Timeline for Periods of Coxeter transformation associated to root posets
Current License: CC BY-SA 4.0
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| Apr 5, 2021 at 18:55 | history | edited | F. C. |
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| Aug 30, 2020 at 14:45 | comment | added | Sam Hopkins | You're right. It's proved there in most cases, but not quite all. | |
| Aug 30, 2020 at 14:42 | comment | added | Mare | @SamHopkins Isnt this conjecture 5.5. in the article? | |
| Aug 30, 2020 at 14:35 | comment | added | Sam Hopkins | Another comment: the $h+1$/$2(h+1)$ order also appears in the paper of Yildirim you link to. She shows $\tau^{h+1}=\pm 1$, in the context of minuscule posets (rather than root posets). | |
| Aug 30, 2020 at 14:05 | comment | added | Mare | @SamHopkins That is a nice observation. I will try to see where it might come from in case this identity is true. | |
| Aug 30, 2020 at 14:02 | comment | added | Sam Hopkins | You can see Sage code here: cocalc.com/share/0efb266aca1eef099315e87e1750671ee5737323/… | |
| Aug 30, 2020 at 13:51 | comment | added | Sam Hopkins | Let $M:=-C^{-1}C^T$. Let $D$ denote the permutation matrix of the action of rowmotion. Thus $MD$ is lower-triangular by my previous comment (which is actually proved in the linked to MO question). It seems that we may always have that $(MD)^2$ is the identity. I feel like this must be related to your problem somehow. | |
| Aug 30, 2020 at 10:12 | history | edited | Mare | CC BY-SA 4.0 |
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| Aug 30, 2020 at 9:56 | history | edited | Mare | CC BY-SA 4.0 |
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| Aug 30, 2020 at 9:15 | history | edited | Mare | CC BY-SA 4.0 |
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| Aug 30, 2020 at 9:02 | comment | added | Mare | @SamHopkins I added more terms and some guesses for the types. For type $A_n$ and $D_n$ it seems to behave regular, while for type $B_n$ one might have to split into even and odd cases. | |
| Aug 30, 2020 at 9:01 | history | edited | Mare | CC BY-SA 4.0 |
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| Aug 30, 2020 at 8:52 | history | edited | Mare | CC BY-SA 4.0 |
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| Aug 30, 2020 at 8:41 | history | edited | Mare | CC BY-SA 4.0 |
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| Aug 30, 2020 at 3:41 | comment | added | Sam Hopkins | It seems the first nonnegative entry in the row of $-C^{-1}C^T$ indexed by $i \in L$ is $\mathrm{Row}(i)$, the rowmotion of the order ideal (a.k.a., the Panyushev map). Since root posets are known to have good behavior of rowmotion (see arxiv.org/abs/0711.3353 and arxiv.org/abs/1101.1277), this may be key to your observation (although note that the order of rowmotion is $2h$ or $h$, slightly different than what you see here). Rowmotion appeared in another related Q of yours: mathoverflow.net/questions/343203/… | |
| Aug 30, 2020 at 3:25 | history | edited | LSpice | CC BY-SA 4.0 |
Minor proofreading
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| Aug 30, 2020 at 0:17 | comment | added | Sam Hopkins | It seems that the period could be $h+1$ or $2(h+1)$. | |
| Aug 30, 2020 at 0:02 | history | edited | Mare | CC BY-SA 4.0 |
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| Aug 29, 2020 at 23:59 | history | edited | Mare | CC BY-SA 4.0 |
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| Aug 29, 2020 at 23:56 | comment | added | Mare | @SamHopkins For $B_3$ it is 7. | |
| Aug 29, 2020 at 23:54 | comment | added | Sam Hopkins | Is the period always $2*(h+1)$ where $h$ is the Coxeter number? | |
| Aug 29, 2020 at 23:48 | history | asked | Mare | CC BY-SA 4.0 |