Here, we will give a different interpretation of the Schur polynomial, however this definition only makes sense in the ring
For a given vector of non-negative integers, define the following determinant, a polynomial in :
For the case where , we also denote The following result is classical.
Lemma. We have Also, for any the polynomial is divisible by
In the above matrix, swapping any distinct results in an exchange of two columns, which flips the sign of the determinant. Thus when , vanishes. Hence is divisible by for all It remains to prove the first statement.
For that, note that and are both homogeneous of degree so they are constant multiples of each other. Since the largest term (in lexicographical order) is on both sides, equality holds. ♦
Definition. Suppose is a partition with ; append zeros to it so that has elements. Now define:
Being a quotient of two alternating polynomials, is symmetric; note that it is homogeneous of degree
Suppose . We have:
It turns out is precisely the Schur polynomial ; this will be proven through the course of the article.
Theorem. Take a partition with and let ; we have:
where is taken over all partitions with obtained by adding squares to such that no two of them lie in the same row.
We need to show
Note that both sides are homogeneous of degree ; hence let us compare their coefficients of where is short for Since both sides are alternating polynomials, we may assume Now expand the LHS:
In , each term is of the form so the exponents of are all distinct. Now, multiplying by increases each exponent by at most 1. Thus to obtain with strictly decreasing exponents, we must begin with strictly decreasing exponents in the first place (i.e. ), then pick such that the exponents remain strictly decreasing.
Thus each corresponds to a binary n-vector of weight r, such that is strictly decreasing. And the latter holds if and only if is a partition.
E.g. suppose we get:
In diagrams, this corresponds to:
i.e. partitions obtained by adding r boxes to such that no two lie in a row. ♦
Theorem. is the Schur polynomial in
We wish to show Successively applying Pieri’s formula gives:
where and the Young diagram for is obtained from by attaching boxes such that no two lie in the same row; the sum is over the set of all such Label the additional boxes in by and take the transpose; we obtain an SSYT of shape and type .
For example, suppose ; here is one way we can successively add squares:
which gives us:
Hence for each , the number of occurrences of in the above nested sum is This gives:
From , we get as desired. ♦
Pieri’s Formulae. Take a partition and let ; we have, in the formal ring :
where (resp. ) is taken over all partitions obtained by adding squares to , such that no two of them lie in the same row (resp. column).
By the earlier Pieri’s formula, the first formula holds in all . Hence it holds in too. Applying to it and using we get the second formula. ♦