Timeline for Group of local complementation as a coxeter group
Current License: CC BY-SA 2.5
23 events
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| Apr 4, 2011 at 11:36 | comment | added | Johannes Hahn | @Mariano: I think the most natural way of looking at it is $lc_i\in Sym(\lbrace all\: graphs\rbrace)$. | |
| Apr 4, 2011 at 6:11 | comment | added | Mariano Suárez-Álvarez | Maybe you can be more explicit inthe question and make explicit the domain and codomain of the $lc$? I think I understand less now what you mean than before :) | |
| Apr 4, 2011 at 5:49 | comment | added | P.H. | @Tracy Hall .. Please see my comment to Mariano. I am sorry for the sloppy comment regarding connectivity of $i$ and $j$. | |
| Apr 4, 2011 at 5:43 | comment | added | P.H. | @Mariano Regarding your first question: sorry for the confusion, I meant the following: Let's say $lc_i$ and $lc_j$ were to act on a specific graph, $G_a$. Then if there exists an edge, $ E_{ij} $ between the two vertices $ i$ and $j$ of $G_a$ then $m_{ij}=3$. Otherwise it is $2$. Now consider the orbit that is generated by all the $\{lc_i|i=1,\cdots,n\}$ (i.e. All the simple graphs related by local complementation), as you say, then $lc_i$ induces an involution on these graphs. Now for $lc_i lc_j$ some of the graphs will have $i$ and $j$ connected and some don't. Hence the $m_{ij}=6$. | |
| Apr 4, 2011 at 4:02 | comment | added | Tracy Hall | Is it even clear that this is a group? If the order of $(lc_i lc_j)$ depends on whether $i$ and $j$ are adjacent, then it changes depending on the action of other generators. How precisely do you decide when a product of generators is trivial, if it sometimes results in the starting graph and sometimes doesn't? | |
| Apr 4, 2011 at 3:50 | comment | added | Mariano Suárez-Álvarez | ($2$-transitively on the generators, that is) | |
| Apr 4, 2011 at 3:48 | comment | added | Mariano Suárez-Álvarez | My point above is: Doesn't the symmetric group on the vertices act $2$-transitively on the group generated by the $lc_i$ by permuting the indices of the generators? It would then follow that all the numbers $m_{ij}$ are equal. | |
| Apr 4, 2011 at 3:43 | history | edited | P.H. | CC BY-SA 2.5 |
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| Apr 4, 2011 at 3:42 | comment | added | P.H. | @ Zaimi Actually that is a question we've wondering about too. These are however what we know: $lc$'s are reflections. The above relations are the only ones mentioned in literature and ones we work with. However the right answer is that we don't know! What we do know is the transformation on the adjacency matrix, $A_G$ of a graph, $G$: $lc_k: (A_G)_{ij} \rightarrow (A_G)_{ij}+(A_G)_{ik}(A_G)_{jk}+Diagonal[(A_G)_{ij}+(A_G)_{ik}(A_G)_{jk}]$. Where the addition is $mod(2)$. | |
| Apr 4, 2011 at 3:37 | comment | added | Mariano Suárez-Álvarez | The $lc_i$ are involutions on the set of all simple graphs on $n$ vertices, right? How does your first comment in this thread make sense? I mean: what does "$i$ and $j$ are disconnected" mean? | |
| Apr 4, 2011 at 3:25 | history | edited | P.H. | CC BY-SA 2.5 |
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| Apr 3, 2011 at 21:30 | comment | added | Mariano Suárez-Álvarez | Please add the clarifications Jim suggested to the body of the question itself. | |
| Apr 3, 2011 at 21:17 | comment | added | Gjergji Zaimi | @P.H. That is not what I asked. I asked if there are any other relations between $lc_i$'s, beside the ones mentioned. | |
| Apr 3, 2011 at 20:53 | comment | added | P.H. | @Holt that property is satisfied as I commented above. | |
| Apr 3, 2011 at 20:51 | history | edited | P.H. | CC BY-SA 2.5 |
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| Apr 3, 2011 at 20:50 | comment | added | P.H. | @Humphreys 1) Local complementation is as Hahn mentions 2) We have a set of generators of a group, which seem to obey the conditions for a coxeter set of generators. So we want to know what group we're dealing with. 3)I am not sure! For now we have only done some numerical work. And we are getting a pattern; only the product of the alternating group and or symmetric group is observed (see my question on "local complementation group of simple graphs" asked before). So we are wondering if there is some graph/group theory property which would clarify what we have. | |
| Apr 3, 2011 at 20:42 | comment | added | P.H. | @Zaimi they can easily be worked out. The $m_{ij}=2$ is for when $i=j$ or $i$ and $j$ are disconnected, and $m_{ij}=3$ when $i$ and $j$ are disconnected. So for a given $(lc_i lc_j)^{m_{ij}}=1$,$m_{ij}=2,3 or 6$. | |
| Apr 3, 2011 at 15:46 | comment | added | Jim Humphreys | Suggestions from a non-specialist: 1) Define "local complementation" for people closer to groups than to graphs, making the involutive property explicit. 2) Indicate what range of examples you've looked at so far, since at first sight there's no reason to expect that you always get Coxeter groups. 3) If there are Coxeter groups in the picture, would that have any immediate implications from the viewpoint of graph theory? | |
| Apr 3, 2011 at 13:28 | comment | added | Johannes Hahn | @Derek: If I understand it correctly a local complementation is the operation of selecting a vertex and changing the subgraph of this vertex and all its neighbors into its complement. Therefore local complementations are involutions by definition. | |
| Apr 3, 2011 at 9:42 | comment | added | Derek Holt | To be a (quotient of a) Coxeter group you would also need the relations $lc_i^2=1$ i.e. $m_{ii}=1$. | |
| Apr 3, 2011 at 4:40 | history | edited | user6976 | CC BY-SA 2.5 |
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| Apr 3, 2011 at 3:08 | comment | added | Gjergji Zaimi | How do you know that those are the only relations among the $lc_i$'s? | |
| Apr 3, 2011 at 2:22 | history | asked | P.H. | CC BY-SA 2.5 |