Lifting and Descending Representations
Let’s recall that a group representation is, among other things, a group homomorphism. This has a few consequences.
First of all, we can consider the kernel of a matrix representation . This is not the kernel we’ve talked about recently, which is the kernel of a
-morphism. This is the kernel of a group homomorphism. In this context, it’s the collection of group elements
so that the image
is the identity transformation. We call this subgroup
. If
is the trivial subgroup, we say that
is a “faithful” representation, since it doesn’t send any two group elements to the same matrix.
Now, basic group theory tells us that is a normal subgroup, and so we can form the quotient group
. I say that the representation
“descends” to a representation of this quotient group. That is, we can define a representation
by
for all cosets
. We have to check that this doesn’t depend on which representative
of the coset we choose, but any other one looks like
for
. Then
, since
is the identity matrix by definition.
I say that is a faithful representation. That is, the only coset that
sends to the identity matrix is the one containing the identity:
itself. And indeed, if
, then
, and so
and
in the first place.
Next, is irreducible if and only if
is. We’ll prove this one by using the properties we proved a few days ago. In particular, we’ll calculate the inner product of the character of
with itself. Writing
for the character of
and
for that of
, we find that
and we use this to calculate:
Essentially, the idea is that each group element in the kernel contributes the same to the sum, and this is exactly compensated for by the difference between the sizes of
and
. Since the two inner products are the same, either both are
or neither one is, and so either both representations are irreducible or neither one is.
We can also run this process in reverse: let be a finite group, let
be any normal subgroup so we have the quotient group
, and let
be any representation of
. We will use this to define a representation of the original group
by “lifting” the representation
.
So, the obvious choice is to define . This time there’s no question that
is well-defined, since here we start with a group element and find its coset, rather than starting with a coset and picking a representative element. And indeed, this is easily verified to be a representation.
If is faithful, then the kernel of
is exactly
. Indeed, if
, then
. If
is faithful, then we must have
, and thus
.
And finally, is irreducible if and only if
is. The proof runs exactly as it did before.
An Alternative Path
It turns out that our efforts last time were somewhat unnecessary, although they were instructive. Actually, we already had a matrix representation in our hands that would have done the trick.
The secret is to look at the block diagonal form from when we defined reducibility:
We worked this out for the product of two group elements, finding
We focused before on the upper-left corner to see that was a subrepresentation, but we see that
as well. The thing is, if the overall representation acts on
and
acts on the submodule
, then
acts on the quotient space
. That is, it’s generally not a submodule. However, it so happens that over a finite group we have
as modules. That is, if we have a matrix representation that looks like the one above, then we can find a different basis that makes it look like
This is more than Maschke’s theorem tells us — not only do we have a decomposition, but we have one that uses the exact same matrix representation in the lower right as the original one. Proving this will reprove Maschke’s theorem, and in a way that works over any field!
So, let’s look for a change-of-basis matrix that’s partitioned the same way:
Multiplying this out, we find
which gives us the equation , which we rearrange to give
as well as
That is, acting on the left of by
and on the right by
doesn’t leave
unchanged, but instead adds a certain offset. We’re not looking for an invariant of these actions, but something close. Incidentally, why are these two offsets the same? Well, if we put them together we find
which must clearly be zero, as desired.
Anyway, I say that things will work out if we choose
Indeed, we calculate
Just as we wanted.
Notice that just like our original proof of Maschke’s theorem, this depends on a sum that is only finite if is a finite group.
One Complete Character Table (part 2)
Last time we wrote down the complete character table of :
which is all well and good except we haven’t actually seen a representation with the last line as its character!
So where did we get the last line? We had the equation , which involves the characters of the defining representation
and the trivial representation
. This equation should correspond to an isomorphism
.
We know that there’s a copy of the the trivial representation as a submodule of the defining representation. If we use the standard basis of
, this submodule is the line spanned by the vector
. We even worked out the defining representation in terms of the basis
to show that it’s reducible.
But what we want is a complementary subspace which is also -invariant. And we can find such a complement if we have a
-invariant inner product on our space. And, luckily enough, permutation representations admit a very nice invariant inner product! Indeed, just take the inner product that arises by declaring the standard basis to be orthonormal; it’s easy to see that this is invariant under the action of
.
So we need to take our basis and change the second and third members to make them orthogonal to the first one. Then they will span the orthogonal complement, which we will show to be
-invariant. The easiest way to do this is to use
. Then we can calculate the action of each permutation in terms of this basis. For example:
and write out all the representing matrices in terms of this basis:
These all have the required form:
where the in the upper-left is the trivial representation and the
block in the lower right is exactly the other representation
we’ve been looking for! Indeed, we can check the values of the character:
exactly as the character table predicted.
One Complete Character Table (part 1)
When we first defined the character table of a group, we closed by starting to write down the character table of :
Now let’s take the consequences we just worked out of the orthonormality relations and finish the job.
We’ve already verified that the two characters we know of are orthonormal, and we know that there can be at most one more, which would make the character table look like:
Do we have any other representations of to work with? Well, there’s the defining representation. This has a character we can specify by the three values
We calculate the multiplicities of the two characters we know by taking inner products:
That is, the defining representation contains one copy of the trivial representation and no copies of the signum representation. In fact, we already knew about the copy of the trivial representation, but it’s nice to see it confirmed again. Subtracting it off, we’re left with a residual character:
Now this character might itself decompose, or it might be irreducible. We can check by calculating its inner product with itself:
which confirms that is irreducible. Thus we can write down the character table of
as
So, why is this just part 1? Well, we’ve calculated another character, but we still haven’t actually shown that there’s any irrep that gives rise to this character. We have a pretty good idea what it should be, but next time we’ll actually show that it exists, and it really does have the character .
Consequences of Orthogonality
We have some immediate consequences of the orthonormality relations.
First of all, since irreducible characters are orthonormal any collection of them forms an orthonormal basis of the subspace it spans. Of course whatever subspace this is, it has to fit within the space of class functions, and so it can’t have any more basis elements than the dimension of this larger space. That is, there can be at most as many irreducible characters as there are conjugacy classes in . And so we know that the character table must have only finitely many rows. For instance, since
has three conjugacy classes, it can have at most three irreducible characters. We know two already, so there’s only room for one more, if there are any more at all.
For something a little more concrete, let be a collection of irreps with corresponding characters
. Then the representation
has character
That is, direct sums of representations correspond to sums of characters. This is just the tip of a far-reaching correspondence between the high-level properties of the category of representations and the low-level properties of the algebra of characters.
Anyway, proving this relation is pretty straightforward. If and
are two matrix representations then their direct sum is
It should be clear that the trace of the direct sum on the left is the sum of the traces on the right. This is all we need, since we can just split off one irreducible component after another to turn the direct sum on one side into a sum on the other.
Next we have a way of reading off the coefficients. Let be the same representation from above, with the same character. I say that the multiplicity
. Indeed, we can easily calculate
Notice that this is very similar to the result we showed at the end of calculating the dimensions of spaces of morphisms. This is not a coincidence.
More generally, if is the representation from above and
is another representation that decomposes as
then the character of is
and we calculate the inner product
In particular, we see that
We see that in all these cases . Just like sums of characters correspond to direct sums of representations, inner products of characters correspond to
-spaces between representations. We just have to pass from plain vector spaces to their dimensions when we pass from representations to their characters. Of course, this isn’t much of a stretch, since we saw that the character
of a representation
includes information about the dimension:
.
This goes even further: what happens when we swap the arguments to an inner product? We get the complex conjugate: . What happens when we swap the arguments to the
functor? We get the dual space:
. Complex conjugation corresponds to passing to the dual space.
Finally, the character is irreducible if and only if
. Indeed, if
is itself irreducible then our decomposition only involves one nonzero coefficient, which is a
. The formula we just computed gives
Conversely, if this formula holds then we have to write as a sum of squares. The only possibility is for all but one of the numbers
to be
, and the remaining one to be
, in which case
, and is thus irreducble.
Irreducible Characters are Orthogonal
Today we prove the assertion that we made last time: that irreducible characters are orthogonal. That is, if and
are
-modules with characters
and
, respectively, then their inner product is
if
and
are equivalent and
otherwise. Strap in, ’cause it’s a bit of a long one.
Let’s pick a basis of each of and
to get matrix representations
and
of degrees
and
, respectively. Further, let
be any
matrix with entries
. Now we can construct the
matrix
Now I claim that intertwines the matrix representations
and
. Indeed, for any
we calculate
At this point, Schur’s lemma kicks in to tell us that if then
is the zero matrix, while if
then
is a scalar times the identity matrix.
First we consider the case where (equivalently,
). Since
is the zero matrix, each entry must be zero. In particular, we get the equations
But the left side isn’t just any expression, it’s a linear function of the . Since this equation must hold no matter what the
are, the coefficients of the function must all be zero! That is, we have the equations
But now we can recognize the left hand side as our alternate expression for the inner product of characters of . If the functions
and
were characters, this would be an inner product, but in general we’ll write
Okay, but we actually do have some characters floating around: and
. And we can write them out in terms of these matrix elements as
And now we can use the fact that for characters our two bilinear forms are the same to calculate
So there: if and
are inequivalent irreps, then their characters are orthogonal!
Now if we can pick bases so that the matrix representations are both
. Schur’s lemma tells us that there is some
so that
. Our argument above goes through just the same as before to show that
so long as . To handle the case where
, we consider our equation
We take the trace of both sides:
and thus we conclude that . And so we can write
Equating coefficients on both sides we find
And finally we can calculate
exactly as we asserted.
