Timeline for Tensor product of simple representations
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 11, 2011 at 1:22 | comment | added | George McNinch | Dear Shenghao, first consider the case where $G$ has a faithful simple representation $L$. The unipotent radical $R$ of $G$ must have a non-0 fixed point in $L$, and $G$ leaves invariant the $R$-fixed points $L^R$. Since $L$ is simple for $G$, conclude $L = L^R$ so that $R$ acts trivially on $L$. Since $L$ is faithful for $G$, $R=1$. In the general case, can view $G$ as a subgroup of $\prod_i GL(L_i)$ for simple $G$-modules $L_i$. Now the image of $R$ in each $GL(L_i)$ is trivial, so indeed $R$ is trivial. | |
| Mar 10, 2011 at 23:43 | comment | added | shenghao | Dear George, thank you for providing the answer (which also solves my concern in earlier comment). Can you explain a bit more about why the unipotent radical of $G$ is trivial as soon as $G$ has a faithful semisimple rep.? | |
| Mar 10, 2011 at 23:40 | vote | accept | shenghao | ||
| Mar 10, 2011 at 0:23 | history | answered | George McNinch | CC BY-SA 2.5 |