Timeline for Significance of half-sum of positive roots belonging to root lattice?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Dec 7, 2016 at 0:00 | vote | accept | Sam Hopkins | ||
| Dec 4, 2016 at 16:09 | answer | added | Jeffrey Adams | timeline score: 3 | |
| Dec 1, 2016 at 23:12 | comment | added | Nate | The root lattice is the subset of the weight lattice consisting of those weights which occur inside tensor powers of the adjoint representation. In particular $\rho$ appears as a weight in a tensor power of the adjoint representation iff you are in one of those types. Any chance your combinatorial phenomenon is some guise of iterated tensoring with the adjoint representation? | |
| Dec 1, 2016 at 19:01 | answer | added | Jim Humphreys | timeline score: 1 | |
| Nov 30, 2016 at 20:28 | comment | added | spin | So you could say that $\rho \in \mathbb{Z} \Delta$ if and only if the irreducible representation of highest weight $\rho = \sum_i \omega_i$ is a representation for the adjoint group. Which happens if and only if the kernel of this representation for the simply connected group is the center. But this observation is a bit trivial. | |
| Nov 30, 2016 at 18:40 | comment | added | Christian Stump | The later then immediately explains the difference between types $B$ and $C$, as one Cartan matrix is the transpose of the other. | |
| Nov 30, 2016 at 18:37 | comment | added | Christian Stump | One observation you likely know anyways: as this half-sum $\frac{1}{2}\sum_{\beta \in \Phi^+}\beta = \sum_{1 \leq i \leq rk(W)} \omega_i$ is the same as the sum of the fundamental weights $\{\omega_1,\ldots,\omega_{rk(W)}\}$, this is the same as saying that the diagonal of the weight lattice lies in the root lattice. Which is then again the same as saying that sum of the columns of the inverse Cartan matrix is an integer vector. | |
| Nov 30, 2016 at 18:22 | history | edited | Sam Hopkins | CC BY-SA 3.0 |
edited body
|
| Nov 30, 2016 at 17:58 | history | asked | Sam Hopkins | CC BY-SA 3.0 |