Timeline for Understanding the Weyl Character Formula
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 14, 2021 at 14:45 | comment | added | Ben Webster♦ | @VítTuček For generic $\lambda$ in any semi-simple Lie algebra, the terms of the BGG resolution are all simple, and there are no non-trivial maps from one to any other. For $\lambda$ having integral inner product with some roots but not others, then yes, there are some maps, but not enough to make all the non-zero cohomology all concentrate in one degree (so "what goes wrong" is not exactness: it's just components of the maps stop existing). | |
| Jul 9, 2015 at 2:17 | vote | accept | EPS | ||
| Jun 25, 2015 at 9:39 | comment | added | Vít Tuček | Ah yes, because they are actually simple. In higher rank, do we at least get a "truncated" BGG resolution? | |
| Jun 24, 2015 at 10:48 | comment | added | Ben Webster♦ | @VítTuček No, there are really no maps. Think about $\mathfrak{sl}_2$. You have the Verma $M(n)$ and the Verma $M(-n-2)$. The space $Hom(M(-n-2),M(n)$ is 1-d if $n$ is an integer with $n\geq -1$, and 0 dimensional otherwise. | |
| Jun 23, 2015 at 12:14 | comment | added | Vít Tuček | Just to be clear, the maps between various Verma modules in the "wish to be BGG resolution" exists, but the problem is that one cannot cook up any differential that would give exact complex. Right? | |
| Jun 23, 2015 at 4:14 | history | answered | Ben Webster♦ | CC BY-SA 3.0 |