Timeline for Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Oct 25, 2022 at 1:52 | vote | accept | David E Speyer | ||
| Oct 16, 2022 at 0:45 | answer | added | Geoff Robinson | timeline score: 6 | |
| Oct 15, 2022 at 7:12 | answer | added | spin | timeline score: 4 | |
| Oct 15, 2022 at 7:09 | comment | added | spin | Have you compared your approach to that in "B. Bolt, T. G. Room and G. E. Wall, On the Clifford collineation, transform and similarity groups. I, J. Austral. Math. Soc., 2 (1961-62), 60-79." ? | |
| Oct 14, 2022 at 22:56 | comment | added | Geoff Robinson | @L Spice: that ${\rm Sp}(2n,p)$ is generated by elements of order prime to $p$ ( except when $p= 3$ and $n=1$), so it suffices to understand how automorphims of order prime to $p$ are represented. ( I had completed the comment, but was chided by the system for taking more than 5 minutes to edit,then ran out of steam. I may also complete a post I started to write ,if I get to the point that I think it sheds any new light. | |
| Oct 14, 2022 at 22:13 | comment | added | LSpice | @GeoffRobinson, that ...? | |
| Oct 14, 2022 at 20:56 | comment | added | Geoff Robinson | Your approach is more geometric than most I have seen (but I am an algebraist, after all), and is very clean. The idea of using unimodular matrices to induce the right automorphisms( especially automorphisms of order prime to $p$ in this context) has occurred incharacter theory and representation theory of finite groups to define "canonical"extensions which impose some rigidity which is useful in trying to extend further. This observation may have been made first by Marty Isaacs. For odd $p$, a key point is that | |
| Oct 14, 2022 at 6:00 | comment | added | spin | For the $\Gamma$ you have constructed, why is the map $\Gamma \rightarrow Sp(V)$ an isomorphism? | |
| Oct 14, 2022 at 3:55 | comment | added | LSpice | Specifically, have you compared your approach to that in Thomas - The Maslov index as a quadratic space? (I have to admit I'm just going on vague cosmetic similarities, and it may be off base.) Also, I am disappointed that this is so far the only question in weil-representation, but I hope it's not the last. | |
| Oct 14, 2022 at 3:54 | comment | added | LSpice | Could you give a reference for where you've seen some attempts to "elementarise" the description of the Weil representation? I feel like I've seen some fairly explicit descriptions for finite fields, but I can't think of any right now other than Gérardin's paper and Thomas's papers, and it would be good to know before I try to hunt down any others that you've already looked at them and that the approach is different. | |
| Oct 14, 2022 at 2:44 | comment | added | David E Speyer | @SamHopkins Thanks. I noticed that and decided it wasn't worth fixing but, now that you mention it, I have. Trying to use $k$ for the rank of the symplectic group now. | |
| Oct 14, 2022 at 2:44 | history | edited | David E Speyer | CC BY-SA 4.0 |
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| Oct 14, 2022 at 2:31 | history | edited | YCor | CC BY-SA 4.0 |
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| Oct 14, 2022 at 2:25 | history | edited | David E Speyer | CC BY-SA 4.0 |
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| Oct 14, 2022 at 2:05 | history | asked | David E Speyer | CC BY-SA 4.0 |