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6
votes
1answer
521 views

When is a conjugacy class of matrices an embedded submanifold?

Let $M_{n\times n}$ denote the set of $n\times n$ real matrices and let $GL_n$ be the subgroup of invertible matrices. $GL_n$ acts on $M_{n\times n}$ smoothly by conjugation, which means that each ...
1
vote
2answers
1k views

Group action, Fixed point set and Orbit Space

I want to know to what extent is the group action determined by its fixed point data and orbit data, i.e. if $G$ acts on $M$ in two ways with the same fixed point set and orbit space, on what ...
8
votes
3answers
521 views

Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?

I want to know if every smooth (finite)group action on $\mathbb{R}^n$ is conjugate to some linear action.Thank you!
4
votes
1answer
268 views

Fixed points sets of pushouts

Let $G$ be a group and $X \to Y, X \to Z$ morphisms of $G$-sets with pushout $P=Y \cup_X Z$. Is then $P^G$ the pushout of $X^G \to Y^G, X^G \to Z^G$? This is not clear from general category theory, ...
5
votes
3answers
580 views

Hamiltonian circle actions and Lefschetz pencils

Suppose that $M$ is a symplectic manifold with a Hamiltonian circle action. Is there a topological Lefschetz pencil on $M$, $f\colon M-A \rightarrow S^2$, such that the fibers are symplectic ...
16
votes
4answers
2k 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 $\mathfrak{S}_n$ ...
2
votes
1answer
203 views

Do subgroups respect the orbit-closure relation?

Suppose G is a Lie group (or algebraic group) acting on a manifold (or scheme) X, and H⊆G is a subgroup. Let x,y∈X be points such that x is in the closure of the orbit H⋅y (but not in ...