Dynamics on Graphs 25

22 views May 25, 2023

We review the important notion of reflection in a vector space equipped with a symmetric bilinear form. There are two closely related reflections determined by a vector: one in the vector and the other in the hyperplane perpendicular to that vector. In the Mutation and Numbers games story, the latter is the crucial notion.

It turns out that the reflection in the hyperplane determined by a singleton population in the space of populations equipped with the Tits quadratic form is just the mutation associated to that vertex. This gives a strong geometric manifestation of mutations with many consequences in the theory of Coxeter groups, roots systems and representation theory of simple Lie algebra and Lie groups.

To ground the discussion, we illustrate the ideas with the A2 graph, which is crucial for understanding SL(3) and SU(3) and the eigthfold way in modern particle physics.

This is part of the Dynamics on Graphs series. A pdf for the lecture may be found at https://www.dropbox.com/s/wqkla7t99el...

Video Contents:

00:00 Mutations in a geometric way

01:13 Example with mutation matrices

02:44 Reflections in vectors and hyperplanes

05:36 Projection of a vector onto another vector

12:43 Mutations are reflections

16:16 Formula for a mutation matrix

21:00 Example of a hyperplane

23:22 Reflections for all roots (positive)

Mutations in a geometric way

0:02

hello everyone today we're going to look

0:04

at mutations

0:07

in a very geometrical way very important

0:09

geometrical way and reveal them as

0:11

actually Reflections but their

0:14

reflections with respect to a somewhat

0:16

non-standard geometry not really

0:19

euclidean geometry but given by a

0:21

different quadratic form called the tits

0:23

quadratic form that we've already talked

0:25

about

0:26

and I'm going to illustrate what happens

0:28

for the A2 situation which connects all

0:32

very nicely with the standard model and

0:35

the Eightfold way of the particle

0:37

physicists so we're starting it however

0:40

in a rather General fashion so let's

0:42

just let X be a general graph okay for

0:44

which the carton Matrix C which I remind

0:47

you is 2 times the identity Matrix minus

0:49

the adjacency Matrix a for which that is

0:52

invertible

0:54

so our go-to example is the A2 example

0:57

here's our graph and here are two

1:00

mutation matrices S1 is this one here

1:02

and S2 is this one here and the carton

1:04

Matrix is two minus one minus one two

1:07

now I remind you that we have sort of

1:09

two different pictures we have

1:10

populations like this one here

1:12

represented by row vectors a1a2 and say

Example with mutation matrices

1:15

p in the space of populations is p and

1:19

weights

1:20

represented just a little bit

1:21

differently with the graph having open

1:23

vertices say Lambda 1 Lambda 2 expressed

1:27

as a column Vector that's a weight of

1:29

the space of those is w and we've seen

1:33

that these mutation matrices very

1:35

generally have this important

1:38

basically an isometry property with

1:40

respect to the quadratic form the tits

1:43

quadratic four built up from this carton

1:46

Matrix in other words s i times c times

1:48

S I transpose is equal to C and you can

1:52

just invert that and sort of

1:54

equivalently s I transpose C inverse s i

1:58

equals c inverse and that's really

2:00

determining the geometrical structure on

2:03

the the weight side so these two sides

2:05

and we know how to connect them with the

2:07

PSI map okay so we're going to

2:09

concentrate mostly on the populations

2:10

today so the tits quadratic form on the

2:13

populations is just given by P dot Q

2:16

equals p c q transpose for row vectors p

2:19

and Q well on the Dual side that's dual

2:23

quadratic form on the weight space is

2:24

given by w dot V equals W transpose C

2:28

inverse V okay so now we've seen that

2:32

these mutation matrices preserve the

2:35

quadratic form in the sense that if we

2:37

take two

2:39

populations p and Q then P dot Q is the

2:42

same as P times s i dot Q times s i so

Reflections in vectors and hyperplanes

2:46

apply s i to both things preserves the

2:48

quadratic form that means that s i is an

2:50

isometry and the same thing kind of

2:52

thing happens on the Dual side so these

2:55

isometries are fundamental objects in a

2:58

linear geometrical situation but we want

3:00

to establish something stronger

3:03

that these operations are actually

3:04

