Generically multiple transitive algebraic group actions
| dc.creator | Popov, Vladimir L. | |
| dc.date | 2004-09-02 | |
| dc.date | 2005-03-05 | |
| dc.date.accessioned | 2026-07-07T07:49:32Z | |
| dc.date.available | 2026-07-07T07:49:32Z | |
| dc.description | With every nontrivial connected algebraic group $G$ we associate a positive integer ${\rm gtd}(G)$ called the generic transitivity degree of $G$ and equal to the maximal $n$ such that there is a nontrivial action of $G$ on an irreducible algebraic variety $X$ for which the diagonal action of $G$ on $X^n$ admits an open orbit. We show that ${\rm gtd}(G)\leqslant 2$ (respectively, ${\rm gtd}(G)=\nobreak 1$) for all solvable (respectively, nilpotent) $G$, and we calculate ${\rm gtd}(G)$ for all reductive $G$. We prove that if $G$ is nonabelian reductive, then the above maximal $n$ is attained for $X=G/P$ where $P$ is a proper maximal parabolic subgroup of $G$ (but not only for such homogeneous spaces of $G$). For every reductive $G$ and its proper maximal parabolic subgroup $P$, we find the maximal $r$ such that the diagonal action of $G$ (respectively, a Levi subgroup $L$ of ~$P$) on $(G/P)^r$ admits an open $G$-orbit (respectively, $L$-orbit). As an application, we obtain upper bounds for the multiplicities of trivial components in some tensor product decompositions. As another application, we classify all the pairs $(G, P)$ such that the action of $G$ on $(G/P)^3$ admits an open orbit, answering a question of {\sc M. Burger}. | |
| dc.description | Revised version, 30 pages | |
| dc.identifier | https://arxiv.org/abs/math/0409024 | |
| dc.identifier | http://arxiv.org/abs/math/0409024 | |
| dc.identifier | in: Proceedings of the International Colloquium ``Algebraic Groups and Homogeneous Spaces'', January 2004, Tata Inst. Fund. Research, Mumbai, India, Narosa Publ. House, 2007, pp. 481--523. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/124883 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14L30, 14L35, 14L40, 20G05 | |
| dc.title | Generically multiple transitive algebraic group actions | |
| dc.type | text |