Timeline for LGV scheme for lattice paths that move in non-unit spatial positive steps
Current License: CC BY-SA 4.0
27 events
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| Nov 30, 2022 at 21:51 | vote | accept | Thomas Kojar | ||
| Aug 22, 2022 at 2:06 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
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| Apr 29, 2021 at 0:05 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
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| Nov 29, 2020 at 23:55 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 29, 2020 at 23:49 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 29, 2020 at 22:30 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 29, 2020 at 16:28 | answer | added | Sam Hopkins | timeline score: 1 | |
| Nov 29, 2020 at 11:51 | comment | added | darij grinberg | The Kim paper doesn't talk about lattice paths, but if you try to prove the Jacobi-Trudi formula he proves there through LGV instead, you'll naturally get lattice paths with jumping past points. His previous paper actually uses this kind of argument. The reason why it doesn't explicitly appear anywhere is that no one has made it to work directly. | |
| Nov 29, 2020 at 6:28 | comment | added | Thomas Kojar | @darijgrinberg I should mention the work "Vertex models, TASEP and Grothendieck polynomials", where they relate dual Grothendieck polynomials and TASEP (from our setting). So maybe there are some connections after all. | |
| Nov 29, 2020 at 6:01 | comment | added | Thomas Kojar | So from what I understand this idea of "jumping past nodes" has not been formalized and it has only made its presence in various settings. If more people want to share too, it would be very helpful for our field where the presence of the LGV-analogy is still wrapped in mystery. | |
| Nov 29, 2020 at 5:58 | comment | added | Thomas Kojar | That 3d-LGV diagram is beautiful. The formulation is still a bit hard but I could see how a projection of 3d lattice paths can generate 2d lattice paths that jump past nodes depending on which plane you project on. | |
| Nov 29, 2020 at 5:55 | comment | added | Thomas Kojar | @darijgrinberg thank you. At least in the Kim paper, can you point me to some part where the similar issue happens of "jumping past nodes" in the language of partitions as in Kim's paper ? | |
| Nov 28, 2020 at 5:14 | answer | added | Amritanshu Prasad | timeline score: 0 | |
| Nov 28, 2020 at 0:19 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 27, 2020 at 23:29 | comment | added | darij grinberg | Ah, much better! So you're asking for an explanation of results that look like they follow from LGV but don't because paths can jump past nodes. Something similar is secretly happening with the Jacobi-Trudi-like formulas for dual stable Grothendieck polynomials ( arxiv.org/abs/2008.12000 ), at least if you try to approach them the LGV way. Amanov and Yeliussizov have a proof of one of these formulas ( arxiv.org/abs/2003.03907 ) that uses some kind of more sophisticated variant of LGV (too sophisticated for me, I'm afraid). | |
| Nov 27, 2020 at 23:20 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 27, 2020 at 20:49 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 27, 2020 at 11:33 | comment | added | darij grinberg | Where are you seeing versions of LGV in the references you've cited? | |
| Nov 27, 2020 at 6:35 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 27, 2020 at 6:30 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 27, 2020 at 5:55 | comment | added | Sam Hopkins | The usual formulation of LGV I know of does not involve "time" in any way. It just concerns tuples of nointersecting paths in directed graphs (usually planar directed graphs, to be most useful). Maybe you can say more. | |
| Nov 27, 2020 at 5:53 | history | edited | Thomas Kojar | CC BY-SA 4.0 |
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| Nov 27, 2020 at 5:46 | history | asked | Thomas Kojar | CC BY-SA 4.0 |