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 […]

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 […]

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

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 […]

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 […]

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 […]

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 […]

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 […]

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?

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 […]

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