Object position duality for the symmetric group: Dynamics on Graphs 8

Wild Egg Maths

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What exactly is a permutation? That depends a bit on the context: it could be a bijection from a set to itself, or it could be a physical rearrangement of a collection of objects. The second description suggests that the position of the objects are also playing a role, and in this video we give a view of the symmetric group acting both on objects as well as on positions.

This will be important to us as we explore Weyl groups, Coxeter groups and explore both the Mutation and the Numbers Games, along with many other fascinating structures related to dynamics on graphs.

00:01

hello everyone i'm nolan walberger and

00:02

thanks for joining me on this adventure

00:05

towards exceptional structures in

00:06

mathematics and physics through dynamics

00:09

on graphs

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we've been talking about the symmetric

00:12

group i'm going to today have a little

00:14

bit of an informal discussion i want to

00:15

introduce a really important idea of

00:17

duality

00:19

in the theory of symmetric groups which

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is a topic that is usually not covered

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or at least not covered very much in

00:26

traditional courses

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and

00:29

this is a really important aspect of the

00:31

subject from our point of view

00:34

and it will turn out to have significant

00:38

um repercussions for our

00:41

discussion of the mutation game we're

00:42

going to see that as a consequence of

00:44

this kind of thinking there's actually a

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dual game to the mutation game called

00:48

the numbers game so there are actually

00:50

these two games floating around that we

00:53

have to get our heads around and how

00:54

they interact is really interesting

00:56

okay but today i'm going to look at this

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from a very elementary point of view

01:01

and i'm going to illustrate everything

01:03

with

01:04

some examples here okay so we have four

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objects okay and they are kind of

01:09

increasing in size so they're naturally

01:11

ordered and i'm going to put them from

01:14

left to right but of course that's a

01:16

sort of subjective point of view from

01:18

your point of view this will be left and

01:20

you know this is left to right but

01:22

here's let's say object one okay it's

01:25

the smallest one here's object two

01:26

here's object three here's object four

01:29

and i'm interested in the symmetric

01:30

group with

01:33

on four things okay that we can deal

01:36

with by

01:37

moving these things around okay now

01:41

in uh the usual formulation what one

01:43

does is one introduces cycle notation

01:46

okay so these sort of algebraic ways of

01:49

of describing you know a particular

01:51

representation so that one there for

01:53

example what i've just done is i've

01:55

swapped objects one and three

01:58

um

02:00

okay that's i've done that to that

02:02

operation um but another way of saying

02:05

that is i've i've swapped the uh

02:08

the objects in positions one and three

02:10

because there's sort of four positions

02:12

here and i'm interchanging the things in

02:15

these two positions so in terms of this

02:17

cycle notation uh the standard or the

02:20

simplest kinds of

02:22

permutations are the transpositions that

02:24

just swap two things like i've just

02:26

shown you and in fact because we have

02:29

the four things ordered the simplest

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transpositions are the ones between

02:33

adjacent elements so for example this

02:36

transposition here

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or or this transposition here

02:40

or this transposition here that's a

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simple kind of uh you know

02:45

sort of basis or generating set for the

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symmetric group and in fact

02:51

those three matrices correspond

02:53

exactly to the mutations of

02:55

a3 which are really

02:58

the same as the symmetric group s4 okay

03:02

remember that the while group for a n is

03:04

essentially s n plus one so that's it's

03:08

the

03:09

graph with the three vertices the a3

03:11

graph which is really uh generating this

03:14

this groups situation here

03:17

okay so now i want to

03:19

talk about where is the duality that i'm

03:21

i want to introduce the duality is that

03:24

hidden here like let's suppose i do a

03:26

couple of trans

03:27

informations so suppose i'm going to do

03:30

the transformation 1 2

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and then followed by the transformation

03:33

2 3 and then followed by the

03:35

transformation 3 4. okay so what does

03:38

that amount to

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so the transformation 1 2 is obvious

03:44

we all agree that's 1 two

03:46

now what's the transformation two three

03:50

okay at this point you might pause and

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say okay

03:53

two three interchange the second thing

03:55

in the third thing yeah but what what am

03:57

i talking about when i say thing because

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there's two kinds of things involved

