Dynamics on Graphs 29

Dynamics on Graphs 29 | More D5 mutation, root and firing frames | Wild Egg Maths

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Frames are combinatorial objects associated to sequences of mutations or firings on a graph that (strictly) increase populations. On the Mutation Game side, these are called mutation frames or root frames, as they move us from positive root to positive root by mutations, starting with a singleton population. On the Numbers Game side, these are called firing frames, and they start with a singleton weight, namely a dual population supported just on one vertex.

These objects are typically posets, or partially ordered sets, or better yet pomsets, or partially ordered multisets. In fact they provide excellent motivation to start studying pomsets more generally. While the rich theory of posets has been a mainstay of modern combinatorics, the corresponding multiset version of pomsets has not gotten very much attention at all. That might be about to change!

The pomsets that we obtain on the Numbers Game side, namely the firing frames, have quite remarkable symmetry properties. Even more surprising is that they end up playing a very big role: perhaps even a crucial role (!) in the representation theory of simple Lie algebras and simple Lie groups. This is a development that we hope to say a lot more about in the future.

In this video we focus on the various possibilities for the ADE graph called D_5, which is associated to the Lie group SO(10). It plays an important role in various grand unified theories (GUTs) of particle physics which try to unify quarks and leptons. Some of these theories also address the asymmetry between matter and anti-matter in the universe.

Amateur researchers can have lots of fun investigating the analogs of these structures for not just the other ADE graphs, but also more general graphs.

0:01

Hello everyone I'm Norman Wildberger thanks for joining me on this adventure we are looking

0:06

at Dynamics on Graphs we've talked about various frames combinatorial objects that we can associate

0:15

with certain graphs by playing both the mutation and the numbers games and especially for the

0:23

Ade graphs the combinatorial structures that we thereby create turn out to be a hugely interesting

0:31

they connect with interesting counter torques they connect with interesting physics and it turns out

0:39

turns out rather surprisingly that they become key objects in trying to understand the representation

0:44

theory of the associated Lie algebras and Lie groups that's a very big story which I hope to

0:52

sort of explain but the first order of business is to become more familiar with this rich variety

0:58

of these combinatorial objects they are partial ordered sets often they are lattices distributive

1:05

lattices in fact and in fact we'll want to sort of go a little bit beyond that theory and think

1:12

about partially ordered multi-sets okay actually that concept will play an important role we'll

1:19

start seeing that as we go so today I want to illustrate what's going on by looking at D_5 which

1:26

is not really small but not as big as the E_n graphs and D_5 is a pleasant kind of example we

1:35

can work through basically what happens with with all the stories we've already talked a little bit

1:41

about D_5 so I'm not going to include the ones that we've already talked about but I'm going

1:44

to look at the vertices that we haven't talked about before and look at the mutation frames the

1:49

root frames or root layers and the firing frames that we can associate to each of the vertices and

1:57

we're thinking about the mutation game acting on populations okay usually represented by a

2:05

Vertex with closed nodes and also the numbers game playing on weights okay sometimes we refer

2:13

to those as firings or dual mutations and that will typically be denoted by these graphs with a

2:19

open vertices okay so there's sort of two aspects that we want to cover here so let's start with a

2:29

Singleton population okay so we're on this side here where there's just a Singleton at the vertex

2:35

labeled two so here's the vertex labeling zero one two three four so we have a single vertex

2:41

population rate at one there and the associated weight okay which I remind you uh captures how

2:51

much change there is at each vertex if we did a corresponding mutation at that vertex to this

2:57

particular population so if we mutated a Vertex one here there'd be a change of one similarity

3:03

also vertex zero or vertex three that's why there's ones here and if we actually mutated

3:08

at the vertex two from this position the one would change to a minus one that's the change of

3:12

negative two denoted by this two bar here okay and mutating at this Vertex 4 is not going to

3:19

change anything so there's a zero I don't always record okay and so this correspondence between a

3:26

population and the associated weight that's sort of our PSI function um that's what we're going to

3:32

record actually simultaneously here so you can sort of see what's happening the advantage of

3:37

looking at things on the weight side is that it becomes very clear which mutations will increase

3:43

the total population that's generally what we're looking at we're looking at starting

3:47

with a single and then increasing the population and looking at all the possible ways that we can

3:52

do that okay and having the the weight diagram in front of us makes that easy because it's the

3:59

strictly positive entries there that are telling us which mutations will increase strictly increase

4:05

the population so for example at this stage here at the very first stage we can see that there's a

4:11

one one here these three vertices, so s0 11 or s3 those three mutations will increase the population

4:19

now since these are not adjacent these mutations actually commute and so we can sort of all do

4:25

them all at once all right so that's what we're doing over here we're doing all three mutations

4:31

at once so we get a one there a one there and a one there and then the corresponding weight

4:36

over here changes in the following way the ones changed to minus ones and each of the neighbors

4:43

of each of the things that we fired at sends a copy of its population to those neighbors

4:49

so each of these ones contributes a one to the central vertex so that goes from being negative

