If we multiply two elementary symmetric polynomials and , the result is just , where is the concatenation of and sorted. Same holds for However, we cannot express in terms of easily, which is unfortunate since the Schur functions are the “preferred” basis, being orthonormal. Hence, we define the following.
Definition. A skew Young diagram is a diagram of the form , where and are partitions and for each
E.g. if and then
Note that the same skew Young diagram can also be represented by where and These two Young diagrams are considered identical.
Definition. A skew semistandard Young tableau (skew SSYT) is a labelling of the skew Young diagram with positive integers such that each row is weakly increasing and each column in strictly increasing. Now is called the shape of the tableau and its type is given by where is the number of times appears.
E.g. the following is a skew SSYT of the above shape. Its type is (4, 1, 1, 1).
Skew Schur Polynomials
Definition. The skew Schur polynomial corresponding to is given by:
where is . E.g. the above diagram gives
The proof for the following is identical to the case of Schur polynomials.
Lemma. The skew Schur polynomial is symmetric.
Indeed, one checks easily that the Knuth-Bendix involution works just as well for skew Young tableaux.
So the number of skew SSYT of shape and type is unchanged when we swap and
For and , we have:
The following result explains our interest in studying skew Schur polynomials.
Lemma. The product of two skew Schur polynomials is a skew Schur polynoial.
For example, we have:
It remains to express as a linear combination of , where
Recall that we have Pieri’s formula , where is summed across all diagrams obtained by adding boxes to such that no two are on the same column. Repeatedly applying this gives us:
where and is obtained from by adding boxes such that no two lie in the same column. Hence, the number of occurrences for a given skew Young diagram is the number of skew SSYT’s with shape and type
If and , here is one way of appending 4, 2, 1 boxes in succession:
which corresponds to the following skew SSYT:
This gives us the tool to prove the following.
Theorem. For any with , we have:
so the linear map is left adjoint to multiplication by
It suffices to prove this for all where Since is the basis dual to , the LHS is the coefficient of in expressing in terms of monomial symmetric polynomials. By definition of , this is equal to the number of skew SSYTs with shape and type ; we will denote this by the skew Kostka coefficient.
By the reasoning above, when is expressed as a linear combination of Schur functions, the coefficient of is also . Since the Schur functions are orthonormal, we are done. ♦
The theorem is still true even when does not all hold, if we take . Indeed, by reasoning with Pieri’s rule again, every Schur polynomial occuring in must have for all
Definition. When we express:
the values are called the Littlewood-Richardson coefficients.
By the above theorem, this equals One can calculate this using the Littlewood-Richardson rule, which we will cover later.