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11
votes
1answer
257 views

Basis for $\Bbb Z[x_1,\cdots,x_n]$ over $\Bbb Z[e_1,\cdots,e_n]$

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 ...
3
votes
1answer
61 views

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 ...
2
votes
1answer
87 views

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 ...
1
vote
2answers
194 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 ...