Timeline for Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 29 at 11:04 | vote | accept | Zoltan Fleishman | ||
| Mar 28 at 9:17 | comment | added | Dan Petersen | All irreducible algebraic representations of the symplectic group, full stop, are contained in some tensor power of $V_1$. | |
| Mar 28 at 8:24 | answer | added | Gro-Tsen | timeline score: 6 | |
| Mar 27 at 22:00 | comment | added | paul garrett | (Of course, this is only talking about finite-dimensional repns...) | |
| Mar 27 at 21:40 | comment | added | Zoltan Fleishman | @Gro-Tsen: Thanks a lot for the answer and reference. If you put it as an answer then I can accept it. | |
| Mar 27 at 21:27 | comment | added | Gro-Tsen | Ah, here's a reference: Fulton & Harris, Representation Theory: A First Course (Springer GTM 129), theorem 17.5. | |
| Mar 27 at 21:21 | comment | added | Gro-Tsen | Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial representation, then $\bigwedge^k V_1 = \bigoplus_{0\leq\ell\leq k,\;\ell\equiv k\pmod{2}} V_\ell$ (no multiplicities) for $0\leq k\leq n$. This is certainly well-known, but sadly I don't have a reference. | |
| S Mar 27 at 20:52 | review | First questions | |||
| Mar 27 at 20:58 | |||||
| S Mar 27 at 20:52 | history | asked | Zoltan Fleishman | CC BY-SA 4.0 |