Report
$G$-complete reducibility and saturation
| العنوان: | $G$-complete reducibility and saturation |
|---|---|
| المؤلفون: | Bate, Michael, Böhm, Sören, Litterick, Alastair, Martin, Benjamin, Roehrle, Gerhard |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Representation Theory, Mathematics - Group Theory, 20G15, 14L24 |
| الوصف: | Let $H \subseteq G$ be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic $p> 0$. In our first main theorem we show that if a closed subgroup $K$ of $H$ is $H$-completely reducible, then it is also $G$-completely reducible in the sense of Serre, under some restrictions on $p$, generalising the known case for $G = GL(V)$. Our proof uses R.W. Richardson's notion of reductive pairs to reduce to the $GL(V)$ case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if $K$ is $H$-completely reducible, then the saturation of $K$ in $G$ is completely reducible in the saturation of $H$ in $G$ (which is again a connected reductive subgroup of $G$), under suitable restrictions on $p$, again generalising the known instance for $G = GL(V)$. We also study saturation of finite subgroups of Lie type in $G$. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case $G = GL(V)$. Comment: 15 pages; v2 minor changes; v3 18 pages, various changes; new is Proposition 4.8 which shows that saturation is compatible with standard Frobenius endomorphisms; v4, 19 pages, introduction rewritten, substantial reorganization of material |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2401.16927 |
| رقم الانضمام: | edsarx.2401.16927 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |