Isomorphisms preserving invariants
| dc.creator | Schwarz, Gerald W. | |
| dc.date | 2008-04-21 | |
| dc.date | 2009-03-06 | |
| dc.date.accessioned | 2026-07-07T12:49:08Z | |
| dc.date.available | 2026-07-07T12:49:08Z | |
| dc.description | Let $V$ and $W$ be finite dimensional real vector spaces and let $G\subset\GL(V)$ and $H\subset\GL(W)$ be finite subgroups. Assume for simplicity that the actions contain no reflections. Let $Y$ and $Z$ denote the real algebraic varieties corresponding to $\R[V]^G$ and $\R[W]^H$, respectively. If $V$ and $W$ are quasi-isomorphic, i.e., if there is a linear isomorphism $L\colon V\to W$ such that $L$ sends $G$-orbits to $H$-orbits and $L\inv$ sends $H$-orbits to $G$-orbits, then $L$ induces an isomorphism of $Y$ and $Z$. Conversely, suppose that $f\colon Y\to Z$ is a germ of a diffeomorphism sending the origin of $Y$ to the origin of $Z$. Then we show that $V$ and $W$ are quasi-isomorphic, This result is closely related to a theorem of Strub \cite{Strub}, for which we give a new proof. We also give a new proof of a result of \cite{KrieglLosikMichor03} on lifting of biholomorphisms of quotient spaces. | |
| dc.description | Minor changes, 6 pages | |
| dc.identifier | https://arxiv.org/abs/0804.3363 | |
| dc.identifier | http://arxiv.org/abs/0804.3363 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/222300 | |
| dc.subject | Representation Theory | |
| dc.subject | Group Theory | |
| dc.subject | 20G20, 14L30 | |
| dc.title | Isomorphisms preserving invariants | |
| dc.type | text |