4:55

two to one and also this one here sends one to its neighbor so there ends up being a one over there

5:02

now at the stage we can see that we can do an s2 or an s4 and that's recorded here and you

5:07

can go through just check that what I've done after the s2s4 then we can do a single s3 and

5:16

then after we've done the s3 then the weight becomes like this and there are no strictly

5:21

positive entries anymore telling us that no more mutations are going to increase the population

5:27

that's so sort of a maximal population which is this one here okay that's a relatively Simple

5:33

Story and if we record the various mutations that we did partially ordered set recognizing

5:41

that some of them have to come before the others but others commute then we get a one zero three up

5:46

at the top and then a two and a four and then a three so this is a little post set we call

5:51

this a mutation frame M plus D5 to the graph and the vertex that we're starting at is vertex two

5:59

Associated to that is a certain positive root layer or frame and that's this one here

6:06

obtained by starting with the Singleton population that we've started with and doing all these

6:13

various possible mutations but now sort of at all possible orders so if we just do a mutation one we

6:19

get to this one if we just do a mutation three we get to this one we do notation zero we get to this

6:24

one we do a one and then followed by a zero well then we get this one over here okay so we get this

6:29

um so a slice of all the positive Roots basically the positive roots that you can get to by

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increasing starting from this particular single tune this one up here at the top okay and we call

6:45

this slice of the positive Roots R plus D5 2. and this is a very interesting thing it's a lattice

6:54

and I've circled in red the join irreducibles in this lattice

7:00

are the elements which are not joins of two or more other ones and also which is not the the

7:08

initial one the initial one is not included so you can think about these ones as being the elements

7:14

for which there's sort of one Edge coming in from from the top and there's these uh six

7:21

and you see that they form a partially ordered set which turns out to be in fact if you think

7:27

about it uh pretty well the same as this one here structurally okay there's three guys up on

7:32

top those three there and there's this one over here and this one over here which is sort of um

7:41

this one here arises from having all three of them okay and then uh there's this one here in

7:47

the bottom and what we're seeing here is sort of Birkhoff's law the join irreducible is there

7:54

um connect the this lattice with this prior simpler poset the lattice is just the lattice of

8:04

ideals of the mutation frame

8:11

and so each element of R plus D5 2 corresponds to an ideal of the mutation frame and for example if

8:19

we looked in R plus D5 2 this particular element this positive root okay which is that one right

8:26

there okay it corresponds to which ideal well you just look at all the joint irreducibles

8:33

which are above it in this in this story you see these four over here and so it corresponds

8:39

over here to the ideal consisting of these four elements here the one the zero the three is four

8:46

that's that's the way it works it's a beautiful thing

8:53

now on the other side so still thinking about that vertex too there's something else that we

9:01

can do starting with the vertex two which is to start with the Singleton weight not a Singleton

9:06

population just a Singleton weight okay that's not the corresponding thing to the Singleton

9:12

population right this Singleton population had corresponding to it this rather complicated weight

9:18

what we're doing now is saying let's not consider this one just for a second and just consider the

9:22

weight that has a Singleton at two there it is right there let's call it Lambda 2. okay and then

9:31

we can sort of do the same kind of thing to it we can look at the frame that we get by mutating in

9:36

in the same kind of fashion still only mutating at places where there are strictly positive entries

9:44

so the first one there's only one strictly positive entry at this vertex two so we could

9:48

do an s2 s2 will send that one to a minus one and ones to all the neighbors then at this stage there

9:55

are three strictly positive entries this one this one this one so we can do any of them or all of

10:00

them in fact if we do all of them then we get this situation here and notice that what happens is

10:05

that the ones coming from these three change this minus one to a plus two so we get a two appearing

10:12

that's of of Interest and then from there the positive ones are this 2 and this one so we

10:20

can do an s2 or an s4 to get this one here and then here you can do an s1 an s0 or an s3 and

10:33

uh the effect of that will be this please check all these computations and then at

10:38

this stage we do an s2 and an s4 I get this one and then we can do an s0 and s1 s3 again

10:46

from here and then finally at this stage here there's only an s2 to do and after we've done that

10:52

then there are no more strictly positive entries okay and then if we go back to what we've done

10:59

here and sort of record all these various things in a post set we get what we're calling the firing

11:06

post set or firing frame and f plus D5 2 and there it is there and it's a little bit interesting

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because there are a couple of twos that occurred okay and we denote those record those by putting

11:23

a little circle around those things so this circle around the two means there's actually

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a multiplicity of two happening there so strictly speaking this is going a little bit beyond the

11:33

usual post-set world okay this is really in the world of partially ordered multi-sets

11:40

so that's pretty interesting from a commentator out point of view because those things have gotten

11:45

far less attention than partially ordered sets and so there's some interesting questions to ask about

11:53

you know how does how does that go in any case we admire the the symmetry of this thing it's a it's

12:00

a beautiful kind of thing and it has an obvious sort of symmetry property and it has a somewhat

