From the previous article, we have columns j < j’ in the column tabloid U, and given a set A (resp. B) of boxes in column j (resp. j’), we get:
where is the column tabloid obtained by swapping contents of A with B while preserving the order. For example, we have the following modulo ker(π):
Here is our main result.
Main Theorem. Each of the following spans as a vector space:
- the collection of over all sets B in column j’ and (j, j’) such that j < j’;
- as above, except B comprises of the top k squares (for all k) of column j’ and j’ = j+1.
We have shown that the first set spans a subspace of ker(π); since the second set is a subset of the first, it remains to show that Q = ker(π), where Q is the vector space spanned by the second set.
Lemma. Consider as varies through all SYT with shape . These span the vector space
Create a new total ordering on the set of all fillings T of shape λ. Given T’, T, we consider the rightmost column in which T’ and T differ; in that column, take the lowest square in which T’ and T differ; we write T’ > T if the entry in that square of T’ is larger than that of T:
Suppose T is a filling which is not an SYT. We claim that modulo Q, we can express [T] as a linear sum of [T’] for T’ > T.
- Let T’ be obtained from T by sorting each column in ascending order; then [T’] = ±[T] and T’ ≥ T, so we assume each column of T is increasing.
- If T is not an SYT, then we can find row i and column j of T such that Taking j’ = j+1 and the top i squares B of column j’ of T, we can replace (modulo Q) by where each term is [T’] for some T’>T.
Applying this iteratively, since there are finitely many fillings we eventually obtain any [T] as a linear combination of [T’] where T’ is SYT. ♦
Proof of Main Theorem
We know that so we get a surjection
Hence the dimension of is at least On the other hand, by our lemma above, the space is spanned by [T] for all SYT T so its dimension is at most the number of SYT of shape λ, i.e. Thus is a bijection so as desired. ♦
As a side consequence, we have also proved that the set of [T] for all SYT T gives a basis for
This concludes our discussion of representations of the symmetric group . Our next target is the representation theory of the general linear group