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

**Proof**

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