Timeline for Modular representations
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 28, 2017 at 19:04 | answer | added | Nate | timeline score: 4 | |
| Feb 28, 2017 at 18:30 | comment | added | Jim Humphreys | @ L.Spice: Yes, the idea of reduction modulo $p$ starts with a suitable integral structure in an ordinary irreducible representation (say over a $p$-adic field). It is a basic but subtle fact proved by Brauer that every choice leads to the same composition factor multiplicities, which allows one to define decomposition numbers unambiguously. | |
| Feb 28, 2017 at 18:25 | answer | added | Jim Humphreys | timeline score: 3 | |
| Feb 28, 2017 at 18:21 | comment | added | LSpice | What is the reduction modulo $p$ of a complex representation? (Oh, I guess you're assuming that it preserves an integral structure?) | |
| Feb 28, 2017 at 18:05 | comment | added | Dipendra Prasad | For the moment I seem to have no more information on G. It is a certain Galois group, so I guess pretty arbitrary finite group. May be one should begin with just the first question: what forces there to be no nontrivial complex rep'n which has trivial rep'n in its reduction mod p? | |
| Feb 28, 2017 at 17:44 | comment | added | Jim Humphreys | For an arbitrary finite group $G$, I wouldn't be at all optimistic about finding so much information relative to an arbitrary prime $p$ dividing its order. Is there any special class of groups (and primes) you want to understand? For example, groups of Lie type for the defining characteristic $p$ can lead to very complicated situations; but some special cases are worked out. (Note too that your "semisimplification" translates into the decomposition matrix for a given finite group and prime.) | |
| Feb 28, 2017 at 17:33 | history | asked | Dipendra Prasad | CC BY-SA 3.0 |