Reflections in suitable hyperplanes so

3:08

in order to

3:10

set that up we have to have a prior

3:12

discussion about Reflections more

3:15

generally in in linear spaces that are

3:19

imbued with some geometrical structure

3:21

given by a DOT product or symmetric

3:24

bilinear form okay so let's just enlarge

3:27

ourselves a little bit here so just a

3:29

review of some basic geometry

3:32

linear algebra suppose V is a vector or

3:35

linear space and it has a symmetric

3:38

bilinear form which will represent by B

3:40

dot w and let me remind you okay that

3:42

this means that it's

3:44

it's uh linear okay so V1 plus V2 dot w

3:49

is V1 dot w plus v2.w and that also

3:54

holds another Direction uh like here and

3:56

it's linear also in the sense that if we

3:58

multiply by a scalar say Lambda Lambda V

4:01

dot w is the same as Lambda times V dot

4:03

w which is the same as V Dot Lambda W

4:05

and there's a symmetric property That V

4:09

Dot W equals w dot V so those are the

4:11

defining characteristics of what we call

4:13

a symmetric bilinear form and in

4:16

practice once you've chosen bases

4:18

symmetric bilinear forms are really

4:20

almost the same as symmetric matrices

4:23

okay so symmetric by linear form is

4:25

essentially like a symmetric Matrix

4:28

now the most familiar one

4:30

that you study mostly in undergraduate

4:33

days is the standard one where this

4:36

Matrix is the identity Matrix so you get

4:38

the usual dot product you know so like

4:42

X1

4:44

times y1 plus X2 times Y2 plus X3 times

4:47

Y3 that's the dot product of X1 X2 X3

4:50

and y1 Y2 Y3 but that's really just a

4:52

very special case of this more General

4:54

story of symmetric binary forms and

4:56

that's crucially what we want to work in

4:58

in this case here okay so in terms of

5:01

the values of the binary form these are

5:03

in whatever field that the vector space

5:05

is defined over and for us the default

5:07

is mostly the rational numbers but we do

5:10

want to keep in mind especially

5:11

interestingly that finite fields are a

5:15

potential environment for doing all of

5:17

this okay so now we have to introduce

5:19

the projection of a vector onto another

5:22

Vector okay so now we want to introduce

5:24

the projection of a vector v onto a

5:27

vector W here's the picture here's a

5:28

vector W there's a vector B the

5:31

projection of V onto W is another Vector

5:34

in the direction of w namely this green

Projection of a vector onto another vector

5:36

one let's say call it U okay which

5:39

happens to have the property that the

5:41

the difference this V minus U Vector is

5:45

perpendicular to W so in a

5:49

space with a symmetric bilingual form

5:52

perpendicularity is given by a DOT

5:56

product equaling zero so the condition

5:59

that we want here geometrically is that

6:02

V minus U is perpendicular to W okay and

6:05

that's the same as V minus U dot w

6:08

equals zero so here's the formula for

6:11

this projection and what is it

6:13

it's first of all a multiple of the

6:15

vector W and here is a scalar okay and

6:19

it has in the numerator V dot w and in

6:24

the denominator w

6:26

dot w

6:28

so I see over here I've said v9 this

6:32

this condition should be W not equal to

6:34

zero we want to make sure that the W is

6:37

non-zero so that the denominator here is

6:40

um

6:40

not zero okay so this is a multiple of w

6:45

and this is the definition of the

6:46

projection of V onto W and to verify

6:51

that it really does satisfy this kind of

6:53

property we want to make a calculation

6:55

so let's calculate V minus U dot w

6:59

so by the linearity this is the same as

7:01

B dot w minus U dot w equals V dot w now

7:05

U is equal to this thing here so we plug

7:07

that in there's B dot w over w dot w

7:09

times W and we have to dot it with W now

7:13

because of the bilinearity the scalar

7:16

sort of comes out and we just have the

7:18

scalar here times w dot w but that's

7:21

exactly what happens to appear in the

7:23

denominator of the scalar so that's

7:25

going to cancel with that denominator

7:26

and we're just going to get V dot w

7:28

minus V dot w which is indeed zero so

7:31

algebraically yes this is a valid uh

7:36

kind of definition because we really are

7:38

talking about the projection of one

7:40

vector onto another

7:41

great okay

7:44

now

7:46

um so in this case here we say that that

7:50

U okay and V minus U are components of

7:54

