We will describe another construction for the Schur module.
Introduce variables for . For each sequence we define the following polynomials in :
Now given a filling T of shape λ, we define:
where is the sequence of entries from the i-th column of T. E.g.
Let be the ring of polynomials in with complex coefficients. Since we usually take entries of T from [n], we only need to consider the subring .
Let Recall from earlier that any non-zero -equivariant map
must induce an isomorphism between the unique copies of in the source and target spaces. Given any filling T of shape , we let be the element of obtained by replacing each entry k in T by , then taking the wedge of elements in each column, followed by the tensor product across columns:
Note that the image of in is precisely as defined in the last article.
Definition. We take the map
where belongs to component
E.g. in our example above, is homogeneous in of degree 5, of degree 4 and of degree 3. We let act on via:
Thus if we fix i and consider the variables as a row vector, then . From another point of view, if we take as a basis, then the action is represented by matrix g since it takes the standard basis to the column vectors of g.
Proposition. The map is -equivariant.
The element takes by taking the column vectors of g; so
where T’ is the filling obtained from T by replacing its entries with correspondingly.
On the other hand, the determinant gets mapped to:
which is . ♦
Since contains exactly one copy of , it has a unique -submodule Q such that the quotient is isomorphic to The resulting quotient is thus identical to the Schur module F(V), and the above map factors through
Now we can apply results from the last article:
Corollary 1. The polynomials satisfy the following:
- if T has two identical entries in the same column.
- if T’ is obtained from T by swapping two entries in the same column.
- , where S takes the set of all fillings obtained from T by swapping a fixed set of k entries in column j’ with arbitrary sets of k entries in column j (for fixed j < j’) while preserving the order.
Indeed, the above hold when we replace by Now apply the above linear map. ♦
Corollary 2. The set of , for all SSYT with shape λ and entries in [n], is linearly independent over
Indeed, the set of these is linearly independent over and the above map is injective. ♦
Consider any bijective filling T for . Writing out the third relation in corollary 1 gives:
More generally, if satisfies and , the corresponding third relation is obtained by multiplying the above by a polynomial on both sides.
Example 2: Sylvester’s Identity
Take the SYT by writing in the left column and in the right. Now is the product:
In the sum , each summand is of the form , where matrices M’, N’ are obtained from M, N respectively by swapping a fixed set of k columns in N with arbitrary sets of k columns in M while preserving the column order. E.g. for n=3 and k=2, picking the first two columns of N gives: