Dynamics on Graphs 26

36 views Jun 1, 2023

We now enlarge the study of mutation frames, originally introduced in the video on E6, to consider what happens with a bipartite graph X when we start with a singleton weight (in the dual population space) and apply canonical sequences of even and odd mutations. These even and odd mutation matrices are created by taking the products of all mutations corresponding to even and odd vertices simultaneously, exploiting the fact that mutations for non-adjacent vertices always commute. The situations for small ADE graphs are explicitly worked out, including E6. We see that we obtain interesting generalizations of posets which we call partially ordered multisets, or pomsets, although our usage here is somewhat different from many occurrences in the literature. There are huge opportunities for amateur mathematicians to explore to what extent the remarkable properties of the pomsets we obtain generalize to more arbitrary bipartite graphs. Even the cases of E7 and E8 are quite fascinating challenges. Programmers have much to potentially engage with here. A pdf for this lecture can be found at https://www.dropbox.com/s/g8tpyeei7z8...A big thank you to all Members of this channel for your support, and also to my Patreon supporters. Video Contents More mutation frames for bipartite graph Example of E Product of mutation matrice Singleton weight Singleton weights at vertex Corresponding mutation frame with symmetric property

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More mutation frames for bipartite graphs

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Hello everyone I'm Norman Wildberger thanks for joining me on this adventure in Dynamics on graphs

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today more mutation frames richer bigger much more general by going to bipartite graphs which

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includes the ADE story but includes much more else besides and thinking about the the mutation and

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the numbers games actually we're concentrating on the numbers games today and creating mutation

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frames in a canonical fashion that's something that we can do in the bipartite graph situation

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and the the general story that we're outlining here has you know just massive um extensions

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possible so you can you can analyze what I'm doing here today I'm going to just go through a few of

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the ADE graphs but you can analyze what's going on for a very general bipartite graphs and what

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exactly happens in that story is is a a major mystery okay so huge amounts of exploration

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possible in this direction okay so what is a bipartite graph X well it's one whose vertices

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can be partitioned into 2 subsets say the even ones and the odd ones so that all the edges in

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the graph connect only vertices from the even set to the odd set okay so amongst the even vertices

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there are no edges amongst the odd vertices there are no edges so all trees are bipartite

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that's it's sort of an easy exercise and uh so for your example is E8 and here is the way of making

Example of E8

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it bipartite so let's say start with this 0 vertex call it say an even one and then the other even

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ones are going to be 2, 4 and 6 and the odd ones that are going to be 1, 3, 5, 7 and you can see

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that indeed all the edges are between even and odd vertices now since the vertices say in E are not

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neighbors none none of those are our neighbors it means that the products of any of the E mutations

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from those vertices will commute remember you have 2 vertices are not neighbors in the graph then the

Product of mutation matrices

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corresponding mutation matrices commute so if we take the product over all the vertices in say the

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even set of the mutations s sub x we're getting an element which is well defined in other words

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it doesn't depend on the order of the elements in the order in which we multiply those matrices

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right usually the order of matrices matter but since all of these sort of mutually commute it

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doesn't matter and we get a well-defined element which we can call s sub E and of course on the

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other side very similarly the product of all the s sub y's or y is in the V sub O the the odd

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vertices that forms also a well-defined element called s- O so what we're going to do now is

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we're going to think about creating these mutation frames um but I'm wanting to play this on the Dual

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side on the Dual mutation side in other words the numbers game so let's uh start with a basis lambda

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1 lambda 2 lambda n of the weight space W X so the weight space are just like populations okay but

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they're defined on the the version of the graph where we have open nodes instead of closed nodes

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that's what we mean by a weight and we'll assume that these lambda l's are singleton weights in

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other words they're they're dual populations which are equal to 0 generally except at the vertex

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I in question where there's a value of 1 okay so what we're going to do is we're going to start

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with 1 of these very simple sort of basis weights just and 1 vertex 0 elsewhere and we're going to

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sort of attempt to create an increasing sequence by mutating wherever there's positive values so

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when I say positive I mean greater than or equal to 0 so what we're going to see is

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that we're going to be able to do this sort of systematically so instead of just having

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to choose you know which ones we're going to be able to just mutate all of the even ones at the

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same time and then mutate all of the odd ones at the same time so we're really only applying these

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2 basic operators S E and S O that we've just defined and we're going to apply them alternately

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so we're only going to fire advertencies where the weight is positive

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okay so then we can record the dual mutations that do increase the population so we're going

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to end up with a partially ordered set but you're going to see that a very interesting new feature

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kind of is emerging here where we're recording not just the mutations that we're doing but also

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how much those mutations increase the population by the corresponding population on the other side

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and that's really just the same as what is the actual value of the weight at the um

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at the vertex where we're firing so what we're getting is something which I'd like to call a

