on the topic of Schur-Weyl Duality for GLn ( directly GLn, not through the lie algebra ) and the double commutant theorem. The section on this material from Fulton and Harris’ “Representation Theory” book should suit my needs, but I find the proofs in the book […]

Geometric intuition for linearizing algebraic group action

I am reading an overview of geometric invariant theory and find myself stuck when we begin linearizing the action of an algebraic group on a variety. The definition given in my notes is that given an action $\sigma: G\times X\rightarrow X$ of an algebraic group $G$ and a line bundle $\pi:L\rightarrow X$, a linearization of […]

The disease problem

Students are sitting in a n * n grid. There’s a disease spreading among them in a particular fashion. At start, there a ‘k’ students infected(At random). After every time step(equal intervals), the number of students infected, increase. The manner of increment after each time step is – The students that were previously infected, are […]

How does Molien series describe polynomial invariants?

As I understood from wiki page, Given a finite group acting on a vector space, Molien series gives a generating function, although I am not sure what this means. And how is this related to the polyomial invariants of the group? Could anyone help me here?

Basis for $\Bbb Z$ over $\Bbb Z$

I’m reading the introductory bits in Procesi’s Lie Groups, and on p. 22 we have (paraphrasing) Theorem 2. $\mathcal{B}=\{x_1^{\large h_1}\cdots x_n^{\large h_n}: 0\le h_k\le n-k\}$ is a basis for the ring $\Bbb Z[x_1,\cdots,x_n]$ considered over $\Bbb Z[e_1,\cdots,e_n]$, where $e_i$ are the elementary symmetric polynomials in the $x_i$. I haven’t been able to see why this […]