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here implicitly there are the objects

04:05

themselves and there are also the

04:07

positions of the objects

04:09

now i haven't been that explicit about

04:11

that you know i haven't said well there

04:12

are these four positions but i've i've

04:14

indicated that by the way i've put these

04:16

things down on my little chess board

04:18

here okay there's really position here

04:20

there's a position here there's a

04:21

position here there's a position here

04:22

and i'm going to interchange things in

04:24

those positions

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so

04:26

at the level of objects if i was going

04:28

to

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interchange the the objects two and

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three so this is the second object this

04:34

is the third object interchanging the

04:35

object two and three would be this

04:38

okay however

04:40

interchanging things in the positions

04:42

two and three here's position one

04:44

position two position three position

04:46

four would be this

04:49

okay so we see that there's a a a

04:52

distinction to be made between

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objects and positions

04:58

okay and this is a really important this

05:00

can be really important for us and i'm

05:02

going to try to

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make that clear here in this

05:06

physical situation by introducing

05:08

formally the positions so objects zero

05:11

and four will you please depart the

05:13

stage momentarily while i introduce the

05:15

positions

05:16

okay

05:18

they're off the stage and now ladies and

05:20

gentlemen

05:21

the positions

05:29

oh let's use these ones okay

05:34

maybe i'll make this in the smallest one

05:35

okay

05:36

it's a question of um

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choices to which uh

05:41

which containers now we have containers

05:43

okay so the objects you know can fit

05:45

inside the containers

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so the containers are like the positions

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and let's agree that this is

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position

05:52

one this is position two this is

05:55

position three this is position four

05:59

all right

06:00

and

06:01

now objects one to four uh can you

06:04

please return to the stage so object one

06:06

will put in position one object two in

06:09

position two object three in position

06:11

three object four and position four

06:15

so now i have something which is a

06:17

little bit more subtle than i did when i

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started out with now i have four objects

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in four positions

06:24

objects one two three four positions one

06:26

two three four

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and now when i'm talking about the

06:30

symmetric group i

06:32

can be more explicit about whether i'm

06:34

moving or talking about objects

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and positions

06:39

okay so let's um let's start

06:44

by analyzing the product of these three

06:46

transpositions uh position the

06:49

transposition one two and then two three

06:52

and then three four okay in that order

06:54

i'm gonna do transition one two and then

06:56

the transition two three and then the

06:58

transposition three four but there's now

07:00

two points of view on what that means so

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let's start by

07:05

um

07:06

looking at

07:07

the uh

07:09

the action on positions

07:11

okay so

07:13

the the transposition one two okay from

07:16

from the position point of view is

07:18

is the rule take

07:20

the things in positions one and two

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these are the positions one and two and

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swap them

07:26

okay that would be like having a robot

07:28

arm you imagine having a robot you know

07:30

that just comes in and does something

07:32

like this so this this particular robot

07:35

the one two robot okay just

07:37

automatically moves to these positions

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and whatever is here it just

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interchanges

07:43

okay so i've just done that swap of

07:46

positions okay and now i'm going to

07:50

apply the interchange 2 3

07:53

on on the positions okay so now now

07:55

there's a different robot okay

07:57

this different robot comes to positions

07:59

two and three which are these two

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positions takes whatever's in there

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and swaps them

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and finally

08:09

the

08:11

third transmission is three four

08:13

we move to positions three and four and

08:16

we take whatever in the air and we swap

08:18

them

08:22

so that is the effect okay what has been

08:24

the effect actually okay can we so

08:26

describe the effect

08:28

well remember that we started with the

08:30

object sort of arranged in increasing

08:32

order from my left to right

08:35

so this guy was over here so what's

08:37

happened is that um

08:39

sort of everything sort of rotated

08:40

cyclically in this direction okay so if

08:43

i want to do everything back okay i'd

08:46

move this one back

08:47

and

08:49

put them back like

08:52

this okay

08:55

so the effect of doing

08:56

transposition one two and then

08:58

transmission two three and then

09:00

transposition three four where the

09:02

transpositions are regarded as acting on

09:04

positions was the accumulative effect of

09:06

that was to move everything over to the

09:08

left and this one over the other side so

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it was like a cyclic rotation

