WEBVTT
Kind: captions
Language: en
00:00:14.139 --> 00:00:23.840
Hello so, last time we saw that one can realize
abstract group elements as concrete say matrices
00:00:23.840 --> 00:00:30.849
rotations and all that and I told you last
time that we are going to learn representations
00:00:30.849 --> 00:00:42.760
in the slash mathematical concept of representations
.
00:00:42.760 --> 00:01:02.359
So, what I have is a group and representation
is very simple it is nothing, but a homomorphism
00:01:02.359 --> 00:01:17.979
from g to GL n say complex numbers C is field
of complex numbers . So, that is a representation
00:01:17.979 --> 00:01:23.259
quite simple definition. So, it is a homomorphism
which starts from G and which takes venues
00:01:23.259 --> 00:01:27.210
in GL n C.
So, what is this, this is our attempt to realize
00:01:27.210 --> 00:01:38.770
any abstract element in the group as a matrix,
what is the advantage, we will see after sometime.
00:01:38.770 --> 00:01:51.700
So, what are the examples, some easy examples
we can see. We can think of you remember signature
00:01:51.700 --> 00:02:00.950
map or the parity map which takes value in
I told this is 0 1, 0 is even parity, 1 is
00:02:00.950 --> 00:02:15.670
odd parity and that can be thought of as a
representation because I can think
00:02:15.670 --> 00:02:34.560
of 0 as as say multiplicative identity.
1 as this by minus 1 0 0 1 and that is a sub
00:02:34.560 --> 00:02:47.640
of GL 2 C, that is very straightforward at
some point or simply you could have originating
00:02:47.640 --> 00:03:02.450
from any group trivial map that is also example
of representation when you have rubiks group
00:03:02.450 --> 00:03:10.099
for example, and when you map an abstract
state of rubiks group to a concrete matrix
00:03:10.099 --> 00:03:17.160
which corresponds to the permutation that
is also an example of representation .
00:03:17.160 --> 00:03:32.209
Few more examples this time let me just concretely
take S 3. So, from S 3 to this time I am taking
00:03:32.209 --> 00:03:41.689
values in GL 2 of C. So, I can think of S
3 as permutation of 3 objects or permutations
00:03:41.689 --> 00:03:49.340
of or the rotations of rotations in the flippings
so, symmetries of triangle. So, there are
00:03:49.340 --> 00:03:56.459
6 symmetries are triangle and accordingly
you map it to corresponding matrix.
00:03:56.459 --> 00:04:13.290
So, let me just write it in this form maybe
you have 1, you have 1 2, you have 1 2 3.
00:04:13.290 --> 00:04:27.210
So, 1 2, 2 3, 1 3 and 1 2 3 and 1 3 2 like
this so, I can map it to. So, one I can map
00:04:27.210 --> 00:04:45.039
to identity matrix 2 by 2 identity matrix
1 2 for example, 1 2 I can map to matrix which
00:04:45.039 --> 00:04:52.520
swaps firsthand the second rows so, it is
just
00:04:52.520 --> 00:05:01.140
So, I am having in GL 3 sorry I am having
in GL 3 I 3 I 3. So, this just swapping 2
00:05:01.140 --> 00:05:13.419
rows say 0 1 0 1 0 0 and then you have 0 0
1 like that. So, corresponding permutation
00:05:13.419 --> 00:05:36.730
matrices so, S n goes to in general corresponding
permutation matrices. So, that is also a representation
00:05:36.730 --> 00:05:40.610
there are much more interesting representations
.
00:05:40.610 --> 00:06:22.159
For example, when you have D n which are what
symmetries of regular n gon. So, from D n
00:06:22.159 --> 00:06:35.849
you can give a representation which is a taking
values in GL 2 of C. So, what is this representation
00:06:35.849 --> 00:06:49.899
, it just again it is a kind of permutation
so, when you have
00:06:49.899 --> 00:07:02.590
say for example, the square say n is equal
to 4 case there are 4 states. So, suppose
00:07:02.590 --> 00:07:10.890
this is identity it goes to 1 so, there are
for cyclic right.
00:07:10.890 --> 00:07:17.459
So, suppose this a state where 1 goes to 2.
