Mutation Matrices

Wild Egg Maths

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To understand the Mutation Game on ADE graphs, we now introduce some linear algebra first of all the adjacency matrix of the graph, and then the matrices which encode mutations at the vertices. This gives us a powerful tool - essentially the idea behind "representation theory" - to work with abstract algebraic objects more concretely by representing them as matrices.

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hello everyone i'm norman wahlberger and

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thanks for joining me we are looking at

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dynamics on graphs and today we're going

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to introduce linear algebra into the mix

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linear algebras very powerful

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very modern approach to geometry and

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algebra

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that's all based on

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linear space and transformations of that

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which are linear we're represented by

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matrices so it connects with a powerful

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augmentation of our usual sort of

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arithmetical framework into matrices

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whenever we can do that whenever we we

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can represent

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um a given situation with matrices we

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are happy because there's all kinds of

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things we know about matrices and we

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have great computers

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and programs to to deal with that that's

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what we're going to do today okay so i'm

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going to illustrate everything with a

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couple of examples we'll start with this

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simple example which notably is not an

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ade

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graph okay this is a little thing we've

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seen something like this before so

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there's four vertices and i'm going to

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label these vertices their names are

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going to be one two three and four

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okay

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now it's not a tree because of this uh

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we have this little cycle here triangle

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okay now once we order the vertices of a

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simple graph say like in this case we

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ordering them one two three four in that

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order then a population

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can be identified with a vector

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identified that's not spelled right is

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it

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identify there then populations can be

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identified with vectors for example a

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population p is represented by a vector

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p 1 p 2 p 3 p 4 where of course p i

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represents the population's value at the

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vertex i

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now

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once we have populations represented by

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vectors

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then

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an important

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capturing of the the graph is its

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adjacency matrix

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okay that's going to be a matrix whose

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size is the same as the number of

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vertices in this case four so it's going

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to be a four by four matrix and what

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this does is it records which vertices

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are connected to which other ones

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the edges

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so the vertex one is connected to the

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vertex two so in the one two spot which

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is this one up here there will be a one

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however one is not directly connected to

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three so in the one three spot there is

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a zero

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now this is a simple graph the edges are

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all undirected so this means that this

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adjacency matrix will be

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[Music]

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symmetric matrix

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okay so it's symmetric about this

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diagonal let's just check it so let's

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look at it column by column so this

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first column is really telling us what's

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connected to our first vertex

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1 and the vertex 1 is only connected to

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the vertex 2.

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this next column is telling us what's

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connected to vertex 2. we see that

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vertex 1 vertex 3 and vertex 4 are all

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connected to it

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third column connecting

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or information about the vertex 3 it's

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connected to vertex 2 and vertex 4 and

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finally the last column telling us that

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vertex 4 is connected to vertex 2 and

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vertex 3.

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okay

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so now let's note that we can actually

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do a little bit of

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matrix multiplication here if we take a

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general population like this

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all right and i'm actually going to

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define this thing so let me

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select that and i'm going to define it

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so let's check my definition so i've

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i've defined the matrix a like that

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okay so i'm just going to put the matrix

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a here

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okay and this is then

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doing the matrix multiplication so if

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you take a vector like this multiply it

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by

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four by four matrix you can get another

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vector of the same size this is how it

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illustrates that

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so let's look at what this is doing this

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is replacing each of the entries with

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the sum of the neighbors

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okay so what used to be in the p1 spot

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now there's a p2 there what used to be

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in the p2 spot there's a p1 plus p3 plus

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p4

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here there's p2 plus p4 here there's p2

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plus p3 so as

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a transformation

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this replaces

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each let's say

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each p sub i

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with the sum of the neighboring values

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let's say

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okay so i want to exploit this kind of

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thing to be to try to get at

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the mutations

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so let's

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then

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try to

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find the matrices

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that

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represent

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the

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mutations

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and the mutations are s1 s2 s3

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and s4

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so let me remind you what the mutation

