Background
Let's talk about Schur functions: a nice basis for the ring of symmetric functions.
Recall that a Young tableau with shape , is a Young diagaram with shape
where each box is filled in with some number.
Definitions: A weak composition of
is a set of numbers such that
,
for
and
. A Young tableau
with content
and shape
is a Young diagram of shape
whose boxes are filled in with
.
As an example, let be a weak composition of
and
be a partition of
. To construct the (only) Young tableau with content
and shape
we first draw the Young diagram of shape
and then we fill in the boxes with the number
coming from
in such a way that the rows are weakly increasing and the columns are strictly increasing. The only tableau that works is the following:
Definitions: For each partition we can associate what is called a Schur function. The Schur function of
is given by:
where
is a weak composition of
.
As an example, let us find . In particular, we will first only look at weak compositions of
which are contained in the set:
Since
all of our tableau must have shape:
In fact, we end up having the following tableau for each of the following weak partitions above:
This implies that we have the following (in order) for the Schur function:
Notice that if we continue with all possible weak compositions then we would have a symmetric function. For example, we would have and
by using the weak compositions
and
respectively.
Notice that the coefficients of the monomials are given by the Kostka numbers.
Relationship with other bases
Jacobi-Trudi identity
The first relationship we will look at is with complete homogeneous symmetric functions . We call this identity the Jacobi-Trudi identity:
where
and
whenever
.
This is best seen in an example. Let . Then
since there are
entries in
and we construct the
matrix as defined by above:
Next we take the determinant of this matrix.
The key thing to notice here is that our partition appears on the diagonal and then we just add
each time we go to the right and subtract
each time we go to the left.
Pieri Rule
There are two other relationships with other bases that we will discuss next. In particular, we will look at what happens when we multiple a Schur function with either a complete homogeneous symmetric function or a elementary symmetric function. These two rules are known as the Pieri rules. Before we define the following two sets for ease of notation:
Here, horizontal strip means that each column in the Young diagram has at most one block and similarly vertical strip means that each row in the Young diagram has at most one block.
Then our two Pieri rules are:
Let's look at an example for the Pieri rules. Suppose that and
. Recall that the Young diagram for
is the following:
We first look at . Since we want to find all
such that:
is contained in
(
),
- we only add
new blocks (
), and
- no column as more than one new block added (
is a horizontal strip)
This gives us exactly four different possible Young diagrams (the green boxes are the new boxes):
By the Pieri rule we have:
Now let's look at . Recall that we want our
to be almost exactly the same as in
but instead of horizontal strips we want vertical strips. So no row should get more than one new block added. This also (by chance) gives us exactly four new tableau:
By the Pieri rule we have:
Littlewood-Richardson rule
In the Pieri rule we multiplied a Schur function by one of the other symmetric function bases. What happens when we multiply it by another Schur function? This is given by the Littlewood-Richardson rule. Let and
be two partitions. Then the Littlewood-Richardson rule states:
where
is the number of semi-standard Young tableau of (skew) shape
with weight
such that the reading word
of
has the property that any initial segment has at least as many
's as
's where the reading word is read from right to left and bottom to top. Such a tableau is often called a Littlewood-Richardson tableau.
Let's look at an example. Let and let
. Then
are going to be all partitions of
. This gives us the following possible Young diagrams:
We have five different possible diagrams. In particular we have:
so it only remains to find each of the coefficients.
Let's first look at . Recall that this it the number of semi-standard Young tableau of skew shape
whose weight is
such that the reading word has a particular property. There is only one way we can fill in this diagram:
Recall that reading word is read from right to left and bottom to top, so we have as a reading word
. But notice that the initial section
has one
and zero
s which means this is not a valid tableau! So
.
Next we look at . In this case, there are two different ways we can fill in the skew diagram
in order to get a (skew) semi-standard Young tableau:
The reading word (right to left, bottom to top) of the left tableau is:
and the reading word of the right tableau is
. Just as in the previous example,
is not a valid reading word(!), but
is. Notice that the initial section
is ok,
is also ok, and
is ok since when we hit the first
we already have two
s. Therefore
.
Continuing, we look at . In this case, there is only one way to fill in the skew diagram
to get a (skew) semi-standard Young tableau:
The reading word for this tableau is
which is a valid reading word since the initial section
is ok,
is ok since there is one
and one
, and
is ok since we are adding another
. Therefore,
.
Let's next look at . As in the
case, there is only one way to fill in the skew diagram
to get a (skew) semi-standard Young tableau:
This gives us a reading word of
which is valid. Therefore,
.
Finally, let's consider . Notice that there is no way to fill in the (skew) diagram
with two
s since we would have to place one
above another and the columns must be strictly increasing. Therefore,
.
This implies (finally) that
Although this computation takes a while to compute by hand, you can use sagemath (which uses Anders Buch's Littlewood-Richardson Calculator) in order to calculate these numbers quickly and efficiently without all of the above work. Documentation for this calculator on sagemath are found here.