All Questions
Tagged with ac.commutative-algebra group-actions
10 questions
2
votes
0
answers
92
views
Action of torus on Laurent polynomials
Let $F$ be a field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for suitable scalars $a_i$.
...
- 376
3
votes
1
answer
180
views
How to find equations of $\mathbb{C}^*$-curves
Fix positive integers $t_1,t_2,t_3$.
Suppose we have a $\mathbb{C}^*$ action on $\mathbb{C}^3\setminus\{0\}$ defined by
$$\mathbb{C}^* \times \mathbb{C}^3 \setminus \{0\} \to \mathbb{C}^3 \setminus \{...
- 1,981
4
votes
2
answers
391
views
Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$
For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$.
For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$...
- 1,209
4
votes
1
answer
108
views
Sufficient conditions for secondary invariants
Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension ...
- 768
0
votes
0
answers
242
views
Quotient by augmentation ideal
Let $p$ be a prime number. Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers and let $R = \mathbb{Z}_p [[X_1, \ldots, X_n]] / (f_1, \ldots, f_d)$.
Assume that a finite abelian group $G$ of order ...
- 183
0
votes
0
answers
79
views
Stable analytic manifold under simple action
For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m} - 1
$$
on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
- 1
2
votes
0
answers
264
views
2 questions on Nagata's counterexample; $k[f_1,...,f_r]=k[g_1,...,g_s]$ vs. $k(f_1,...,f_r)=k(g_1,...,g_s)$
Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...
- 339
8
votes
1
answer
434
views
Free group actions on varieties and algebras of coinvariants
Suppose $k$ is an algebraically closed field of characteristic zero and $A$ is a finitely generated commutative associative reduced $k$-algebra.
Suppose the group $\mathbb{Z}_2$ acts on $A$ in such a ...
- 618
14
votes
2
answers
1k
views
Symmetric group action on squarefree polynomials
The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest.
Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ...
- 15.5k
32
votes
7
answers
5k
views
Invariant polynomials under a group action (hidden GIT)
Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$).
Now the symmetric group $\mathfrak{S}_n$ ...
- 1,993