09:11

great

09:12

and now i'm going to do the same thing

09:15

with the objects now a different

09:18

exercise algebraically it kind of looks

09:20

the same so i'm going to start out with

09:22

the transposition one two but now

09:25

thought of as moving objects one and two

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so where our objects wanted to here

09:30

objects one and two

09:32

i'm gonna swap them

09:35

so in this preliminary state

09:38

that transposition had the same effect

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as

09:42

interchanging positions one and two

09:45

okay but that's not going to happen more

09:47

generally so now we move to swapping two

09:50

and three but objects two and three so

09:53

where's object two and three here's

09:55

object two here's object three

09:57

okay so now

09:59

the transition two three acts like this

10:04

and finally

10:05

the next one

10:07

swapping three and four but now

10:10

positions three and four here's three

10:13

here's four

10:14

let's swap them

10:22

what is the cumulative effect

10:24

the cumulative effect

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on the level of the objects

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is that

10:31

the

10:33

uh the objects have all been cyclically

10:35

rotated but in the opposite direction

10:37

from what we started with

10:40

the opposite direction from previously

10:41

right this one this way this one moved

10:44

here this one moved here and this one

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went over there

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so we can see that there definitely is a

10:50

distinction between

10:52

thinking about transpositions or

10:54

permutations more generally as acting on

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objects

10:57

or on positions

11:01

okay so

11:03

let me give you a variant on that okay

11:06

so

11:08

i'm gonna i'm gonna do that that second

11:10

one that i've just shown you but in a

11:12

slightly different way which is maybe a

11:13

little bit more symmetrical so i'm going

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to go through it again so let's let's

11:17

swap positions one and two

11:20

okay so here's position one and here's

11:22

position two here's object one and two

11:25

now here's another way of swapping

11:27

positions one and two okay

11:30

i'm gonna keep the objects where they

11:32

are

11:33

okay poised there in the mid-air for a

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second and here are the positions i'm

11:37

just gonna swap the positions

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and put the object straight back down so

11:42

the objects are no longer moving but the

11:45

positions have moved the containers have

11:47

moved if you like

11:49

okay and then what about um

11:53

two

11:54

three

11:55

okay so uh there's position two this is

11:58

this one here there's position three so

12:01

a way of doing that is to materially

12:03

lift these guys up let me just put them

12:04

back here for a second

12:06

and i'm just gonna swap the positions

12:11

and put these things back in the same

12:13

place

12:14

and finally three and four

12:17

three and four

12:19

there was three there was four positions

12:21

so here are the objects in positions

12:23

three and four

12:24

i'm going to move them out there

12:25

momentarily and swap the positions

12:30

put the objects back

12:32

so with this view of things the objects

12:35

are not moving and it's the underlying

12:38

containers which are moving

12:42

okay this uh it makes the symmetry maybe

12:44

even more clearer okay now it's

12:46

completely symmetrical between objects

12:48

and positions we can either

12:50

fiddle around with the objects or we can

12:52

fiddle around with the positions

12:56

and of course is this is this the same

12:58

thing that i you know

13:02

started out describing

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so to get sort of back to the uh

13:07

the the original thing okay

13:10

i should probably uh go like this

13:15

yes

13:18

all right so that was that was the thing

13:20

that i started by describing the the the

13:24

motion on on positions okay and then

13:27

that was a cyclic rotation in in this

13:29

direction

13:31

okay so i want you to think about this

13:34

this is going to have really interesting

13:36

consequences but a challenge for us now

13:38

is to think about how are we going to

13:40

deal with this algebraically so we want

13:41

to incorporate this understanding into

13:43

the mathematics okay and set it up so

13:46

that this symmetry is kind of

13:49

very accessible to us and we can make

13:51

computations with it and it's going to

13:53

be really important for us to be able to

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to

13:56

transform from the object point of view

13:59

to the position point of view and and

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back and forth all right

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so you might like to think um before the

14:06

next video how we're going to do that

14:08

okay that's a really nice little um sort

14:10

of algebraic exercise