the vector v so here's an arbitrary

7:55

vector v and one of the components is in

7:58

the W Direction and the other component

8:00

is in the perpendicular Direction

8:04

Okay so now we're going to use this

8:07

projection to Define two Reflections

8:10

there are two actually different kinds

8:12

of Reflections and one is a reflection

8:15

in a vector and the other is a

8:18

reflection in the hyperplane

8:19

perpendicular to a vector okay so here's

8:22

our diagram again here's W and u v is

8:26

before okay here's the vector U as

8:28

before

8:30

so one thing that we could Define is the

8:33

reflection of v in W that's this one

8:37

down here so R sub W of V reflection in

8:40

W of V so how do we get this thing well

8:43

we take the Specter here and and this

8:46

difference is V minus U so what we want

8:48

to do is we want to put another copy of

8:49

that on the other side okay so

8:52

algebraically what we're taking is we're

8:54

taking the vector v and we're

8:55

subtracting two times the vector v minus

8:59

U okay

9:01

now just algebraically that becomes V

9:04

minus 2 V plus 2u or all together minus

9:09

V plus 2u

9:11

and Since U is that projection that

9:14

we've just defined we can write the

9:16

formula purely in terms of V and W as

9:18

this reflection r w of V is equal to

9:21

minus V plus 2 times V dot w over w dot

9:27

w times w

9:29

great now there's however another kind

9:32

of reflection that we can do so here's

9:34

the vector W and here this thing here

9:37

this plane that we see here is to

9:39

represent the hyperplane perpendicular

9:42

to W so this doesn't require us to be in

9:45

a two or three dimensional space this

9:47

could be in an arbitrarily High

9:48

dimensional space and then this

9:51

hyperplane is to be thought of as a

9:53

hyperplane in the sense of having

9:55

Dimension one less than the ambient

9:57

space because of the ambient space is

9:59

three-dimensional this will be two

10:01

dimensional it'll consist of all those

10:03

vectors which are perpendicular to W

10:05

that's a single linear condition so that

10:08

defines a hyperplane so now what we're

10:11

talking about is reflecting the vector v

10:13

in this hyperplane so we want to go sort

10:15

of towards the hyperplane and out the

10:17

other side an equal amount well this

10:20

Vector here okay is

10:23

in the direction of w well that's the

10:26

the W component of V so that's the what

10:29

we were calling U before so the

10:32

direction that we want to go in is we

10:33

basically want to start from V and we're

10:35

going to go

10:36

two times in the direction of minus U

10:40

okay so the reflection of v in this

10:43

hyperplane let's call it h sub w

10:45

is given by this formula it's V minus 2u

10:51

and notice that's just exactly the

10:54

negative of what's appearing here okay

10:56

so it's the same as minus r w v so in

10:59

other words this first Vector that we

11:01

got by reflecting V in W is the negation

11:05

of this second Vector that we're getting

11:07

which is the reflection of v and the

11:09

hyperplane determined by w

11:12

okay so that allows us to give a very

11:15

quick formula and what is that formula

11:17

it's that this reflection in the

11:20

hyperplane of V is V minus 2 V dot w

11:25

over w dot w times w

11:28

so

11:29

in the mutation game story it's these

11:32

Reflections in hyperplanes which are the

11:34

key sort of geometrical Transformations

11:37

and this formula here uh plays a very

11:41

important role in the whole story of

11:43

coccider groups and and root systems and

11:46

you know an Associated sort of

11:48

representation Theory or Lee algebraz

11:51

so

11:53

um and I just want to make a a little

11:55

note here sort of a simplification that

11:57

if it happens that our original Vector W

12:00

has the property that its quadrants is

12:02

equal to two in other words that w dot w

12:04

is equal to two well then this thing

12:06

simplifies because the 2 in the

12:08

denominator here will cancel it with the

12:10

two in the numerator and we just get V

12:12

minus V Dot W Times w

12:15

okay so now I want to show you that

12:19

mutations are actually reflections okay

12:22

so let's read this so suppose X is an

12:24

Ade graph with carton Matrix C which is

12:27

2i minus a and

12:31

remember that for X vertex in the graph

12:35

the Singleton population let's say

12:38

denoted by Delta sub X has quadrants

12:40

Delta sub x dot Delta sub X so remember

Mutations are reflections

12:43

what we do is we

12:45

take the Delta sub X and Delta sub X

12:48

transpose on the other side of the

12:50

carton Matrix C and that's going to be 2

12:53

since C has twos on the diagonal

12:56

okay so this thing here is a vector it's

12:58

just a vector with all zeros except for

13:00

a single one and so what you're doing

13:02

here is you're just picking up the

13:03

certain diagonal element of C but C has

13:06

all diagonal elements two okay so uh the

13:10

the starting Singleton populations that

13:13

we started the mutation story with all

13:16

have quadrants two

13:18

now since mutations Preserve

13:23

the quadrants that means that any root

13:26

population that we can obtain from

13:27

Singletons that's the definition of a

13:29

root population it's a population

13:30

obtainable from a Singleton by sequence

13:35

of mutations it follows that any such

13:37

root population p is also going to have

13:41

quadrants two in other words that P dot

13:46

P equals two also

13:48

that means that so here's the diagram

13:50

now so p is that we thought of as one of

13:53

our root populations

13:55

and if say Q is another

13:57

general population then the reflection

14:01

of Q in this hyperplane H sub p is by

14:05

the formula that we've just seen Q minus

14:08

Q dot p

14:10

times P because in this case P dot P

14:13

equals two so we get this simplified

14:14

formula for this reflection taking this

14:17

Vector Q to this Vector r p perp of Q

14:26

now something else to observe is that in

14:28

the special case when p is a Singleton

14:30

say Delta sub X is Singleton that if you

14:34

have some other population yeah

14:37

Q okay Q dot P okay so that's Q dot

14:42

Delta X so what are we doing here we're

14:44

taking Q times the carton Matrix times

14:46

Delta X transpose

14:49

but

14:50

that's

14:52

Q times W sub X where W sub X is c times

14:57

Delta X transpose so if you take a

15:00

matrix and you multiply it on the right

15:03

by a vector which is all zeros except

15:06

for a one in the I spot

15:09

or in this case perhaps better to say in

15:11

the X spot then what you're picking up

15:14

after you do that matrix multiplication

15:15

is just the ith or the X column of C

15:20

is the call is the X column of C

15:25

so uh the The Columns of c are naturally

15:29

uh playing uh an important role it

15:32

follows then that

15:34

this reflection in the

15:37

in the hyperplane

15:40

perpendicular to this Delta

15:45

population

15:46

is Q minus q w x times Delta X and what

15:52

we're seeing here is that it's a

15:54

particular form for things right where

15:57

in order to get the reflection of Q

15:59

we're taking q and we're changing in how

16:02

well this is just a multiple of Delta X

16:04

Delta X is this

16:06

Vector with zeros everywhere okay except

16:10

for a single one in the X spot

16:14

so the the population Q is only changing

Formula for a mutation matrix

16:17

in the X position

16:20

and

16:21

the

16:23

you're changing it by Q times w x in

16:27

other words you're taking the uh the the

16:29

X column of the carton Matrix okay and

16:34

you're just essentially multiplying Q by

16:38

that and if you think about that that X

16:41

column is well there's a 2 on the on the

16:44

diagonal but all the other ones are are

16:46

negative ones corresponding all to all

16:48

the neighbors of X in the graph

16:51

okay so if you think about this for a

16:55

minute or two you see this is

16:56

essentially the formula for a mutation

16:58

Matrix

16:59

Singleton populations are just

17:03

the mutations

17:06

so let me illustrate all this story here

17:08

so to make it more concrete by looking

17:10

at the case of A2 okay so there's the

17:13

carton Matrix two minus one minus one

17:15

two uh say if x is equal to X1 then the

17:19

singles in population as a row Vector is

17:21

one zero and W X 1 is the First Column

17:26

of C which is 2 minus one so if we have

17:29

a general population Q then the effect

17:32

of this

17:34

reflection in the hyperplane

17:38

defined by Delta X1 okay it's going to

17:42

be Q minus here's q1 Q2 and we have to

17:44

multiply by this

17:46

Vector wx1 just 2 minus 1. and that's

17:51

multiplying by the delta

17:55

function Delta population at X1 so what

17:57

is this amount to well here's the vector

17:59

Q okay we do this dot product we get 2

18:02

q1 minus Q2 times 1 0.

18:06

okay and then looking at components the

18:09

first component will be Q 1 minus two Q

18:11

one minus Q2 that's minus q1 plus Q2 and

18:15

the second component is just Q2 because

18:17

there's no contribution here so we're

18:19

getting minus q1 plus Q2 Q2 which we see

18:22

is same as q1 Q2 times this Matrix here

18:25

which indeed is the mutation Matrix S1

18:29

so this illustrates exactly that the

18:31

mutation matrices are coming about as

18:34

Reflections in hyperplanes

18:37

determined by these Singleton

18:40

populations with respect to the tits

18:42

quadratic form

18:45

and the story is the same for x equals

18:47

X2 uh the same story uh exactly the same

18:51

kind of thing you you check that you're

18:53

getting the second mutation Matrix S2

18:57

okay so in general the above formula for

18:59

the reflection exactly coincides with

19:02

our prior description or construction

19:03

and mutation matrices let me remind you

19:05

that that procedure was to take the

19:08

identity Matrix and replace the X column

19:11

of it with the X column of the adjacency

19:14

Matrix a except that the diagonal entry

19:17

is um is now a minus one

19:22

and another observation also note that

19:24

the hyperplane HP of vectors Q for which

19:26

Q is perpendicular p is fixed under the

19:28

reflection R sub P perp because the dot

19:32

products are zero so those vectors are

19:34

not going to to change

19:37

so for x equals A2 again and P a

19:40

population P1 P2 a more General Vector

19:44

the hyperplane HP is given by P dot x

19:47

equals zero so that's this condition

19:49

here there's the vector P there's the

19:51

carton Matrix there's the vector X it's

19:54

actually it's transpose and then you

19:56

write this out this condition is this

19:58

linear condition on X1 X2 2p1 minus P2

20:01

times X1 plus minus P1 plus 2p2 times X2

20:07

so that's the formula for the hyperplane

20:10

determined by a general

20:13

population p

20:15

. so now for the positive Roots one zero

20:18

zero one and one one

20:21

the the respective hyperplanes then if

20:23

we just use this formula here and

20:26

look at the case when p is equal to 1 0

20:28

and then the case when p is equal to

20:30

zero one in the case when P equals one

20:31

one we get these three respective

20:34

hyperplanes two x one minus x two equals

20:36

zero minus X1 plus two x two equals zero

20:39

and X1 plus X2 equals zero and these are

20:42

shown in this diagram here so have a

20:44

look at this so

20:46

here's the plane X1 X2 here okay so

20:51

these are the usual axes for populations

20:54

here is the line given by 2x1 minus X2

20:58

equals zero so that's what reflection in

Example of a hyperplane

21:02

Vector one zero

21:04

correspondent corresponds to a

21:06

reflection in this hyperplane this is

21:09

the perpendicular hyperplane to this one

21:10

this Vector one zero and the

21:12

perpendicular hyper plane to the vector

21:14

is zero one is this line here with

21:16

equation of minus X1 plus 2x2 and the

21:20

third Direction the

21:22

population of one one it's perpendicular

21:25

hyperplane is X1 plus X2 equals zero so

21:29

this is an example here okay of uh

21:33

reflecting in these various hyperplanes

21:36

so if we start with one one okay then we

21:40

apply S1 we're going to get this point

21:43

here which is

21:45

0 1 that's exactly a reflection in this

21:48

thing here in the sense that we're going

21:52

in the one zero Direction but we're sort

21:55

of going up to the hyperplane and then

21:58

going equal amount in the other

22:00

direction while S2 is reflection in this

22:03

hyper plane that sends this point to

22:05

that and then a further S three so for

22:08

example S3 uh is a reflection in this so

22:10

here is then the the orbit of the vial

22:13

group the orbit of all the powers of S1

22:16

and S2 the group The W generated by S1

22:19

and S2 which has six elements

22:22

so crucially this is a different

22:24

geometrical picture from what you see in

