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 […]

Intereting Posts

Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$
What is meant by “The Lie derivative commutes with contraction”?
Venn diagram for conditional probability property of Independent Events
Neumann series expansion for the resolvent
Asymptotic analysis of the integral $\int_0^1 \exp\{n (t+\log t) + \sqrt{n} wt\}\,dt$
Hilbert dual space (inequality and reflexivity)
Why this proof $0=1$ is wrong?(breakfast joke)
the approximation of $\log(266)$?
Injective functions with intermediate value property are continuous. Better proof?
Does every number not ending with zero have a multiple without zero digits at all?
Find all of the homomorphisms $ \varphi: \mathbb{Z}_{15} \to \mathbb{Z}_{6} $.
Justifying $\log{\frac{1}{P_{X}(x)}}$ as the measure of self information
Why can a quadratic equation have only 2 roots?
Dual of a dual cone
Difficulty in understanding a part in a proof from Stein and Shakarchi Fourier Analysis book.