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does if we go back to our previous thing

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so i have this formula up here for

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mutation somewhere this is the the key

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formula here i just grab that all right

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come back down here

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so recall

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it's probably good to remind us recall

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that

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if you have some population p and you're

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mutating at x then this is the formula

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for the

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mutation of p at x at a new vertex y

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and most of the time it leaves the

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values unchanged but it just alters the

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value at x as i've mentioned many times

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okay we want to now incorporate this

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formula into a matrix

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so let's

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let's uh let's look at for example so

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in this example

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suppose we take

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this thing here

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and we

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apply say s sub 1 to it

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what's that going to do

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well it's going to leave p2 p3 and p4

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unchanged and what's it going to do to

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p1 well it's going to negate it first of

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all so that's step one to negate it and

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then step two is to add the values at

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the adjacent

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vertices namely p sub 2.

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okay so that's the effect of s1 on this

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general thing so how can i represent

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that with a matrix i want to write that

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as this thing times some matrix

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okay

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so what matrix should i use well it

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should be more or less the identity

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matrix and to get an identity matrix i'm

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going to go compute matrices fill matrix

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four by four is fine and identity matrix

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okay

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let me just check if i actually make

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that multiplication i'm going to just

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get what

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i started with but i'm not going to use

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this i'm going to modify this because

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what we have to do is we have to arrange

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that the first entry here is negative of

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the original p1 plus p2 i'm going to get

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that by just putting a minus 1 here

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and a plus one here

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let me check that

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it's p2 minus p1 that's the same thing

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comma p2 p3 p4

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so

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how did i get this matrix this matrix

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here is sort of capturing

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um capturing

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the

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the

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the mutation s1

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and

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so more generally

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let's

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let's look at uh

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say s2

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not really more generally but okay

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let's let's look at s2

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and

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so i i want to write this

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as

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this thing here

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times sum matrix so what matrix should i

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use here

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well the first order of business is

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probably to use the identity matrix and

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let me grab the identity matrix by

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taking this thing and

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i'll just

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turn it back into an identity matrix

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there's the identity matrix

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okay

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but that's not what

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all that we do we also need to

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to to modify this identity matrix by

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doing what by changing this entry

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in

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well this case in in the second entry

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because we're going to change the p2 to

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minus p2 and then we're also going to

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have to change

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essentially this column here because

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it's this column here that's affecting

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the second

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the second entry of our vector

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okay so what i'm going to do is i'm

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going to

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add another matrix

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and maybe to do that i might just

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grab a

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zero matrix

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there's a zero matrix let's grab the

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zero matrix okay so what i'm going to do

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with this zero matrix is i'm going to

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modify it first of all by putting a

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minus two there

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the effect of that

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is that instead of having a plus p2 i'll

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have a plus p2 minus 2 p2 which is a

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total of minus p2

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so that has the effect of negating the

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second thing

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and the other thing that i'm going to do

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is i'm going to

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have to add the neighboring populations

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and i can figure out what those are from

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the adjacency matrix by coming up here

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and looking at this second

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column

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okay that second column is responsible

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for

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summing up the neighbors so if i replace

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this thing here with one

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one and one

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and i might just

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evaluate that okay that's that in the

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particular matrix so to be consistent uh

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we'll put an equal sign here

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equals

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there's the vector times this

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so this

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matrix i think is going to have the

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effect of the mutation s sub 2. and i'm

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going to check that by just doing ctrl

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enter and seeing what i get

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oh that didn't work what was i doing

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wrong there um

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maybe there was a space actually okay

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there probably was a space let me try

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this

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ah that's more like it okay

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so is this right

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um if we take this population here and

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we apply s2 then the p2 negates okay and

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then we sum

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the entries of the adjacent neighbors

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which in this case are all of the other

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vertices

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that's

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correct

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so now you see we have the potential of

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replacing these somewhat abstract

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mutations with concrete matrices