So, 1 is here, 2 is here, 3 is here, 4 is
00:07:17.459 --> 00:07:30.010
here. So, it goes to minus of identity. So,
this one is actually 2 by 2 identity matrix
00:07:30.010 --> 00:07:40.550
minus i is this scalar minus i, 0 0 minus
i or you could have then it i also (Refer
00:07:40.550 --> 00:07:55.330
Time:07:42) i or minus i and then now 1 goes
to so, 1 2 3 4 like that. So, here it is minus
00:07:55.330 --> 00:08:10.890
1 which is minus 1 0 0 minus 1 and similarly
you have 1 here 1 2 3 4 and that goes to say
00:08:10.890 --> 00:08:23.280
i which is i 0 0 i and then the flipping 1
the 1 then this is flipped.
00:08:23.280 --> 00:08:37.260
So, I have this so, flipping could be 2 here,
1 here, and 3 here and 4 here and that goes
00:08:37.260 --> 00:08:54.750
to say flipping off this thing. So, this goes
to 0 1 1 0 and then other you can obtain by
00:08:54.750 --> 00:09:03.690
appropriate composition . So, whatever is
actually happening for the definition the
00:09:03.690 --> 00:09:10.640
way D n is constructed just by taking the
regular n gon and considering symmetries first
00:09:10.640 --> 00:09:18.330
you put all the cyclic orientations all cyclic
configurations and then flip it and then once
00:09:18.330 --> 00:09:27.680
more all cyclic configurations and the way
GL is defined using that you can define a
00:09:27.680 --> 00:09:33.430
representation.
Here is an important concept for representations
00:09:33.430 --> 00:09:52.860
which is that of character of a representation
. So, what do we mean by character, character
00:09:52.860 --> 00:10:08.800
is a map from G to C. So, I have a representation
let me call it f the representation given
00:10:08.800 --> 00:10:14.760
this representation I can associate with a
map maybe I will call it chi f it is quite
00:10:14.760 --> 00:10:23.090
character of f what it does is given g to
it . So, given g it associates to it so, you
00:10:23.090 --> 00:10:41.230
can see just say this fg, fg is a matrix.
So, I can think of trace of this matrix right
00:10:41.230 --> 00:10:45.100
and that comes with the various interesting
properties.
00:10:45.100 --> 00:10:56.010
So, this is point character so, g going to
trace of the corresponding matrix character
00:10:56.010 --> 00:11:06.340
is quite useful in understanding groups in
understanding conjugacy classes of groups,
00:11:06.340 --> 00:11:34.360
useful in understanding groups and it is conjugacy
classes and in fact, this notion of what are
00:11:34.360 --> 00:11:46.300
called irreducible representations is
00:11:46.300 --> 00:11:53.363
very central to our theory of a characters
to the theory of representations
00:11:53.363 --> 00:12:02.310
So, what is irreducible representation, so,
I have say this representation . So, I can
00:12:02.310 --> 00:12:14.300
think of GL n C as matrix or I can think of
it as automorphisms of the vector space C
00:12:14.300 --> 00:12:22.230
n. So, C n is a vector space I could have
said r n but theory over r is slightly
00:12:22.230 --> 00:12:29.430
complicated theory over complex numbers is
quite straightforward . So, you can think
00:12:29.430 --> 00:12:51.850
of it as a linear map from C n to C n . So,
I can say that when do I say there are representation
00:12:51.850 --> 00:13:15.940
this f is irreducible, if you consider a subspace
in this. So, f is irreducible , if a
00:13:15.940 --> 00:13:37.660
subspace which satisfies that f g W is contained
in W, so called stable subspace.
00:13:37.660 --> 00:13:56.820
So, if stable subspace is either C n or t
will 1. So, it is very clear that if I have
00:13:56.820 --> 00:14:06.440
a subspace 0 subspace then any matrix over
0 is going to be 0 any matrix over the whole
00:14:06.440 --> 00:14:15.810
thing is again going to take it to C n within
C n, but if you can get some non trivial element
00:14:15.810 --> 00:14:21.380
non trivial subspace which satisfies this
property. So, non trivial stable sub space
00:14:21.380 --> 00:14:24.510
then the representation is not going to be
irreducible.
00:14:24.510 --> 00:14:29.360
So, it is irreducible if whenever you have
stable subspace then there are only 2 options
00:14:29.360 --> 00:14:34.800
for stable subspace either this or this. So,
it is so turns out they representation is
00:14:34.800 --> 00:14:40.290
made of these irreducible representations,
every representation is actually what is called
00:14:40.290 --> 00:15:03.360
a direct sum of irreducible representations.
So, I am not going to say all that
00:15:03.360 --> 00:15:17.420
in detail every representation is direct sum
of irreducible representations . And here
00:15:17.420 --> 00:15:31.880
is very interesting thing theorem quite interesting
thing, which says that number of irreducible
00:15:31.880 --> 00:16:02.910
representations of finite group G equals number
of conjugacy classes of G, what are conjugacy
00:16:02.910 --> 00:16:11.980
classes , you have a group into break it into
partitions on what basis 2 elements are in
00:16:11.980 --> 00:16:19.640
same partition if they can be obtained from
each other by conjugation.
00:16:19.640 --> 00:16:24.670
So, any 2 elements here are conjugate to each
other and any 2 elements here are conjugate
00:16:24.670 --> 00:16:29.362
to each other and like that with that kind
of equivalence relation you partition your
00:16:29.362 --> 00:16:37.800
group and each partition is conjugacy class.
So, how many conjugacy classes are there is,
00:16:37.800 --> 00:16:52.730
actually same as number of irreducible representations
and in fact, easy thing for us is to realize
00:16:52.730 --> 00:17:00.460
that if you have a character evaluate it on
the element or if you evaluate it on a conjugate
00:17:00.460 --> 00:17:06.929
element.
Answer is same, same complex number is there
00:17:06.929 --> 00:17:12.760
because if you have 2 different matrices if
they are conjugate to each other than their
00:17:12.760 --> 00:17:22.800
traces are also same, chi f is just about
take the trace or theorem is that 2 elements
00:17:22.800 --> 00:17:45.740
g 1 and g 2, they are conjugate to each other
if and only if chi f g 1 is same as chi f
00:17:45.740 --> 00:17:55.429
g 2 for every irreducible representation here.
So, for all characters all irreducible characters
00:17:55.429 --> 00:18:06.299
if you are having same value of characters
for on these 2 elements g 1 and g 2 then they
00:18:06.299 --> 00:18:13.990
have to be conjugates.
So, one can think of what is called character
00:18:13.990 --> 00:18:30.679
table and what does character table do. So,
here you have characters, here you have conjugacy
00:18:30.679 --> 00:18:37.090
classes, say here is a conjugacy classes of
g, here is a character corresponding to representation
00:18:37.090 --> 00:18:45.150
irreducible representation g. So, these are
irreducible characters characters which are
00:18:45.150 --> 00:18:52.030
corresponding to irreducible representations.
So, here your entry will be chi f g in like
00:18:52.030 --> 00:18:59.950
that you will make a square matrix what why
is it square matrix? The square matrix because
00:18:59.950 --> 00:19:06.059
of this equality number of rows the same as
number of columns, number of rows is number
00:19:06.059 --> 00:19:10.050
of irreducible characters and number of columns
is number of conjugacy classes.
00:19:10.050 --> 00:19:17.760
So, therefore, to each group you can associate
this data which is called character table
00:19:17.760 --> 00:19:23.039
there are various interesting properties of
character table for example, character table
00:19:23.039 --> 00:19:29.410
is actually an invertible matrix you can think
of it as a complex matrix right and this is
00:19:29.410 --> 00:19:39.100
a invertible matrix quite interesting property
. A important thing is that you can read lot
00:19:39.100 --> 00:19:47.300
about a group that understand various properties
of groups via character tables in via in general
00:19:47.300 --> 00:20:03.970
theory of characters.
