Timeline for Innocent question on tensor products of modular representations
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2012 at 12:33 | comment | added | darij grinberg | Thanks a lot for the links and proofs (and Passman's paper, which is new to me, although most results in it are not)! | |
| Dec 28, 2012 at 5:24 | comment | added | Benjamin Steinberg | math.wisc.edu/~passman/balgebra.pdf is a good reference for the Hopf ideal result and the fact that each Hopf ideal in a group algebra is generated by the elements g-1 ranging over some g in some normal subgroup N. The moral is that the largest Hopf ideal contained on the radical is generated by g-1 with g in the p-radical. | |
| Dec 28, 2012 at 5:20 | comment | added | Benjamin Steinberg | The last map follows because g-1 is in the radical iff g is in the kernel of each irrep. By above this occurs iff g is in P. The kernel of $KG\rightarrow K[G/P]$ is generated by the elements g-1 with g in P. | |
| Dec 28, 2012 at 5:17 | comment | added | Benjamin Steinberg | There are 2 proofs on mathoverflow.net/questions/69039/… that every normal p-subgroup is contained in the kernel of each irrep. Now consider the regular rep of G. It can be written in block triangular form with the diagonal blocks irreducible reps. The kernel of the projection to the diagonal is precisely the intersection of the kernels of the irreps. But the kernel is unitriangular hence a p-group. | |
| Dec 28, 2012 at 2:54 | comment | added | darij grinberg | I also don't understand how you construct the last map in the sequence. | |
| Dec 28, 2012 at 2:51 | comment | added | darij grinberg | ... the proof of this fact that I know doesn't carry over.) | |
| Dec 28, 2012 at 2:51 | comment | added | darij grinberg | Oh, so what you call the $p$-radical is the $p$-core, as far as I understand. I fear I need some more proofs or references here. I've got a reference for the fact that completely reducible reps are closed under tensor products if and only if the Jacobson radical is a Hopf ideal (Satz 5.3 in Theresia Nolte's diploma thesis math.rwth-aachen.de/~Gerhard.Hiss/Students/… ). But I'm missing a proof that $P$ is the intersection of the kernels of all irreps of $G$ over $K$. (This generalizes the fact that all irreps of a $p$-group over $K$ are trivial, but ... | |
| Dec 27, 2012 at 20:55 | history | answered | Benjamin Steinberg | CC BY-SA 3.0 |