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okay so let's introduce

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so now let's

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introduce

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the

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corresponding

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mutation matrices okay

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and the first

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one s sub 1

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is

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what did we get this this one here was

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the matrix that was representing s1

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right

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and

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s2

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was

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this one this was representing

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s2

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and

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what's going to represent s3 i'm going

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to do the same kind of thing i did over

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here i'm going to take the identity

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matrix

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and i'm going to add

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something which is almost zero i should

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get a zero matrix let me get a zero

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matrix once and for all

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uh matrices fill matrix

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zero

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okay this is zero matrix i'm gonna store

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it somewhere temporarily up here maybe

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another line there okay

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all right let's go down here so okay so

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what do i have here so s3 so there's the

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identity matrix and now i'm going to do

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the mutation at

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3. so that means i have to put a minus 2

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here but actually first i might take the

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third

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i'm going to take this third column from

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the adjacency matrix all right and i'm

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going to replace

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this third column here with that third

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column in the adjacency matrix and i'm

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also going to put a minus 2 here to make

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sure that the third

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entry actually gets negated

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so what do we get all together out of

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here we get this

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and i'm going to just check

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that that actually is correct so down

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here i would check

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that if i take our general population

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and i take this matrix here and i apply

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it

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uh am i getting the right thing so p3 is

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negated and then we're summing the

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values of p2 and p4 which i think is

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correct

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okay and finally

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what was the other thing we were doing

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we were taking

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s4

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so we're taking the identity matrix

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there it is there

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grab that

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sorry that was that wrong grab this one

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yes and then we're adding

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where's our zero matrix

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there it is there

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and how am i modifying that i am

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modifying it by taking the fourth

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the fourth

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column of my adjacency matrix and

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putting it there

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and also putting

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-2 in that

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column in the diagonal position

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and that evaluates to this

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and i'm going to check that also

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check

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if i take this vector

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and apply

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this thing to it

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that does have the effect of mutating

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at the fourth vertex so p4 goes to minus

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p4 plus the sum of the neighbors

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okay so let me clean this up having done

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all these things so s4 was just

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this thing

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got s1 we've got s2 we got s3 we've got

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s4 we've done some checks but we don't

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need that anymore

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and

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i guess we don't really need our zero

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matrix anymore either

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so in this example yes more generally

15:35

okay

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i'm sorry this is

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this is a

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summary of what we've done

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okay

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what i'm going to do now is i'm going to

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actually define these things so i'm

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going to make this as a definition

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i'm going to make this

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as a definition

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make this

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as a definition i'm going to make this

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as a definition

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just check

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we have a we have s1 we have s2 we have

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s3 and we have s4

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okay now i'm going to check some of the

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relations what are the kinds of

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relations that we would want well we

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want things like s1 squared should be

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the identity so

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scientific workplace is perfectly

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capable of dealing with these matrices

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it knows these are matrices and it knows

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how to do matrix

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arithmetic so yes indeed s1 squared

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equals

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the identity i'm just going to quickly

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check s2 squared also the identity s3

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squared also the identity and s4 squared

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also the identity so those are that

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those properties are valid for all of

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them

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okay what

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um

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what are some other identities that we

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um we are interested in well remember

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the braid relation

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braid relation with something like s

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sub 1 s sub 2 s sub 1.

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let's see what that thing does

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s1 s2 s1

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if we want to know what what is that

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actually doing on

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a

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population

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then we can just take a population and

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apply

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our matrix to it and we see what it's

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doing

17:30

okay in this case it's leaving p3 and p4

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unchanged which is not a surprise and

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it's performing this particular

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combination on

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um on the first two entries in

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particular this is a new matrix that we

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haven't seen before so we we've sort of

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enlarged our list of matrices

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this way

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now the braid relation says that this

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should be the same as s2 s1 s2

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okay we can just evaluate that and see

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if it is the same and we see that it's

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the same

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so these are equal

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you see and now we could

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in principle

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use the underlying matrix

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capability of of my computer to