So, some remarks I would mention, these are
00:20:03.970 --> 00:20:24.760
accomplishments of representation theory or
character theory. So, there are various problems
00:20:24.760 --> 00:20:34.770
in group theory which are otherwise difficult
to deal with, but using the concept of representations
00:20:34.770 --> 00:20:41.540
of characters one can easily deal with those
problems and then just tell some historical
00:20:41.540 --> 00:21:02.350
example what is called Burnside's pq theorem
what it says that if you have 2 primes and
00:21:02.350 --> 00:21:08.850
you have a group whose order is p to the power
a, q to the power b; that means, in the factorization
00:21:08.850 --> 00:21:12.080
of group they are not more than 2 distinct
primes which are occurring.
00:21:12.080 --> 00:21:17.990
So, this is just information of the order
of the group and nothing else what is surprising
00:21:17.990 --> 00:21:26.830
is that from this you can conclude that your
group is having very special property of being
00:21:26.830 --> 00:21:39.299
what is called soluble group, soluble groups
can be understood in terms of what is called
00:21:39.299 --> 00:21:50.260
a chain a descending chain of groups the derived
chain of group. But let me just say that this,
00:21:50.260 --> 00:22:00.530
the solubility has something do with in the
solubility of certain polynomial .
00:22:00.530 --> 00:22:10.290
So, this is the property of groups, solubility
is the property of groups, important property
00:22:10.290 --> 00:22:24.960
of groups that has genesis in theory of equations,
theory of polynomial equations quite important
00:22:24.960 --> 00:22:31.550
property also. So, once side p p q theorem
actually this is statement proving in group
00:22:31.550 --> 00:22:37.960
theory, but one can prove it using corrected
theory. Another interesting problem which
00:22:37.960 --> 00:22:52.799
I will mention is quite curious, we take G
group such that the order of the group is
00:22:52.799 --> 00:23:01.920
odd and then you say n is number of conjugacy
classes .
00:23:01.920 --> 00:23:16.000
In general computing number of conjugacy classes
is not straightforward for a group, statement
00:23:16.000 --> 00:23:28.780
is that if you consider G and you consider
n this is a relation G is actually congruent
00:23:28.780 --> 00:23:38.830
to n mod 16 quite curious number of conjugacy
classes and number of elements the group the
00:23:38.830 --> 00:23:49.290
difference is always divisible by 16 provided
your order the group is odd. This makes use
00:23:49.290 --> 00:24:22.070
of water called real representations rather
real characters and so, called real conjugacy
00:24:22.070 --> 00:24:29.321
classes.
I would not say much in detailed, but the
00:24:29.321 --> 00:24:38.429
message that I want to put forth is that you
have a group, it is abstract group you find
00:24:38.429 --> 00:24:43.990
ways to understand it you find different representations
of it sometimes those representations could
00:24:43.990 --> 00:24:52.600
be tangible, they could be there in the real
life all the examples I have given earlier
00:24:52.600 --> 00:24:57.710
through puzzles and toys and some symmetry
considerations and sometimes you can realize
00:24:57.710 --> 00:25:05.000
you can represent those groups in terms of
matrices and if you analyze those matrices
00:25:05.000 --> 00:25:11.640
close enough then you can that then it can
reveal lots of interesting properties or groups
00:25:11.640 --> 00:25:15.080
Representation theory is quite a wide subject
00:25:15.080 --> 00:25:21.740
I just try to give a flavor of that and in
fact, not just of groups, but for other objects
00:25:21.740 --> 00:25:30.880
mathematics as well is the representation
theory and it has it is own ways of characterizing
00:25:30.880 --> 00:25:36.300
reducible representations and all that and
purpose of any kind of representation theory
00:25:36.300 --> 00:25:41.760
association to understand your original object
much better and quite often in some combinatorial
00:25:41.760 --> 00:25:46.830
way or in a way were making computations is
quite handy.
00:25:46.830 --> 00:25:57.380
So, I hope you put all your somewhat in coming
lectures what we are going to understand is
00:25:57.380 --> 00:26:05.700
something more about this particular group
and as I said this is a group of rotations
00:26:05.700 --> 00:26:11.840
in r 3 some interesting things are coming
up and those things will have a some they
00:26:11.840 --> 00:26:17.750
are coming up as a nice application of group
actions so, be with me.
00:26:17.750 --> 00:26:18.830
Thank you