Timeline for Chip-firing clocks
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 12, 2021 at 19:21 | vote | accept | James Propp | ||
| Apr 6, 2021 at 3:14 | vote | accept | James Propp | ||
| Apr 12, 2021 at 19:21 | |||||
| Apr 5, 2021 at 17:45 | answer | added | Sam Hopkins | timeline score: 3 | |
| Apr 5, 2021 at 16:02 | comment | added | Sam Hopkins | @lambda: yes, I think you are right! | |
| Apr 5, 2021 at 15:16 | comment | added | Sam Hopkins | @Josh: If $H = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$ then, with $k=2$, $(v_1,v_2)\mapsto v_1$ is such a function $f$. | |
| Apr 5, 2021 at 15:12 | comment | added | Josh | James, do you have any example of such a function/graph that you are willing to share? | |
| Apr 5, 2021 at 14:07 | comment | added | lambda | Is this the same as a solution to $Hx \equiv \mathbf 1 \pmod k$? | |
| Apr 5, 2021 at 2:48 | answer | added | Will Sawin | timeline score: 3 | |
| Apr 4, 2021 at 23:12 | history | edited | James Propp | CC BY-SA 4.0 |
clarified that G is a directed graph
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| Apr 4, 2021 at 23:12 | comment | added | James Propp | Yes, I meant to write "directed graph". I'll fix that. | |
| Apr 4, 2021 at 21:56 | comment | added | Sam Hopkins | One more (rather trivial) observation: we need that $\alpha(v) = 0$ for any $v \in \mathrm{ker}(H)$. You say "outdegree-regular graph" which maybe suggests you are thinking of directed graphs. But if $G$ is undirected, then for instance one element of the kernel is the all $1$'s vector $\mathbf{1}$. And $\alpha(\mathbf{1}) = 0$ if and only if $k$ divides $n$. If $G$ is moreover connected then $\mathbf{1}$ generates the kernel of $H$ so this is the only requirement that we get from consideration of the kernel. But I don't see any reason why if $k$ divides $n$ then such an $f$ must exist. | |
| Apr 4, 2021 at 19:19 | comment | added | Sam Hopkins | If $\alpha$ were instead the zero morphism, then indeed these $f$ would just be maps from $\mathrm{coker}(H)$ to $\mathbb{Z}/k\mathbb{Z}$. | |
| Apr 4, 2021 at 19:18 | comment | added | Sam Hopkins | Another way to phrase this question in terms of lifts of group homomorphisms: we have the linear map $H\colon \mathbb{Z}^n\to \mathbb{Z}^n$, and the linear map $\alpha\colon\mathbb{Z}^n\to\mathbb{Z}/k\mathbb{Z}$ for which $\alpha(v)$ is the sum of its coordinates modulo $k$; then the $f$ in question are the lifts of $\alpha$ along $H$. But this doesn't really answer your question because I think in general questions about lifts of group homomorphisms are hard/might involve group cohomology. | |
| Apr 4, 2021 at 18:56 | history | asked | James Propp | CC BY-SA 4.0 |