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calculate rather

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uh complicated

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combinations of mutations okay

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so we can now

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perform

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complicated

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combinations

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of mutations

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algebraically

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using these underlying

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matrices

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so let me just give an example so let me

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just make it an example up so suppose i

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have say say i'm interested in applying

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s1 and then i'm interested in playing s4

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and maybe i'll apply s2

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then maybe s3 and then maybe another s1

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okay

19:02

so i'm applying five different mutations

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and i want to know what is the combined

19:06

effect well here's the combined effect

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it's represented by

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this matrix

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so its effect will be so if i want to

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sort of know what it actually does

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so this has

19:20

so

19:21

this has the effect

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which is what well let's take

19:30

this thing here

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and

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multiply by this matrix and we'll see

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exactly what

19:38

the combined effect of these five

19:40

mutations

19:41

is on the original population p1 p2 p3

19:44

before we get this rather complicated

19:46

thing

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and

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it's certainly suggestive that if we

19:51

just cook up more and more mutations

19:56

we're going to get more and more

19:57

complicated matrices

19:59

and for this

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example that i've been looking at

20:02

starting with this particular matrix

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that is what's going to happen

20:07

that's not a surprise to us because we

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know that

20:11

this particular graph does not have a

20:14

finite

20:15

set

20:16

of roots so if we start with a singleton

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and we perform mutations randomly we're

20:21

going to get bigger and bigger

20:22

populations

20:25

that's

20:27

pretty well telling us that we are not

20:29

going to get a finite number of

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possible

20:33

um transformations

20:35

by starting with these four mutations

20:38

and performing them arbitrarily because

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if we only got a finite number of of

20:42

transformations then we're not going to

20:44

be able to jack the population up as

20:46

much as we want and we know we can't do

20:48

that for this matrix because it's not an

20:50

ade

20:52

so knowing already that this thing has

20:53

an unbounded number of positive roots

20:56

implies that

20:58

the

20:59

the collection of matrices that we can

21:02

generate

21:04

by taking more and more combinations of

21:06

these sis is also going to be unbounded

21:12

okay so since

21:14

the

21:15

positive roots

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for this

21:21

graph

21:22

are unbounded

21:25

it follows that

21:30

there will

21:32

be an unbounded

21:34

number

21:35

of

21:37

such transformations

21:40

that we can obtain

21:43

by taking

21:44

products

21:46

of

21:46

our four basic mutations s sub 1 s sub 2

21:51

s sub 3 and

21:54

and s

21:55

sub 4.

22:00

so we're touching base here now with

22:02

group theory because it turns out that

22:04

we have a group here

22:07

these transformations are all linear

22:09

transformations so

22:10

any combinations of them are going to be

22:13

linear transformations

22:15

and they're all invertible

22:17

okay so it's it's it's a group of

22:19

transformations but in this case it's an

22:21

unbounded group

22:23

and in fact probably trying to describe

22:25

this group is a major uh undertaking

22:28

it's like a you know a master stasis try

22:30

to get your head around what this group

22:32

is like my estimation it's not obvious

22:35

at all

22:36

okay but

22:38

we are

22:40

mostly interested in the aede graphs at

22:42

least initially because they're the

22:43

simplest so now we have a technology

22:46

that allows us to investigate these aede

22:49

graphs

22:50

not just from this

22:52

combinatorial point of view but also

22:53

from a group theoretical point of view

22:56

we can now ask what are the

22:58

associated groups

23:00

generated

23:01

by

23:02

the basic mutations for the ade graphs

23:07

so a key question

23:10

what are the groups

23:13

generated

23:14

by the mutations

23:17

s sub i

23:19

for

23:20

er

23:20

for an ade graph

23:24

okay

23:25

these are called

23:28

bile groups

23:31

named after hermann weil one of the most

23:33

prominent

23:34

20th century mathematicians

23:37

so that's a whole new domain of

23:41

interesting algebraic objects that we're

23:43

going to now make contact with