12:05

sort of a spindle shape it's kind of getting bigger as we go towards the middle yeah roughly

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and um okay so that's a pleasant thing and we see in particular uh in connection with

12:18

the previous examples that this thing here is not isomorphic as a partial order set to R plus D5 2

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okay so now we can go to the next vertex let's consider the vertex

12:37

0 that's this one up here so we start with the Singleton population now back to the mutation

12:42

game Singleton population at vertex zero and then here are the sequence of possible limitations here

12:51

are the corresponding weight diagrams okay and so we do an s2 and then an s1 s3 and then an s2s4 and

12:59

then an s3 and at that stage there are no more positive strictly positive entries on this side

13:06

so things are finished and we can then record the mutation frame it's a relatively simple one

13:12

two one three two four three and the associated positive root layer these are the positive roots

13:21

that you can build up starting with this positive root and Performing increasing mutations and here

13:27

they are also somewhat familiar sometimes I I put this thing here over on this side so you

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it doesn't really matter but the shape is a little bit different okay so if I

13:37

put this over on this side then it would look a little bit more like a like a wedge

13:41

maybe we've seen that on other situations so this is R plus D5 0 and circled are the

13:48

join irreducibles and if you look at that as a poset you'd see that you know essentially yes

13:55

that's again Birkhoff's theorem that the this lattice of root layers is

14:02

same as the lattice of ideals of M plus D5 0. I'm going to put it another way these

14:12

join irreducibles form a poset which is isomorphic to this thing here this little rectangular thing

14:21

okay excellent and what about on the Dual side so if we start with just a a Singleton weight

14:30

at the vertex zero okay and then we're going to go from here to here to here to here to here to

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here to here I've been record all those things uh starting with zero and then a two and then a one

14:42

three a two four zero three two and one so that's sort of firing right that's a firing frame F plus

14:50

D5 ) and again that's a pleasant kind of thing that we've seen before and in this case F plus of

14:59

D5 0 is isomorphic to the root frame because these two things are the same if we if you swap this one

15:09

over into the other side there then structurally this poset and this poset look the same

15:17

cool

15:18

okay what about starting with a Singleton population at the vertex three this one right here

15:25

well here's a record of these things so it's fun to do this you know it's fun to just

15:31

go through and and do this and check this into something that you can do yourself and you can

15:36

do it also for other graphs and sort of see what happens it's a huge interesting world

15:41

that creates structure sort of automatically and the structure really does have a dynamic aspect

15:47

like it really feels like you're following some dynamical process which is you know percolating

15:53

along around the graph it's almost like parallel particles which are zipping around the path and

15:57

they go out and then they come back and they cancel or whatever so there's a there's an

16:02

aspect almost an evolutionary aspect one one might say even though these are very simple examples of

16:09

um a sort of a process that's unfolding here that ends up being recorded with

16:14

these frames okay so starting with this thing and then we did s2s4 s0 s1 s2 s3

16:22

and then we get this mutation frame okay and the associated positive root layer frame

16:31

these are the positive roots that um I can start starting with that Singleton population and then

16:39

just doing the various mutations here the join irreducibles in red and that's the same posset as

16:47

I showed you before so again we see Birkhoff's theorem again connecting these two things and

16:58

on the Dual firing side starting with a Singleton weight right there then you

17:03

go from here to here to here to here to here to here and here and here and

17:12

there's a two that appears there there's a two that appears there

17:15

and to get this firing frame F plus D5 3 which is this Pleasant little thing foreign

17:22

you know and perhaps you're wondering why are these things sort of symmetrical like

17:27

these these firing frames in particular seem to be quite symmetrical and and sort of pleasant

17:32

and actually have other interesting properties too that we'll talk about at some later point

17:37

okay and then finally our last vertex vertex four here's the story starting with the

17:44

mutations starting with the Singleton population right there s3 s2 s0 s1

17:49

Etc and you can follow it along and we get this pretty simple mutation frame three two one zero

17:56

two three Associated positive root layer or frame is this one here those are all the

18:03

positive roots that you build up starting with that Singleton here are the adjoin irreducibles

18:09

The join irreducibles are isomorphic as opposed to that one I just showed you and the numbers side

18:18

here's the associated firing frame we start with the Singleton weight here go here here

18:23

here here here here here and here and we record what we've done and we get this thing here um

18:33

F plus D5 4 if I've done things correctly

18:38

okay so that's kind of fun um and these these combinatorial objects okay are are interesting

18:46

from a purely combinatorial point of view they have interesting properties having to do with

18:51

symmetry is having to do with their shape and we do with other properties that combinatorialists

18:55

like to study like Sperner properties and such but ultimately they're really important

19:02

because they open a door to understanding the representation theory of the Lie algebras and

19:08

Lie groups in a very novel way okay a way which is not appreciated by the the research community

19:15

at this stage okay so there's a whole there's a whole really important application of these things

19:22

they're not just interesting sort of exceptional remarkable combinatorial structures they also

19:28

have this very deep and interesting connection with representation theory I'm Norman Wildberger