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partially ordered multi-set or partially ordered mset and this is a combinatorial chapter which I

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think has largely been completely unexplored but we're going to see that we have really

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good cause to start thinking about partially ordered themselves so there's a whole world

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of carbonatorics okay here that we're going to essentially have to build up in order to

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understand the richness of what's going on Okay so the posets that we're creating here are similar

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to ones that we've seen before especially in the earlier video on E6 you may remember

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but here we're consistently starting with dual mutation so we're not starting with populations

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back there we started with sort of singleton populations now we're starting with these

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singleton weights okay now what I'm outlining here I think is is a great great place for

Singleton weights

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amateur mathematicians to contribute and it's also a really enticing arena for programmers

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because you know you can get a graph you can put a graph very concretely and you can sort of

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generate what's going on very algorithmically what we're doing here is very cut and dried we're just

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starting with the singleton weight and then we're performing these various s_E's and s_O's and then

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sort of recording what happens so this is perfect stuff for um programmers and I encourage some

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of you to to try that and especially to move not just from the ADE story but also into the general

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bipartite grass and try to explore what happens there okay so let's start with say A4 so here's

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A4 and we've divided it into 2 groups of vertices say odd ones even ones and let's start with

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um a Singleton weight say at this vertex 1 okay so we have a one 0,0 so that's our initial weight and

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I remind you these are open versus that's alerting us to the fact that this is a dual population

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okay so what are we going to do well this thing is only positive at s_1 okay and so we're going

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to apply s_1 but in fact we rather think about applying s odd okay in other words we're going

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to apply the product of s_1 and s_3 now s_3 doesn't do anything so the effect is the same

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as applying as well but I just want you to think about applying s odd and then s even

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and then s odd and then s even etc so if we apply s odd in other words s_1 we fire here so that 1

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turns into a minus 1 and the neighbor gets the 1 okay and then at this stage here the the positive

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elements are all in the even vertices 2 and 4 so we're going to apply s_E so we're going to mutate

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or fire actually at vertex 2 and at vertex 4. the vertex for firing doesn't do anything so it's just

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firing it at s_2 it turns that into a minus 1 the 1 there cancels with that minus 1 we get a

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0 and it moves over here to give us a 1 okay and then we do s_3 or in other words s_O again that 1

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turns into a minus 1 we also get a 1 over here and finally an s_4 this is a 1 here turns that

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into a minus 1 and that's a 0 and then there's 0 otherwise and at this stage here there are no

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more positive elements that are that are that are strictly positive so there's no way of increasing

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the corresponding population strictly increasing by any more firing so this is sort of where the

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story ends and now we record what the various uh firings that we've done are in this poke set

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so we've fired at vertex 1 and then the vertex 2 and then the vertex 3 and then at vertex 4 okay

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pretty simple and now here is uh the story where we're starting with the singleton weight at vertex

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2 okay at the 1 here so please check we fire s_2 we get this 1 and then we fire uh now this

Singleton weights at vertex 2

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is the first stage here where there's 2 positive elements so we you know so ordinarily we would

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say well we could fire at vertex 1 or vertex 3 but what we're doing here is we're firing them

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both sort of simultaneously because they commute with each other right so the effect here is just

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to go from here to here we've applied actually two mutations or dual mutations and that's really

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the same as applying this operator s_O the 1 turns into minus 1 this 1 turns into a minus 1 the 2 ,1

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contributed with this minus 1 to give us a plus 1 and this 1 over here gives us a

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1 there and then at the next stage we do an S sub even we get this thing here you can check

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and then finally an s sub odd and we get this 1 here and at this stage here there's no more

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strictly positive entries so things start so things finish and now if we record which

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firings we've done well we did the firing 2 okay and then at the next stage we did

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1 and 3 so that's this row here okay this this row 1 and 3 together after the 2 we did 1 and

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3 together and then after that we did 2 and 4 together and then after that we did 3 and here

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are the various relations uh between them so we get this mutation frame associated to this 1 here

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now the other 2 vertices are similar to to this 1 so there's no need to uh to do them separately

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okay so what about a 5 okay there's A5 divided up into 2 things so I'm just gonna list what you

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get you can just check it's sort of similar if we start with the singleton weight 1 then here

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is the mutation frame if we start with 1 at 2 we get this 1 here we start with 3 okay then

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we get this sort of rectangular thing it's a product of chains and at 4 symmetrically

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this 1 and then 5 symmetrically this 1 here and once you've done this you can kind of see what

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the picture is going to be for in general A sub n graph okay it's going to be some rectangular

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thing that looks like this in sort of an obvious way okay what about the D's next interesting

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situation well D4 is a great place to start in D4 the even vertices there's only 1 of them and the

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other 3 are all say odd so s_even just consists of s_2 and s_odd consists of s_0 s_1 and s_3 okay

