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.
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.
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:
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:
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.