Timeline for What is the sandpile torsor?
Current License: CC BY-SA 4.0
13 events
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| Feb 10, 2023 at 13:29 | history | edited | Carl-Fredrik Nyberg Brodda | CC BY-SA 4.0 |
Fixed broken link in [1]
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| Feb 10, 2023 at 12:24 | answer | added | Lilla Tothmeresz | timeline score: 3 | |
| Apr 2, 2022 at 1:16 | answer | added | Alex McDonough | timeline score: 6 | |
| Aug 15, 2013 at 3:25 | vote | accept | JSE | ||
| Aug 14, 2013 at 6:29 | answer | added | Tom Church | timeline score: 32 | |
| Aug 22, 2012 at 4:41 | comment | added | Richard Montgomery | If you apply your divisor $D(v)$ formally to a function $f$ on the graph, you get $(\Delta f)(v)$, the value of the graph's Laplacian $\Delta$ applied to $f$, evaluated at $v \in G$. I don't know where to go with this remark, beyond pointing out that there is an enormous literature on the graph Laplacian, with many analogues to the Laplacian in Riemannian geometry. See the book by Colin de Verdiere, Spectre des Graphes, for example. | |
| Mar 25, 2012 at 20:35 | comment | added | JSE | Dima -- yes, "torsor" and "principal homogeneous space" are synonyms, at least in my usage. | |
| Mar 25, 2012 at 5:44 | comment | added | Dima Pasechnik | something along these lines: each spanning tree $T$ defines a bijection $f_T$ from the set $\mathcal{T}$ of spanning trees to the sandpile group. Take the compositions $f_{T'}^{-1}\circ f_T$, for $T,T'\in\mathcal{T}$. They generate a permutation group on $\mathcal{T}$, isomorphic to the sandpile group. Maybe actually nobody proved anything this? | |
| Mar 25, 2012 at 5:37 | comment | added | Dima Pasechnik | I thought it is known that the sandpile group acts f.p.f. on the set of spanning trees, thus making it into a principal homogeneous space. Are the torsors you talk about the same objects as principal homogeneous spaces? | |
| Jan 15, 2012 at 17:40 | history | edited | JSE | CC BY-SA 3.0 |
added 1 characters in body
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| Dec 16, 2011 at 5:32 | comment | added | JSE | Cool -- I think I agree about C_4 (but didn't do it carefully enough to be sure) and then I didn't brave the 5-edge graph. The K_4 should be a very interesting example, since it has a big automorphism group to play with! | |
| Dec 16, 2011 at 5:13 | comment | added | Tom Church | I haven't made any progress on proving or disproving the more precise question --- but I sure had fun playing around with these permutations! I highly recommend it as a holiday diversion. If I haven't made any mistakes, the m.p.q. holds also for the 4-cycle C_4 and the square-with-diagonal K_4-minus-one-edge (at least with the usual cyclic ordering). | |
| Dec 15, 2011 at 19:08 | history | asked | JSE | CC BY-SA 3.0 |