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so here I'm showing you what happens with if we start with the singleton at 0 okay so you get 0

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and then the way to read here is we're going left to right and then we go down to the next line like

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this like this like this etc okay so this 1 here uh we start here we do an s_0 and then we do an

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s_2 or an s_E and then uh an s_0 which gives us a firing it so at 1 and 3 from here to here and

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then an s_E that firing a 2 and finally the firing s_0 to give us a 0 so we get this nice symmetric

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mutation frame and what about if we start with the singleton weight at 2 well at this stage

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something a little bit more interesting happens so we start with 2 okay there it is there we do

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the s_E so the 1 changes into a minus 1 we get ones all around and now we do an S 0 so we get

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minus ones on the outside but the 3 coming from them contribute with the minus 1 to give us a 2

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so I on our next firing okay we're going to do a firing of 2 but you see there's a 2 there

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up till now we've always been firing at places where there's a 1 so we can sort of record the

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fact that there's a 2 there so on the on the population side that would correspond to a

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bumping of the population by by by 2 not by 1 okay so we can sort of record that by doing something

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with this too maybe circling it okay so I'm sort of tentatively circling it there to alert us to

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the fact that right here we're doing a firing at a vertex which has a 2 in it okay after that we get

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to this 1 here and then we fire one 0, 3 get here and another 2 to get here and then things stop

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so again we get a nice symmetric thing but we are sort of moving into this situation where

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we actually have a partially ordered multi-set because there's actually a multiplicity here

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multiplicity of 2 for that Central object that central vertex 2.

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okay so um it's quite interesting to explore what happens say for D5 and D6 and try to get a sense

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of what happens for the general D n case and I'll leave that to you as an exercise a fun exercise

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to see with you can see what the frames appear that there are so I'm going to go now to E6 and

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here is E6 divided into even and odd vertices okay and if I start with a 1 over here I'm not

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going to go through the whole thing we're going to get something that we've already seen before

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it's this familiar beautiful kind of thing which we've called the bat on some other occasion and

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here you can go through it one by 1 if you want to check okay so that's great but another possibility

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is to start with a 1 at this position okay so again there's arrows here in the obvious way

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so we read left to right and then we go down here and over and then down okay so if we follow this

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thing here then this is what what you get and you see that at 1 stage here this 3 at vertex

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3 there's a 2 that's right at this stage here this stage here there's a 2 appearing at vertex 3

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and then after we've fired then on the next level there's also a 2 at vertex 4 so I'm

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circling Now sort of concretely vertex 3 and vertex 4. and you see that there's some more

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twos floating around so there's more things to circle okay so we're getting what's really not

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just a partially ordered set it's actually a partially ordered multi-set okay this is

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really interesting this is a combinatorial kind of object which has not seen enough attention

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but we're going to see that there's plenty of reason for us to start thinking about partially

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ordered multi-sets okay they're appearing very naturally here okay what's the next one to look at

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um uh it's where we start with the one say at that central vertex there okay there's

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the pattern you can check that I've done it correctly hopefully and here's the corresponding

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mutation frame that we get it has some elements of multiplicity 2 okay but interestingly it

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also has these 2 elements of multiplicity 3 okay the 3 Cube and that's just another way

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of writing it um so this this 3 cube means that correspondingly up here

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uh you can see some threes appearing okay there's a 3 there appearing 3 they're appearing etc

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and notice that this thing here even though it's a little bit complicated it still has the sort

Corresponding mutation frame with symmetric property

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of nice symmetric property uh that we're kind of used to and it actually has some

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other additional and important properties which we're gonna have to talk about uh more generally

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we'll have to learn a little bit and discuss a little bit more generally about properties

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of process so that we can explore more deeply uh the the interesting structure of the things

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that we're seeing here and there's one more in the E6 story where we start with the one

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up on on this side okay and uh and we just do that and what we get is is this 1 here

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a little bit simpler okay there's all kinds of interesting questions

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um naturally you'll want to know what happens for E7 and E8 and that's something that you can

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do yourself so you can generate these things yourselves and there's actually quite a lot

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of possibilities there right so for example for E8 there's eight possible starting points

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8 possible places where we could start with that initial singleton weight so there can

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be eight such diagrams okay and what are the properties of these diagrams you know I mean uh

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do they do they share the kinds of sort of symmetrical and pleasant features that that we're

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seeing here and and because all of this that I'm telling you now in principle at least potentially

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goes for a more general bipartite graphs we could ask you know choose some more general

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bipartite graph and what happens when we try to generate mutation frames this way so we're we're

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getting I think it's fair to say a remarkable combinatorial structures okay but they're not

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exactly posted so we have to enlarge our view of that that branch of combinatorics and there's all

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kinds of possibilities for exploration or what happens with more general bipartite graphs okay