Timeline for The dimension of the Grassmannian cohomology ring $H^*(\mathrm{Gr}_{n,d})$ and the fundamental $\frak{sl}_n$-representation $V_{\pi_d}$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2023 at 20:38 | comment | added | LSpice | Names of papers from @IgorMakhlin's comment: Nhok Tkhai Shon Ngo - $\operatorname{GL}_n$-structure and principal $\mathfrak{sl}_2$-triple on the cohomology ring of complex Grassmannian; Gatto and Salehyan - The Cohomology of the Grassmannian is a $\mathit{gl}_n$-module. | |
| Feb 14, 2023 at 16:32 | comment | added | Vladimir Dotsenko | @IgorMakhlin - interesting! I was not aware of these papers. Particularly arxiv.org/abs/2111.08754 might be a good candidate for a conceptual answer to the OP's question. | |
| Feb 14, 2023 at 15:48 | comment | added | Igor Makhlin | @Vladimir One can find papers such as this or this that seem to consider actions where root vectors act as derivations of the cohomology ring which shift the grading. This is kind of what I had in mind, not that I claim to have made sense of either paper. | |
| Feb 14, 2023 at 11:56 | comment | added | Vladimir Dotsenko | @IgorMakhlin the ring is graded, and the $sl_n$ action is going to mess it up, so something non-trivial will have to happen. I don't know what it could be. | |
| Feb 14, 2023 at 0:50 | comment | added | Igor Makhlin | Do you know of some nice natural way of defining a structure of an irreducible $\mathfrak {sl}_n$-module on this ring? AFAICT, the standard action induced from the Grassmannian is trivial. (Not sure if this was part of the question but kinda curious.) | |
| Feb 13, 2023 at 14:11 | comment | added | Vladimir Dotsenko | @SamHopkins here is a speculative idea. Coordinates of the Plücker embedding should separate different points. If we wish that the embedding is defined in a universal way, we perhaps should wish that it separates points over $\mathbb{F}_1$, which naively makes one think that the number of coordinates should be given by the number of $\mathbb{F}_1$-points, that is $\binom{n}{d}$. Someone who is well versed in geometry over $\mathbb{F}_1$ can say if this can be made rigorous, but as a heuristics I certainly find it rather useful. | |
| Feb 13, 2023 at 10:44 | comment | added | Sam Hopkins | I suppose so, yes. | |
| Feb 13, 2023 at 10:29 | vote | accept | Didier de Montblazon | ||
| Feb 13, 2023 at 10:00 | comment | added | Vladimir Dotsenko | @SamHopkins well, "the coordinate ring" is something that is not as well defined as the cohomology - you are presumably talking about the coordinate ring of the Plücker embedding - so your question is why the Plücker embedding exists, or not? | |
| Feb 13, 2023 at 3:23 | comment | added | Sam Hopkins | Is there a conceptual explanation for the relation with this dimension and the dimension of the linear part of the coordinate ring? | |
| Feb 13, 2023 at 2:40 | history | answered | Vladimir Dotsenko | CC BY-SA 4.0 |