Timeline for Representation ring of the symmetric group $S_n$ in the limit as $n \to \infty$
Current License: CC BY-SA 4.0
14 events
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| Dec 15, 2021 at 16:27 | comment | added | Nate | Not quite what you ask but: Character theory roughly says that Rep(G) is dual to the center of the group ring of G. On that side there is the Farahat-Higman ring, which is a sort of limit of the centers of the group algebras of S_n, and it has a pretty nice presentation. | |
| Dec 14, 2021 at 7:48 | history | edited | Saal Hardali | CC BY-SA 4.0 |
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| Dec 12, 2021 at 17:24 | comment | added | YCor | There exists a notion of finitely generated/presented profinite group. This projective limit also has a topology, for which we could have similar finiteness notions. | |
| Dec 12, 2021 at 16:40 | comment | added | David E Speyer | Related mathoverflow.net/questions/10735 | |
| Dec 12, 2021 at 16:21 | comment | added | Anton Mellit | Theory of FI modules is what you may be looking for arxiv.org/abs/1204.4533 . On the other hand, completion with respect to the augmentation, if tensored with $\mathbb{Q}$ becomes one-dimensional, so you don't get anything interesting in this way. | |
| Dec 12, 2021 at 16:00 | answer | added | Will Sawin | timeline score: 7 | |
| Dec 12, 2021 at 15:40 | history | edited | Saal Hardali | CC BY-SA 4.0 |
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| Dec 12, 2021 at 15:18 | history | edited | Saal Hardali | CC BY-SA 4.0 |
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| Dec 12, 2021 at 15:10 | comment | added | Saal Hardali | @WillSawin Good point! Silly me. I guess I might want to complete it somehow. Perhaps w.r.t. the augmentation ideal. | |
| Dec 12, 2021 at 15:08 | comment | added | Will Sawin | Given that this ring is uncountable (it's isomorphic, as a group, to $\prod_{n=1}^{\infty} \mathbb Z$) we would need uncountably many generators. | |
| Dec 12, 2021 at 15:07 | comment | added | Saal Hardali | @GeoffRobinson Can you be a bit more specific? | |
| Dec 12, 2021 at 14:49 | comment | added | Saal Hardali | @SamHopkins I believe this is the case if $S_n$ is replaced with polynomial representations of $GL_n$ (by Schur Weyl duality). I suspect the answer for symmetric groups (if it exists) should be quite different. | |
| Dec 12, 2021 at 14:48 | comment | added | Sam Hopkins | Do you not just get the famous “ring of symmetric functions” in the limit? | |
| Dec 12, 2021 at 14:44 | history | asked | Saal Hardali | CC BY-SA 4.0 |