22:27

every single text and paper on the

22:29

subject okay and

22:31

I have to say you know I've thought

22:33

about this for decades this is a a

22:35

superior way of thinking okay it's it's

22:38

not a euclidean geometrical story people

22:41

it's not okay physicists listen

22:43

carefully it's not euclidean geometry

22:46

here that we're doing it's it's related

22:48

but it's a different quadratic form and

22:51

so you have to respect that that's just

22:53

the way the world is

22:54

and if you're a particle physicist you

22:57

probably can sympathize with that

22:59

because these lines have have a physical

23:01

meaning

23:06

okay so a little bit more General so a

23:09

slightly bigger picture here okay so

23:11

we'll end with this

23:12

um this is the story where we see the

23:15

the root lattice and also the weightlabs

23:18

at the same time so let's start with the

23:21

here the the positive roots in red okay

Reflections for all roots (positive)

23:24

and there's a Delta one and Delta Two

23:26

these are the Singleton populations on

23:30

the vertices one and two and in Black

23:33

here you see the integral

23:36

lattice generated by them so the black

23:38

points here correspond to the the

23:41

populations the integral functions on

23:45

the

23:46

on the uh on the vertices of A2

23:50

and over here we see Lambda 1 and Lambda

23:53

2 okay those are the vectors of the

23:56

image of the weight vectors under our

23:58

PSI inverse map and they generate this

24:02

finer lattice consisting of the black

24:05

ones as well as the Blue Points okay and

24:09

it's it's this this diagram here which

24:12

is really responsible for the Eightfold

24:14

way

24:15

and

24:16

you can see the richness perhaps by

24:19

looking at a slightly more complicated

24:21

example so let's look at say um say say

24:24

this Vector here let's say let's start

24:25

with this one here okay you get this

24:27

vector by going Lambda 1 Lambda 1 Lambda

24:29

one two three Lambda ones and two Lambda

24:33

2s okay

24:34

so in in the weight space diagram this

24:37

would be like the the column Vector

24:39

three

24:40

two and then here are the various images

24:44

of that under the

24:47

Reflections determined by these

24:50

hyperplanes okay so reflecting in this

24:52

hyperplane goes from here to here and

24:56

what happens is that we're going an

24:58

integral number of

25:01

steps of this

25:03

basis Vector Delta one so we're starting

25:06

here and we're basically reflecting and

25:09

we're going an integral number of Delta

25:10

ones in this case two of them to go from

25:13

here to here we're reflecting in uh in

25:17

hyperplane perpendicular to Delta 2

25:19

that's namely this one here and we're

25:21

going in integral number one two three

25:24

copies of Delta two to go from here to

25:27

here okay and so the the vowel group

25:30

then ends up giving us this uh this

25:34

configuration this hexagon here okay and

25:38

uh this entire story is a hugely

25:40

important in in the representation

25:42

theory of of su

25:45

su3 or or sl3 and you know connect

25:49

strongly with a lot of particle physics

25:52

having to do with the strong force

25:55

and so the The crucial thing here is to

25:59

understand is so that these mutations

26:01

which we just started originally is

26:03

thinking about in terms of algebraic

26:05

things algebraic Transformations

26:07

starting from the graph they actually

26:09

have geometrical content and crucially

26:12

their Reflections their reflections with

26:15

respect to this very tight geometry and

26:17

that turns out to have like huge

26:19

consequences and uh in fact so the next

26:22

step is to invoke uh probably the most

26:26

important theorem about Reflections okay

26:28

uh what I call the reflection theorem

26:30

and that's going to shed a whole lot of

26:32

light on on the various possibilities

26:34

that emerge when you apply the vile

26:38

group more generally for larger larger

26:41

situations