Incidentally, this establishes what we suspected when setting up the character table: if and
are inequivalent irreps then their characters
and
must be unequal. Indeed, since they’re inequivalent we must have
. But if the characters were the same we would have to have
. So since inequivalent irreps have unequal characters we can replace all the irreps labeling rows in the character table by their corresponding irreducible characters.
Inner Products in the Character Table
As we try to fill in the character table, it will help us to note another slight variation of our inner product formula:
where our sum runs over all conjugacy classes , and where
is the common value
for all
in the conjugacy class
(and similarly for
). The idea is that every
in a given conjugacy class gives the same summand. Instead of adding it up over and over again, we just multiply by the number of elements in the class.
As an example, consider again the start of the character table of :
Here we index the rows by irreducible characters, and the columns by representatives of the conjugacy classes. We can calculate inner products of rows by multiplying corresponding entries, but we don’t just sum up these products; we multiply each one by the size of the conjugacy class, and at the end we divide the whole thing by the size of the whole group:
We find that when we take the inner product of each character with itself we get , while taking the inner product of the two different characters gives
. This is no coincidence; for any finite group
irreducible characters are orthonormal. That is, different irreducible characters have inner product
, while any irreducible character has inner product
with itself. This is what we will prove next time.
The Character Table of a Group
Given a group , Maschke’s theorem tells us that every
-module is completely reducible. That is, we can write any such module
as the direct sum of irreducible representations:
Thus the irreducible representations are the most important ones to understand. And so we’re particularly interested in their characters, which we call “irreducible characters”.
Of course an irreducible character — like all characters — is a class function. We can describe it by giving its values on each conjugacy class. And so we lay out the “character table”. This is an array whose rows are indexed by inequivalent irreducible representations, and whose columns are indexed by conjugacy classes . The row indexed by
describes the corresponding irreducible character
. If
is a representative of the conjugacy class, then the entry in the column indexed by
is
. That is, the character table looks like
By convention, the first row corresponds to the trivial representation, and the first column corresponds to the conjugacy class of the identity element. We know that the trivial representation sends every group element to the
identity matrix, whose trace is
. We also know that every character’s value on the identity element is the degree of the corresponding representation. We can slightly refine our first picture to sketch the character table like so:
We have no reason to believe (yet) that the table is finite. Since is a finite group there can be only finitely many conjugacy classes, and thus only finitely many columns, but as far as we can tell there may be infinitely many inequivalent irreps, and thus infinitely many rows. Further, we have no reason to believe that the rows are all distinct. Indeed, we know that equivalent representations have equal characters — they’re related through conjugation by an invertible intertwinor — but we don’t know for sure that inequivalent representations must have distinct characters.
As an example, we can start writing down the character table of . We know that conjugacy classes in symmetric groups correspond to cycle types, and so we can write down all three conjugacy classes easily:
We know of two irreps offhand — the trivial representation and the signum representation — and so we’ll start with those and leave the table incomplete below that:
Characters of Permutation Representations
Let’s take to be a permutation representation coming from a group action on a finite set
that we’ll also call
. It’s straightforward to calculate the character of this representation.
Indeed, the standard basis that comes from the elements of gives us a nice matrix representation:
On the left is the matrix of the action on
, while on the right it’s the group action on the set
. Hopefully this won’t be too confusing. The matrix entry in row
and column
is
if
sends
to
, and it’s
otherwise.
So what’s the character ? It’s the trace of the matrix
, which is the sum of all the diagonal elements:
This sum counts up for each point
that
sends back to itself, and
otherwise. That is, it counts the number of fixed points of the permutation
.
As a special case, we can consider the defining representation of the symmetric group
. The character
counts the number of fixed points of any given permutation. For instance, in the case
we calculate:
In particular, the character takes the value on the identity element
, and the degree of the representation is
as well. This is no coincidence;
will always be the degree of the representation in question, since any matrix representation of degree
must send
to the
identity matrix, whose trace is
. This holds both for permutation representations and for any other representation.
The Inner Product of Characters
When we’re dealing with characters, there’s something we can do to rework our expression for the inner product on the space of class functions.
Let’s take a -module
, with character
. Before, we’ve used Maschke’s theorem to tell us that all
-modules are completely reducible, but remember what it really tells us that there is some
–invariant inner product on
(we’ll have to keep straight the two inner products by which vector space they apply to). With respect to the inner product on
, every transformation
with
is unitary, and if we pick an orthonormal basis to get a matrix representation
each of the matrices
will be unitary. That is:
So what does this mean for the character ? We can calculate
And so we can rewrite our inner product
The nice thing about this formula is that it doesn’t depend on complex conjugation, and so it’s useful for any base field (if we were using other base fields).
The catch is that for class functions in general we have no reason to believe that this is an inner product. Indeed, if is some element that isn’t conjugate to its inverse then we can define a class function
that takes the value
on the class
of
,
on the class
of
, and
elsewhere. Our new formula gives
so this bilinear form isn’t positive-definite.